GRE Pchem

Intensive vs. Extensive Properties

*Extensive Properties* are also known as capacity factors. They depend on the amount of matter present in terms of mass (ie total volume or energy of a system). Extensive properties are additive, and are equal to the sum of their parts. *Intensive Properties* are also known as intensity factors. They are independent of the quantity of material present (ie temperature, molar volume, density, pressure, viscosity, refractive index). They are the same for every part of the system at equilibrium, meaning they are not additive.

What is a Partition Function

A partition function is the bridge between the quantum mechanical energy states of a macroscopic state and the thermodynamic properties of the system. These properties can be expressed as a function of the partition function. For the canonical ensemble, examples include U (internal energy), (P) pressure, and (S) entropy, shown.

Phase Equilibria

A phase is any part of a system that is a homogenous. The Gibbs phase rule provides a good tool for studying heterogeneous equilibria: f = c - p + 2 where c = number of components, p = number of phases, f = degrees of freedom. Degrees of freedom in a system is defined by the number of independent variables (such at temperature, pressure, or concentration) that may be varied without altering the number of phases in the system. Consider a one component system. From the Gibbs phase rule, we known that (for this system) c = 1 and f = 3 - p. - If p= 1 (system has one phase) f = 2, system is bivariant - If p= 2 (system has two phases) f = 1, system is univariant - If p= 3 (system has three phases) f = 0, system is nonvariant

Real Solutions

A solution is a homogeneous system with at least two constituents. It may be gaseous, liquid, or solid. In a real solution, interactions between molecules of the different components differ from interactions between molecules of the same components. Take for example a solution of chloroform and acetone. Dipole-dipole interactions occur here, Cl3C-H....O=C(CH3)2 In this case, heat will be given off when the solution is formed so that deltaH mixing < 0. This is because the associations between the molecules restricts their motion. This should give the system a less positive entropy (S) than in an ideal solution.

Ideal Solutions

A solution is a homogeneous system with at least two constituents. It may be gaseous, liquid, or solid. The characteristics of an ideal solution are: (A) The deltaV mixing = 0. In an ideal solution, the volume of the mixed components is equal to that of the unmixed components. (B) The deltaH mixing = 0. There is no change of enthalpy in an ideal solution system when the components are mixed to form the solution. (C) The entropies of each component in an ideal solution are greater than the entropies of the pure, isolated materials.

Equilibrium State

A system is said to be in a *state of equilibrium* when its properties do not vary with time. Three conditions are implied when a system is at equilibrium: A: Thermal equilibrium ( T is the same everywhere in the system) B: Mechanical equilibrium: (P is the same everywhere in the system) C: Chemical Equilibrium (Chemical composition is unvarying throughout the system)

Adiabatic Processes

Adiabatic processes occur when a system is enclosed by an adiabatic boundary, so that its temperature remains independent of the temperature of the surroundings. For example, vessel wrapped in thick glass wool (an insulator) will reach thermal equilibrium very slowly, if at all A system enclosed in a true adiabatic boundary will always remain at a temperature different from that of its surroundings- it will never reach thermal equilibrium. In an ideal adiabatic boundary, the flow of heat is 0

Molecular Distribution as a Function of Temperature

At a given temperature, only a fraction of the molecules will have the required energy to cross the Ea threshold. The attached graph shows the distribution of molecules as a function of temperature. The number of molecules with energy greater than Ea is given by the area under the curve that is past the Ea threshold. This is clearly greater for T2 > T1 than for T1. This means that for T2 > T1, the number of effective collisions is also greater, and so in turn the reaction rate is also greater. Not all collisions between molecules have the required kinetic energy to create new bonds. Only those molecules that have both a favorable orientation and energy > Ea will actually lead to product formation.

Ideal Gases: Boyle-Mariotte's Law

Boyle-Mariotte's law states that the volume of a given mass of gas varies inversely with pressure if temperature is held constant. The equations for this law are expressed as shown.

Magnetic Interactions from MO Diagram

By looking at the highest levels of a molecule's MO diagram, one can predict the magnetic character: Molecules with paired electrons only are diamagnetic, and do not interact with a magnetic field Molecules with any unpaired electrons are paramagnetic, they have magnetic moments and are able to interact with a magnetic field.

Gibbs & Helmholtz Energy

Changes in entropy/enthalpy are only two factors that affect the spontaneity of a chemical reactions. The Gibbs free energy, also known as the molar free enthalpy (G) combines these two thermodynamic factors by: G = H - TS For a process that occurs t a constant termperature, the free energy variation can be given by: deltaG = deltaH - TdeltaS Since H and S are both state properties, it follows that G is a state property as well. This fact allows us to study the influence of temperature on the spontaneity of a given process. In a reaction where the system entropy decreases, the final term -TdeltaS will be positive. The reaction is only spontaneous if deltaH is negative enough and large enough to overcome -TdeltaS. If we want to convert the energy of the reaction to work, there myst be enough energy left as deltaH to be delivered to the surroundings after -TdeltaS has been accounted for. Only energy in excess of -TdeltaS is available for work. *Free energy* is thus a measure of the work that can be done by a system after the entropy demand has been supplied by deltaH. This is the meaning of the "free" part of free enery- it is free to do other work since entropy demand is satisfied. Gibbs free energy (as described above) applies to constant pressure processes. For constant volume processes, the Helmholtz Free Energy (A) is used instead: A = U - TS

Kinetics Basics

Chemical kinetics is the study of the rates and mechanisms of chemical reactions. On the energy profile attached, the transition state is indicated by the maximum point. If a minimum occurs in the curve while proceeding towards the products (as it does in this example), it is called an intermediate. The activation energy (Ea) is used to describe the difference between the energy of the reactants and that of the transition state (activated complex). That is to say, the Ea tells how much energy is needed for the reactants to react the transition state and proceed through the reaction.

Colligative Properties

Colligative properties are the physical properties of dilute solution that depend only on the number of molecules in the solution (not on their chemical nature). Presence of nonvolatile particles dissolved in a solvent can cause the physical properties of the solution (ie, freezing or boiling point) to differ from those of the pure solvent, because the presence of the nonvolatile solute reduces the number of solvent particles present per unit volume. The three most important colligative properties are: (1) Freezing Point Depression (2) Boiling Point Elevation (3) Osmotic Pressure Depression The presence of a solute modifies that freezing and boiling points of a solvent. The phase diagram of a solution when a nonvolatile solute is added is shown. Note that the phase diagram of the solution is generally displaced lower than that of the pure solvent.

Chemical Potential

Consider a system that consists of one phase but has more than one component. The chemical potential of the ith component of the system is given by the equation shown for ui (mu = u), where ni is the number of moles of component i. Then, the chemical potential is the variation of free energy of the component during an increase in its quantity, and the chemical energy is udn. Each type of energy is the product of two factors, an intensity factor (intensive property) and a capacity factor (extensive property), as shown. The intensity factors are potentials, meaning they are the tension or driving force of the type of energy being considered. Temperature, for example, is the driving force of heat transition. When two systems with different potentials interact, equalization of the potentials takes place at the expense of the corresponding capacity factors. When phase equilibrium is achieved, the chemical potential of a component is the same in both phases. Chemical potential is the driving force during mass transfer. If we consider a system consisting of two phases, a & b: - Chemical potentials of component i in each phase are (ui)a and (ui)b respectively. - At constant T and P, a certain amount of i is transferred from one phase to another. If dni moles of i goes from phase a to b, then: deltaGb = (ui)b dni and deltaGa = (ui)a dni - This yields a total change in free energy of: deltaGtotal = [(ui)b - (-ui)a]dni Equilibrium between the two phases is achieved when deltaG total = 0 or when (ui)b = (-ui)a

Ozone Decomposition Reaction Mechanism Example

Consider the decomposition of ozone: 2 O3 (g) --> 3 O2 (g) r = k[O3]^2 / [O2] The suggested mechanism is: (1) O3 <--> O2 + O (fast) where rf =k_1[O3] and rr = k_-1[O2][O] (2) O + O3 --> 2 O2 (slow) where r2=k_2[O][O3] Step 2 is the rate determining step (since it is labelled as the slow step), but its rate equation doesn't agree with the experimental equation for the overall reaction. It also contains [O], which is a reaction intermediate, and intermediates do not show up in the overall equation. This can be explained relatively simply since we can safely assume that rf = rr for step 1, and so: k_1[O3] = k_-1[O2][O] which can be rearranged to see that: [O] = (k_1/k_-1)[O3]/[O2] If we sub this in for the [O] in the rate equation of step 2: r = (k_2k_1/k_-1)[O3]^2/[O2] thus rate = k[O3]^2/[O2] where k = (k_2k_1/k_-1)

Bond Dissociation Energies (BDE)

Consider the reaction: H-H + Cl-Cl --> 2 HCl Dissociation of H-H and Cl-Cl bonds is endothermic, requiring 436 and 242 kJ/mol energy respectively. Formation of the HCl bond, however, is exothermic, releasing 431 kJ/mol of energy. This means that the reactants absorb 678 kJ/mol of energy and are dissociated while the combination of atoms to form 2 HCl liberates 826 kJ/mol of energy. deltaH R for the reaction is simply the difference between the energy absorbed by the reactants and the energy liberated by the formation of products: deltaH*rx= Sum deltaH (bond enthalpies reactants) - sum deltaH (bond enthalpies products) The bond enthalpy is the average energy required to break a particular bond in one mole of gaseous molecules, for example: Cl2 (g) --> 2 Cl (g), deltaHrx = +242 kJ/mol For the original reaction of H and Cl, the deltaH rx is -92 kJ/mol.

Standard Enthalpy of Formation (deltaH*f)

Enthalpy change of a reaction can be evaluated using the standard enthalpies of formation (deltaH*f) of the reactants and the products. The standard state is defined as 1 atm pressure at 25 C, in which the substance has a stable physical state under these conditions. Enthalpies of formation of the elements in the standard state is 0. This means that deltaH*f(H2(g)) = 0 at 25 C. At higher temperatures, it becomes positive, and at lower temperatures, it becomes negative. The standard enthalpy of formation of a compound is the enthalpy change that would occur if one mol of compound were obtained directly from its elements in the standard state. An example of this is: C (graphite) + O2 (g) --> CO2 (g), deltaH*(298 K) = -393.5 kJ Since the standard enthalpies of formation of C and O2 are 0, it follows that the deltaH*f(CO2) = -393.5 kJ. The standard enthalpy of formation for compounds is often referred to as the standard heat of formation. The deltaH*f of most compounds have been measured and tabulated by this point.

Integrals

Explanation here: http://mathforum.org/library/drmath/view/64571.html

The Third Law of Thermodynamics

First stated as the Nernst heat theorum, and now commonly known as the third law of thermodynamics: The entropy (S) of all perfect crystalline substances is the same at absolute 0 (zero Kelvin). The third law deals with the behavior of matter at cryogenic temperatures. Absolute zero (0 K) cannot actually be attained, which is taken for evidence of the third law. At 0 Kelvin, all atoms in a pure crystal are perfectly aligned and unmoving. They have no entropy of mixing because the pure crystal is made up entirely of one element. This means that, if the entropy of each element in a pure crystalline state is taken as 0 at 0 K, then it follows that every substance has a postive, finite entropy.

Fluidity

Fluidity is a term often used in connection with viscosity. The fluidity of a substance (phi) is the reciprocal of the viscosity, equation shown.

Ideal Gases: Gay-Lussac's Law

Gay-Lussac's law describes ideal gases and states that the volume of a given mass of gas is directly proportional to its temperature if pressure remains constant. The equations for this law are expressed as shown.

Hess's Law

Hess's Law states that id the reaction can be broken down into a number of steps, deltaH of the overall process is equal to the sum of the enthalpy changes occurring in each separate step. This holds true because H is a state function, and is thus independent of the reaction path. An example: The deltaH of the following reaction cannot be directly measured: C (s) + 1/2 O2 (g) --> CO (g) Use the following information to calculate the deltaH of the reaction: C (s) + O2 (g) ---> CO2 (g) + 393.5 kJ CO2 (g) + 283 kJ ---> CO (g) + 1/2 O2 (g) Summing up these equations and cancelling out identical terms yields: C + O2 + CO2 + 283kJ -->CO2 + CO + 1/2 O2 + 393.5kJ and finally C(s) + 1/2 O2 (g) --> CO (g) + 110.5kJ Since energy has been released by this reaction (it is a product) we now know that deltaH = -110.5 kJ

Ideal Gas Behavior

Ideal gases are characterized by the absence of intermolecular forces. The internal energy (U) of an ideal gas is independent of volume. Equations showing this attached. It has been shown that, for ideal gases, C_p - C_v = R. We also know that y(gamma) = C_p / C_v, which is sometimes more conveniently expressed by y - 1 = R / C_v. It is also true that the adiabatic, reversible behavior of an ideal gas is expressed by: (T2/T1) = (V1/V2)^y-1.

Ideal Gases: Joule's Law

Ideal gases are characterized by the absence of intermolecular forces. The internal energy (U) of an ideal gas is independent of volume. Thus: deltaU T = 0 At constant temperature, a change in V leads to a change in P. This is known as Joule's Law, which is basically a statement that the internal energy of a perfect gas depends only on its temperature. Equations are shown.

Radiant Excitance

If a tungsten filament is heated, it will begin to glow dull red at 900 K, then bright red, orange, yellow, and finally white at ~2300 K. This glow is known as *radiant excitance*

deltaG and Spontaneity

If deltaG < 0, the reaction is spontaneous If deltaG > 0, the reaction is not spontaneous If deltaG = 0, the system is in a state of equilibrium

Temperature and Spontaneity

If deltaS > 0 & deltaH < 0 ; spontaneous at all temperatures If deltaS > 0 & deltaH < 0; spontaneous at high temperatures If deltaS < 0 & deltaH < 0; spontaneous at low temperatures If deltaS < 0 & deltaH > 0; never spontaneous at any temp

Endothermic vs. Exothermic

If heat is absorbed during a reaction, it is said to be endothermic and deltaU and deltaH are both positive. If heat is expelled during a reaction, it is said to be exothermic and deltaU and deltaH are both negative. It is true that deltaH = deltaU + PdeltaV. The difference between deltaH and deltaU is quite small if the reaction is held at constant pressure and the deltaV over the reaction is slight, as is the case when liquids or solids are involved. If gases are involved, however, PdeltaV can cause a significant difference between deltaH and deltaU. If the reaction produces a net change (deltan) the expression can be written as: deltaH = deltaU + PdeltaV = deltaU + delta(nRT) = deltaU + RTdeltan

Photochemistry & Multiplicity

In most cases, all electrons of the ground state are paired. When this is true, the total electronic spin (S) = 0 and the multiplicity is 2S + 1 = 1, meaning the ground state is a singlet (S_0). In the excited state, if the unpaired electrons have opposite spins, then S is once again 0 and the excited state(s) in question are also singlet (S_1, S_2 etc). If the unpaired electrons in the excited state have identical spins, then S = 1. This means that 2S + 1 = 3, and thus the multiplicity is 3 and the excited states in question are triplet state.

Reversible vs. Irreversible

Irreversible processes cannot be reversed under any circumstances. A process is considered a reversible process if the direction of the process can be changed at any point, even the end.

Kinetics & Rate Law Overview

Less confusing here: http://chemistry.bd.psu.edu/jircitano/kinetics.html

Collision Theory

Most rate constants increase exponentially with temperature. Usually, k doubles for every 10 C temperature increase. Collision theory states that reactants must collide to interact. If the temperature increases, so does the collision frequency and molecular speed. Even with a temperature increase, the rate of reaction is still much smaller than the number of collisions. This means that only a small percentage of collisions actually lead to an effective reaction. According to collision theory, the energy to break bonds comes from the kinetic energy that molecules have before colliding. If sufficient to cross the activation energy threshold, this kinetic energy is transformed into potential energy, which is stored in a transition state called the activated complex. Not all collisions between molecules have the required kinetic energy to create new bonds. Only those molecules that have both a favorable orientation and energy > Ea will actually lead to product formation.

Phase Equilibria Example: Water

Pure water is an example of a one component system. Water may exist in three phases, liquid, ice, and steam. Because water is a one component system, the maxium degrees of freedom = 2. Water and any other one component system can be represented by a 2D diagram. The most convenient variables to plot are P and T, shown. There are three areas of this diagram, each representing a single phase. In these single phase areas, the system is bivariant, meaning that either P or T can be modified (independently) without altering the number of phases. The dividing lines between the single phase areas denote conditions where equilibrium exists. Along these lines, the system is univariant, and has only one degree of freedom. This means that, for a given temperature, there is only one value for pressure at which the two phases may coexist The lines of all sections intersect at a point A, which represents the conditions under which all three phases are simultaneously at equilibrium. This is known as the *triple point*. At the triple point, the system is invariant. This means that there are no degrees of freedom, and thus neither P or T can be altered without causing the disappearance of one of the phases.

Raoult's Law

Raoult's law says that when a solute is added to a pure solvent, the vapor pressure above the solvent decreases: P1 = ix_1 P_1^0 P1 is vapor pressure of the solvent with added solvent, x1 is the mole fraction of solvent, P_1^0 is the vapor pressure of the pure solvent, i is (# moles after solution/# moles before solution) This is a linear equation, and thus the graph of Pi versus x1 gives a straight line that has an elevation equivalent to P^0 of the solvent.

The van der Waals Equation

Real gases have a more complicated state equation because the intermolecular interactions must be accounted for, since molecules in real gases attract each other. The equation for real gases is known as the modified ideal gas law, or the van der Waals equation, where "a" is a constant that accounts for the force of interactions between gas particles and "b" is another constant, which accounts for the excluded volume.

Real Gas Behavior

Real gases show PVT behavior and relationships that are different from an ideal gas. At high pressure, calculations formed with Clapeyron's equation (PV = nRT) deviate by 2-3%. van der Waals attributed the failure of the PV = nRT relationship to the fact that it: (a) Neglects the volume occupied by gas molecules (b) Neglects the attractive forces among molecules The presence of molecules of nonnegligible size in a gas means that a certain volume, called the *excluded volume* is not available for molecules to move in. If "b" represents the excluded volume of a mole of gas, Clapeyron's equation can be corrected to: P(v-nb) = nRT In addition, intermolecular attractions lower the mobility of each molecule. This means that the pressure exerted by the gas is reduced, as if the number of molecules were effectively reduced. Thus, we get a second correction factor which applies to pressure. van der Waals' complete equation then becomes: (P-an^2 / V^2)(V-nb) = nRT where a is a proportionality factor. The factors a and b are characteristic for each gas and temperature.

Schrodinger's Equation

Schrodinger's equation is a complex wave function used to describe the quantum mechanical state of a particle. It is a mathematical construct that cannot be experimentally verified. It is used to describe the probability density of a particle (that's the probability of finding a particle, say an electron, at time t at a given position r = x,y,z in a volume dV). This probability (w) is proportional to the wave function wx,y,z,t = abs value(Ψ)^2 dV The attached image represents an electron "somewhere: in a volume element dx, dy

Standard Free Energy (deltaG*)

Standard free energy can be defined as the deltaG of a process occuring under standard conditions (1 atm, 25 C, 1 M concentration). If we define deltaG*f as the deltaG* that occurs when 1 mol compound in its standard state is formed from its elements (also in standard state) then we can write: deltaG*rx= Sum deltaG*f(products)- SumdeltaG*f(reactants)

Statistical Mechanics

Statistical mechanics tries to predict probable behavior of a large collection of molecules called an ensemble. Ensembles are described by specific macroscopic properties such as volume, potential energy, pressure, and temperature. The individual molecules of an ensemble are distributed over a range of macroscopic states, so they differ from each other. Instead of trying to precisely define the states of all constituent molecules of an ensemble, statistical mechanics describes their most probable states in an attempt to arrive at a good description of the enemble's macroscopic properties.

The Isothermal-Isobaric Ensemble

Statistical thermodynamics also makes use of other ensembles, one being the isothermal-isobaric ensemble, where N, T, and P are fixed values. Each ensemble leads to a particular partition function, and each partition function leads to characteristic thermodynamic functions. For the isobaric-isothermal ensemble, the partition function is delta(N, T, P) and a characteristic thermodynamic function is the Gibb's free energy: G = -KTln delta(N,T, P)

The Microcanonical Ensemble

Statistical thermodynamics also makes use of other ensembles, one being the microcanonical ensemble, where N, V and U are fixed values. Each ensemble leads to a particular partition function, and each partition function leads to characteristic thermodynamic functions. For the microcanonical ensemble, used to describe isolated systems, the partition function called omega(N, V, U) can be related to the entropy of the system: S = Kln omega(N, V, U) The partition function is a summation over all possible quantum states, and therefore this last equation shows that, for an isolated system, the entropy is proportional to the logarithm of the number of states available to the system.

Statistical Thermodynamics

Statistical thermodynamics uses the distribution laws of statistical mechanics to calculate and predict the energies and molecular velocities of ensembles, as well as their most probable energies and velocities. A macroscopic system can be defined by a few properties, such as volume, pressure, density or temperature. From a microscopic point of view, however, there are a great number of quantum states that are consistent with the fixed macroscopic properties. In statistical thermodynamics, to calculate any property, such as energy, one must calculate the value of that property in each quantum state. Then, supposing each quantum state has the same weight, the average value of the quantity is taken. We can then postulate that this average value corresponds to the thermodynamic property in question from a macroscopic point of view.

Boiling Point Elevation (deltaTb)

The boiling point elevation, deltaTb, is given by: deltaTb = Kb x m_solute x i Where Kb is a constant characteristic of the solvent called the molal ebullioscopic constant, and m_solute is the molality of the solute (mol solute/kg solvent). The ionizability factor (i) takes into account how many individual particles each solute molecule forms in the solution.

Bond Order

The bond order of a molecule can be found using: bond order = [ (# e- in bonding MOs) - (# e- in antibonding MOs)] / 2

The Canonical Ensemble

The canonical ensemble is the most basic concept of statistical thermodynamics. A canonical ensemble is an assembly of A identical systems, each of which is characterized by its number of systems (N), its volume (V), and its temperature (T). The systems are in thermal contact with each other, and thus energy can circulate from one system to another. Although the energy of each system varies, the average energy is known. If we consider an ensemble with A systems, in which Ai systems are distributed between macroscopic states (states of the system defined by macroscopic qualities such as T, V, etc) of energy Ei, then the probability of finding a system in a state of energy Ei is given by: Pi = Ai/A = Gi x exp(-Ei/KT)/Sum Gi x exp(-Ei/KT) where Gi is the degeneracy of the state Ei. [exp(x) = e(natural log)^x, so in these equations, exp simply represents the natural log raised to the power of whatever variable happens to come in parentheses after it.]

Partition Function (Z) of a Particle

The denominator of the Boltzmann distribution gives the partition function (Z) of a particle: Z (T, V) = Sum gi x e^(-ei/KT)

Canonical Partition Function (Zc) of a System

The denominator of the Ei probability equation is known as the canonical partition function of the system (Zc): Zc (N, T, V) = Sum Gi x e^(-Ei/KT)

Boltzmann Distribution

The distribution law of a canonical ensemble can be compared with the Boltzmann distribution, which says that for an isolated system with N particles that can occupy various energy levels (ei) with degeneracy (gi), the distribution is: Pi = Ai/A = gi x exp(-ei/KT)/Sum gi x exp(-ei/KT)

Work Given Pressure & Change in Volume

The equation is shown. deltaV = Vf - Vi

Freezing Point Depression (deltaTf)

The freezing point depression, deltaTf, is given by: deltaTf = Kf x m_solute x i Where Kf is a constant characteristic of the solvent called the molal cryoscopic constant, and m_solute is the molality of the solute (mol solute/kg solvent). The ionizability factor (i) takes into account how many individual particles each solute molecule forms in the solution. Some examples are: C6H12O6 (s) --> C6H12O6 (aq) ; i = 1 NaCl (s) --> Na+ (aq) + Cl- (aq) ; i = 2 Ca(NO3)2 (s) --> Ca+2 (aq) + 2 NO3- ; i = 3

First Order Reaction Half Life

The half life of the reaction is the amount of time required for the original concentration of reactants to be reduced by half. Thus for a first order reaction: [A] = [A]_0/2 and t = t1/2 Giving us the half life equation: t 1/2 = ln2 / k_A

Osmotic Pressure (pi)

The osmotic pressure (pi) is defined as the pressure required to stop the migration of solute across a semipermeable membrane. It is the third colligative property and can be described by the equation: pi = CRT where C is the concentration of the solution and T is the temperature in K. We can see from this equation that as C increases, the osmotic pressure increases as well.

Rate of Conversion (J)

The rate of conversion (J) is defined by the attached equations. Since A and B are disappearing and C and D are being formed, the value for J is positive. First equation should be fractional like the rest?

Standard Enthalpy of Reaction (deltaH*rx)

The standard enthalpy of a reaction, deltaH*rx is the difference in enthalpy between products and reactants, when both are in the standard state at 298 K: deltaH*rx= Sum(H*f products) - sum(H*f reactants) This equation is more useful than deltaHrx = H products - H reactants because H cannot actually be measured. Even though H cannot be meausred, in accordance with Hess's law, we can evaluate the deltaH*rx through a succession of steps: First, the determination of deltaH*f of the reactants Next, the determination of deltaH*f of the products. Finally, we can sum these up and plug them into the given equation for the determination of deltaH*rx in the last step. Remember to multiply the deltaH*f of products/reactants by their stoichiometric coefficient to account for each run of the step necessary.

Time-Dependent Wave Function (Schrodinger's)

The time-dependent wave function is used to describe the harmonic wave motion of a free particle: Ψ(r,t) = a e^(i[wt-(k x r)]) where a = amplitude in units of m^-3/2 ; w = frequency ; i = imaginary unit (sqrt-1) ; k = wave number vector ; r = radius vector describing the position of the particle in space.

First Law of Thermodynamics

The total energy of a system and its surroundings is always constant. (Though energy may be exchanged between the system and the surroundings.)

Thermochemistry

Thermochemistry involves the application of the first law of thermodynamics to the study of chemical reactions. The subject deals with the measurement and calculation of the heat absorbed or released in a chemical reaction. Enthalpy is a state property. This means that the enthalpy change over a a reaction depends only on the enthalpies of the initial and final state (it is path independent). If a chemical reaction is represented by: reactants ---> products The changes in internal energy and enthalpy for the reaction can be represented by the equations: deltaU = Uproducts - U reactants and deltaH = Hproducts - Hreactants

The Derivative

This symbol just means "derivative with respect to one of those variables while holding the other variables constant." It essentially refers to a change in a variable, as in the regular lowercase d.

Transport Processes

Transport processes are those processes that bring a system out of equilibrium. They are called transport processes because matter or energy is transported to the surroundings or another part of the system. When unequilibrated forces exist in a liquid, they disrupt the state of mechanical equilibrium. rho = round p, eta = long n, phi = circle vertically halved

Electromagnetic Units

Wavelength: 1mu (micron) = 1000 nm = 10,000 angst = 0.0001 cm = 10^-6 m Frequency: Frequency (v) is the number of waves that pass a given point in unit time. It is related to wavelength by: v = c / gamma, where c = speed of light, 3.0 x 10^8 m/s Wave number: Wave number (vbar) is the number of waves per unit distance, so it has units of reciprocal distance (cm^-1): vbar = 1 / v

Displacement of Equilibrium in a Closed System

When a closed system is displaced from equilibrium, it undergoes a process where its properties change until a new equilibrium state can be reached. This process may be: (1) Isobaric; at constant pressure (2) Isochoric; at constant volume (3) Isothermal; at constant temperature (4) Cyclic; with initial state = final state (5) Adiabatic; occurring with no exchange of heat between the system and its surrounding, Q = 0

Work (W)

Work involves movement of matter from point A to B. Examples of work include pressure-volume (PV) work, which involves the expansion and compression of gases, and electrical or mechanical work, which occurs when a force acting upon a system moves through a distance (like in the piston example). In thermodynamics, work always involves the exchange of energy between a system and its surroundings. If work (W) is done on a system, W is positive. If work (W) is done by a system, W is negative.

Photochemical Processes

{# used for * throughout} (1) Vibrational Relaxation A# is usually produced in an excited vibrational state. Intermolecular collisions transfer part of this vibrational energy to other molecules, allowing A# to relax to the lowest excited vibrational level (2) Internal Conversion A molecule A# in its lowest vibrational state can make a radiationless transition to a different excited electronic state, A# --> A#'. For this process to occur, A# and A#' must have the same energy state. The molecule A#' is usually in a lower electronic state, but in a higher vibrational state than A#. An example of internal conversion is shown. {If A# and A#' are both singlet or triplet states (both the same) then the radiationless process is called internal conversion. If A# is singlet and A#' is triplet (if they're in different states), the radiationless process is called intersystem crossing} (3) Radiationless Deactivation Occurs when A# transfers its electronic excited energy to another molecule, and returns to its ground electronic state: A# + B --> A + B#. The products A and B# can have additional translational, rotational, and vibrational energy. (4) Fluroescence Occurs when light is emitted from an excited electronic state to a lower electronic state without spin change (deltaS = 0). A# can lose its electronic energy by spontaneously emitting a proton, which brings it to the ground state: A# ---> A + hv {If there are collisions between molecules, A# can lose its electronic excited energy and return to the ground state through intersystem crossing or internal conversion. In the absence of collisions, the typical lifetime of a singlet excited state is 10^-8 s} (5) Phosphorescence This process involves the emission of radiation from a triplet excited electronic state to a lower singlet state. This transition occurs with deltaS =/= 0. Since the selection rule for electronic transitions is deltaS = 0, phosphorescence has a very low probability. {If there are intermolecular collisions, the excited electronic energy is lost to internal conversion and intersystem crossing. In the absence of collisions, the typical lifetime of an excited triplet state is between 0.001 to 1 s}

Rate Basics (Stoichiometry)

The rate at which reactants are consumed and products are formed is proportional to the stoichiometric coefficients of the balanced chemical equation, shown by the attached equations. In these equations, t represents time, and n represents the number of moles present.

Reaction Rate (r)

The reaction rate (r) describes the rate of conversion (J) per unit volume. This equation is shown. If V remains constant throughout the reaction, we are able to get the relationship: -(1/a)(d[A]/dt), through the process shown, where [A] is the molar concentration of A.

Types of Systems

There are three types of systems defined on the basis of their matter and energy transfer: (1) Isolated systems: which have no exchange of energy or matter (2) Closed systems: which exchange energy but not matter (3) Open systems: which exchange both energy and matter

The Ideal Gas

An ideal gas has the simplest thermodynamic system. Gases are composed of molecules in random motion. In an ideal gas, collisions between these molecules are perfectly elastic, and the molecules do not attract one another. Pressure (P) of a gas system is defined by the collisions between the molecules and the container walls Temperature (T) of the gas system is directly related to molecular speed and the average kinetic energy of any molecule in the system. Internal Energy (U) is the sum of the energies of constituent molecules. It depends only on the temperature and the number of particles, and is independent of volume. The behavior of an ideal gas at all temperatures and pressures can be expressed by the equations: PV = nRT and U = U(T) However, no real gas exactly satisfies these equations.

Second Order Reaction Type II

Another common type of second order rate law comes from reaction type: aA + bB ---> products, with rate law r = k[A][B] The integrated rate expression takes a more complicated form, shown.

Stokes's Equation

Stokes's law describes the fall of a spherical body of radius r and density rho, falling by gravity through a fluid of density rho1. It is given by F1 in the attached equations. F1 refers to the force acting on the spherical body, and g is the acceleration due to gravity. The force F1 is opposed by a frictional force due to the medium, which increases as the velocity of the falling sphere increases. When a uniform rate of fall has been reached (terminal velocity), the frictional force (F2) becomes equal to the gravitational force F1. Stokes showed that the force due to friction is equal to F2, shown, where v is the terminal velocity of the falling sphere. When gravitational and frictional forces are equal, F1 = F2, and the viscosity coefficient (eta) can be found using the equation listed. Stokes's equation is valid as long as the radius of the falling sphere is larger than the distance that separates the molecules of the fluid.

The Harmonic Oscillator

The harmonic oscillator is a particle that has mass m, which, under the influence of a linearly applied force will move in one of several directions with a frequency (w_0). The Schrodinger equation for a one dimensional oscillator is shown. The eigenvalues of the harmonic oscillator are quantized and equidistant: E_n = hv_0(v + 1/2) with v = 0, 1, 2, 3... Eigenfunction for the harmonic oscillator is shown. The zero point energy, E_0 = 1/2hv_0, is the lowest energy possible for the harmonic oscillator. The harmonic oscillator can be used to model two ideas: (1) The vibrations of atoms and molecules (2) The lattice vibrations of crystalline materials

State Properties

The state of a system at equilibrium is defined by state properties. A state property simply describes the state of a system. All state properties are interrelated- if one varies, at least one other will vary as well. State properties are measurable. Some include: - Temperature (T) - Volume (V) - Pressure (P) - Internal energy (U) - Enthalpy (H) - Entropy (S) A state property is independent of the path taken by a system as it changes states. The differentials (infinitesimal differences or to the derivatives of functions) of all state properties must be exact, such as dV in the case of volume, which remains the same for all processes occurring between the different states reached by the system.

Changing Entropy

The third law lets us obtain absolute values for entropy of chemical compounds using calorimetric measurements. We can obtain the difference in entropy between 0 K and T using reversible additions of heat to a heat reservoir: St - So = integral dQ/T The integration is performed from 0 K to T. If a substance in a given phase is heated from T1 to T2, it gains entropy according to the equation: S2-S1 = integral dH/T = integralC_p dT/T = integralC_p d(lnT) Once again, the integration is between T1 and T2, and is graphed with a plot of C_p/T versus C_p or T versus lnT, if the required C_p values have been measured. However, C_p values are difficult below 15 K, so extrapolation to absolute zero is necessary. From 0 K to 25 C, the system undergoes a number of phase transitions. At each transition C_p changes abruptly. The entropy change at this transition can be calculated using the equation: deltaS transition = deltaH transition/ T transition By adding all of the contributions to entropy from 0 K to 25 C, we can get the standard entropy at 25 C called S* (degree sign).

Time-Independent Wave Function (Schrodinger's)

The time-independent wave function is used to describe the motion of an electromagnetic wave travelling in a vacuum: (h^2/8pi^2m) ▽^2Ψ + [E - V(r)]Ψ = 0 where w = wave function in units of m^-3/2 ; m = mass in kg ; ▽ = Laplace operator in units of m^-2 ; h = Planck's constant V(r) = potential in units of J ; E = energy in units of J

N-th Order Reactions

There are many nth order rate laws. For those that take the form d[A]/dt = -k_A[A]^n, the integrated rate expression is shown, as well as the half life equation.

The Dual Nature of Light

There are other properties of light, such as refraction, which can only be explained by considering light as a wave. This led to the theory on the dual nature of radiation. In 1924, de Broglie proposed that electrons could also behave as waves, with wavelength gamma given by: gamma = h/p =h/mv where p is the momentum of the electron, v is the velocity, m is the mass and h is Planck's constant.

p-p-π Bonding (MO Theory)

There are three p orbitals, with designations px, py, and pz. When two atoms approach one another along an axis x, head on overlap can only occur with their respective px orbitals, leading to a p-p-σ bond, since it is formed by a head on overlap as is characteristic of sigma bonds. When py and pz orbitals approach each other along the x axis, it yields a p-p-π orbital, which unlike the p-p-σ orbital, is not rotation symmetric. Instead p-p-π orbitals are symmetric about a plane that contains the x axis. This means that the overlap between the py or pz orbitals is a side to side overlap rather than a head on one, characteristic of a pi bond.

Understanding Work

We can understand work better by examining the example shown. The main equation is W = -P_ext(V_f - V_i). Given the diagram, we can also write: W = F_ext x dl or W = P_ext x A x dl = P_ext x dV Then, the work done on the system as a piston moves from V_i to V_f is explained using an integral of the changing volume, meaning that it is a continuous summation of the values. This brings us full circle back to the equation: W = -P_ext(V_f - V_i) Which will tell us how much work was done by the piston over the entire process if the proper values are plugged in.

Determining r

What is r for the following reaction: H2 + Br2 --> 2 HBr Answer shown. The key to this is remembering that the equation for r is: -(1/a)(d[A]/dt). The first two values are based on the reactants, and both have a stoichiometric number of 1, so the fraction goes away and only the negative sign remains. The final value is a product, so it has no negative sign, but it also has a stoichiometric number of 2, which is where the 1/x --> 1/2 comes from. This is as far as we can go without actual numbers for the concentrations.

s-s-σ Bonding (MO Theory)

When atoms combine to form molecules, they attempt to achieve the lowest energy level. In covalent bonding, two single atomic orbitals combine to become a molecular orbital. Attached diagram shows s-s-σ bond formation. An example of s-s-σ bond formation is: H + H --> H2 When the two H atoms approach each other, their respective 1s1 orbitals overlap and energy is lost. As they overlap still more, they eventually react their lowest possible energy state, which will dictate the length of the covalent s-s-σ bond. This creates a new orbital (the MO) that contains both bonding electrons and belongs to both atoms. Energy is required to break the bond of the H2 molecule formed. In fact, the same amount of energy that was required to make the bond is required to break it; (435 kJ/mol for H2) and this is known as the binding energy.

Path Functions

When thermodynamic quantities are path dependent, like heat (Q) or work (W) they are not considered state functions. These quantities have inexact differentials (infinitesimal differences or to the derivatives of functions), and only quantities with exact differentials are state functions. Quantities like heat and work are called *path functions* because the quantities Q and W are different for different functions, meaning dQ and dW are inexact differentials. This means they are dependent on the path taken by the system as it changes states.

Dissociation in Photochemistry

{# used for * throughout} A# is often formed in an excited vibrational level. If A# has enough vibrational energy, dissociation may occur: A# --> B + C where the decomposition products (B and C) may react further, especially if they are free radicals. The example on the left is that of an electronic transition in a diatomic molecule where A# has vibrational energy that exceeds the dissociation energy (De). The example on the right shows excitation of a diatomic molecule to a repulsive electronic state (one in which there is no minimum in the potential energy curve), which also leads to dissociation.

Parts of a System

A *system* is any part of the universe that is of interest. Systems of interest are finite and macroscopic. Only measured variables (known as state properties) are used as thermodynamic parameters. The detailed structure of matter in a system is not accounted for. *Surroundings* are everything that is ouside of/around the system. A *boundary* is what encloses a system and sets it apart from its surroundings. It is an imaginary concept, and may: (a) completely isolate a system from surroundings or (b) allow interaction system and surroundings. Two types of transfer can occur between a system and its surroundings: (1) energy transfer (2) matter transfer; through particle movement across the boundary

Broadening Effects

Another factor that will have an effect on band shape is the extent of broadening. Because of the Heisenberg Uncertainty principle, the exact energy levels of a transition cannot be known. This is expressed by: deltaEdeltat ~=~ h / 4π The extent of this uncertainty, known as *lifetime broadening* is inversely proportional to the lifetime delta t, and the broadening can be evaluated using: deltav/cm^-1 ~=~ 2.7 x 10^-12 / deltat Another broadening mechanism results from the Doppler effect, due to the fact that molecules travel at high speeds in every possible direction. This causes doppler shifts in the spectral lines, which broadens them. This effect is due to the fact that an object approaching an observer with speed v, while emitting radiation of wavelength y, appears to be emitting from [1-(v/c)]y, rather than from y.

The Ideal Gas State

At the zero pressure limit, real gases approach ideal behavior. If a real gas is compressed to infinite pressure while retaining ideal gas behavior, the resultant state is known as the ideal gas state. Ideal gas heat capacities (igC) are used to describe gases such as this. igC values are different for different gases and are a function of temperature only. For monoatomic gases such as helium/argon, the effect of temperature on on molar heat capacity in the ideal gas state is negligible. Heat capacities are given by (y= gamma): igC_v = (3/2)R igC_p = (5/2)R y = 1.67 For diatomic gases such as H2, O2, and N2, the heat capacities change very slowly with temperature. Around 20 C, heat capacities are given by (y= gamma): igC_v = (5/2)R igC_p = (7/2)R y = 1.40 For polyatomic gases, like CO2 or CH4, heat capacity varies significantly with temperature. y value is usually less than 1.3

Quantum Yield

Consider the decomposition reaction: A2 + hv --> 2 A The first step of this photochemical reaction is the absorption of a single photon by molecule A2. As a result of this activation, the molecule dissociates. If the products react no further, the number of reacting molecules is equal to the number of photons absorbed. However, if the photochemically activated molecule initiates a sequence of thermal reactions that deactivate the molecule, there is less than one molecule reacting per photon. Quantum yield (phi) is the concept that describes the relationship between the number of reacting molecules and the number of protons absorbed by: phi = (# of moles reacting per unit time) / (# of moles of light absorbed per unit time)

Energy Barriers & Thermodynamic Favorability

Consider the hydrolysis of ATP to form ADP and a phosphate. This is a thermodynamically favorable reaction with a deltaG° = -30.5 kJ/mol at 298 K and pH = 7. However, a solution of ATP at 298 K and pH = 7 remains stable. How could this be explained? The attached energy profile provides a clue. If we define the reaction coordinate (x-axis) as the length of the P-O bond that must be broken in ATP for the reaction to proceed, we know that cleavage of this bond requires a lot of energy, and therefore the reaction must proceed uphill first in order to react the transition state. In other words, ATP has a high energy barrier (Ea) which accounts for its stability under standard conditions, despite the reaction still being favorable overall.

Competing First Order Reactions

Consider the simplest case, A --> B and A --> C, with rate constants k1 and k2 respectively. The rate law is: d[A]/dt = -k1[A] - k2[A] = -(k1+k2)[A] [B] and [C] can then be found as shown, and it becomes clear that [B]/[C] = k1/k2

Work & Heat Relationship

The equation shown expresses the relationship between heat and work, where deltaU is the change is internal energy, Q is heat, and W is work: deltaU = Q + W Energy may be exchanged through heat or work.

Use of Heat Capacities

The equations dU = C_vdT and dH = C_pdT are always valid for an ideal gas. Since real gases approach ideal gas behaviors at low pressure, these equations provide good approximations of real gas behavior at the P --> 0 limit. The following relationship exists between C_v and C_p for an ideal gas: C_v - C_p = R, where R is the gas constant. The ratio of heat capacities is often given by(y = gamma): y = C_p/C_v Combining this with our relationship equation, we get: R/C_v = y - 1

Second Order Reaction Half Life

The half life of the reaction is the amount of time required for the original concentration of reactants to be reduced by half. Thus for a second order reaction: t1/2 = 1/[A]_0k_A

Equation of State

An equation of state relates the thermodynamic properties of a system at equilibrium. The simplest is that of an ideal gas: PV = nRT ; where n is the number of moles and R is the gas constant (8.3143 JK^-1mol^-1)

Zeroth Order Reaction Half Life

The half life of the reaction is the amount of time required for the original concentration of reactants to be reduced by half. Thus for a zeroth order reaction: t 1/2 = [A]_0 / 2k_A

Using Molecular Orbital Theory

A maximum of three covalent bonds are allowed between two atoms: - one p-p-σ bond (px-px-σ) - two p-p-π bonds (py-py-π, pz-pz-π) One example of covalent bonding is in the formation of molecular oxygen, as shown in the attached MO diagram. (RXN: O + O --> O2) Molecular oxygen (O2) consists of two oxygen atoms. Each oxygen atom has the configuration 1s2 2s2 2p4. In the diagram, the corresponding AO (atomic orbitals) and MO (molecular orbitals) are represented by horizontal bars, and their electrons by arrows. The diagram illustrates the bonding between the 8 electrons of one oxygen with the 8 electrons of the other, to form a molecular oxygen with 16 electrons. The lower MO levels are filled out first. Each oxygen has two electrons in its s orbitals (two is 1s, two in 2s) for a total of 4 electrons each. These four electrons go into the bonding s-s-σ and antibonding s-s-σ** MOs of the O2 molecule. Since both bonding and antibonding σ MOs are completely full, the 1s and 2s σ bonding cancel out. The next MO level (p-p-σ) can accomodate two electrons, and the next, p-p-π can accommodate four electrons. The remaining two electrons must go into the p-p-π** antibonding orbital. There is now a total of 6 electrons in the π bonding orbitals, but two electrons are in the p-p-π** antibonding orbital, so two electrons must cancel out in the bonding orbitals. This makes the bond order of molecular oxygen 4/2 = 2. Of the 16 electrons in molecular oxygen, 14 are paired and two are unpaired in the p-p-π** orbital because of Hund's rule, which says that two electrons with the same spin must go into separate orbitals. These unpaired electrons seen in the MO diagram of molecular oxygen allow us to predict that molecular oxygen has a magnetic moment (is paramagnetic), which has been experimentally verified.

Electronic Absorption Spectrum

A spectrum records the dependence of the absorption or emission intensity on wavelength or frequency. Several factors affect the shape of its spectral bands, for example, the selection rules that determine whether or not a transition is allowed. Our example, the absorption spectrum of metalloporphyrin, shown, can be be understood by examining its simplified energy level diagram, also shown. The structure of metalloporphyrin is also noted. Depending on the substituents of the metalloporphyrin ring, it can be assigned a specific point group (the one shown in our spectrum can be approximated in the D4h point group). *Group theory* is used to assign symmetries to the orbitals in which the electrons reside, as well as to the energy levels involved in the transition. It is these symmetry assignments that decided whether a transition is allowed or forbidden. For instance, the transition that occurs between the S0 and S2 levels is a π --> π** transition that involves electrons of the porphyrin ring. This transition is allowed by symmetry, and is observed as a strong band at ~480 nm in the spectrum. The S0 to S1 transition is not allowed by symmetry, and the corresponding band at ~560 nm is accordingly very weak. The other weak band at ~540 nm is a mixture of electronic and vibrational transitions (Qv in the energy diagram).

The Electromagnetic Spectrum

All types of electromagnetic radiation (gamma, xrays, UV, visible, IR, radio waves) travel at the same velocity, 3.8 x 10^8 m/s. Electromagnetic radiation is characterized by two properties, amplitude and periodicity. These properties are described using quantities of wavelength (gamma), wave number (vbar) and frequency (v). The wavelength (gamma) is the distance from crest to crest on adjacent waves. Sample wave with labelled wavelength shown

s-p-σ Bonding (MO Theory)

An atomic s orbital and p orbital can overlap to form a molecular s-p-σ orbital. The wave function sign (+/-) of the s orbital can only be positive (+), and so the overlap is always with the positive (+) lobe of the p orbital. The resulting rotation symmetric molecular orbital is an s-p-σ orbital which yields an s-p-σ bond, diagram shown.

3 Step Reaction Mechanism Example

Consider this reaction which occurs in three steps: 2 N2O5 --> 4 NO2 + O2 r=k[N2O5] Broken up, the reaction is: (1) N2O5 --> NO3 + NO2 r{step1}=k{step1}[N2O5] (2) NO3 + NO2 ->NO + O2 + NO2 r{step2}=k{step2}[NO3][NO2] (3) NO + NO3 --> 2 NO2 r{step3}=k{step3}[NO][NO3] In this example, NO3 and NO2 are *reaction intermediates* meaning that they are a species formed in one step and consumed in a subsequent one. - The NO produced in step 2 is used in step 3, and so for each occurrence of step 3 there must be one occurrence of step 2. - Both steps 2 and 3 consume one NO3 intermediate, however, only one is formed in step 1. This means that there must be two occurrences of step 1 for each occurrence of steps 2 & 3 Since the overall equation is simply the sum of all steps, we now have: 2x step 1 + 1x step 2 + 1x step 3 = overall reaction The number of occurrences each step of a reaction undergoes is called the stoichiometric number of the step. Each step on its own is called an elementary reaction. We can also express the rate of each step in our example in terms of molecularity (how many molecules must collide for the reaction to proceed). Step 1 is unimolecular (just N2O5) while steps 2 and 3 are both bimolecular. Finally, in this reaction we can tell that the rate limiting step of this reaction is step 1. This is because its rate equation as we obtained from its molecularity corresponds exactly to the rate we were given for the overall reaction, which was determined experimentally, and the overall reaction rate can only be as great as the slowest step.

Consecutive First Order Reactions

Consider two consecutive irreversible reactions: A ---> B ---> C The rate constant of the first reaction (A->B) is k1, and the rate constant of the second reaction (B->C) is k2, so the concentrations of each can be found according to the attached equations.

Eigenfunctions

Eigenfunctions are the solutions to the Schrodinger's equation, and they exist only for specific eigenvalues of energy E. The totality of the eigenvalues for E yields the entire energy spectrum of the particle. If the limit of r --> infinity V(r) = 0, then the energy eigenvalues yield a discrete spectrum in the E < 0 range, and a continuum in the E >/= 0 range.

Heat (Q)

Energy may be exchanged through heat or work. Heat (Q) is the thermal energy that flows from a hot body to a cold one. When hot and cold bodies are in contact, the transfer of thermal energy takes place until thermal equilibrium is reached. When energy is added to a system as heat, it is stored as kinetic and potential energy by the molecules of the system. The units of heat are Joules (J) since heat is a form of energy. If heat (Q) is added to a system, Q is positive. If heat (Q) is removed from a system, Q is negative.

Enthalpy (H)

Enthalpy is expressed by the main equation: H = U + PV All variables have units of energy (joules, J), and U P and V are system properties, so it follows that H will also be a system property. Equations for the change in enthalpy for any process as well as constant pressure processes are shown.

Entropy (S)

Entropy (S) measures the disorder of a system. Like internal energy (U), it in an intrinsic property. It is also related to the measurable quantities that characterize a system. For a reversible process, the change in entropy is given by the equation: dS = dQ/T Perfectly reversible processes do not exist in nature, and so all natural processes result in an increase in entropy: deltaS total >/= 0 Where S total = S system + S surroundings The idea of entropy can be summarized by: (1) The total energy of the universe is constant, but the entropy of the universe is always increasing. (2) All natural processes are spontaneous, which means they must occur with an increase in entropy.

Exact vs. Inexact Differentials

Explanation: http://chemistry.stackexchange.com/questions/22171/why-is-du-an-exact-differential-and-dq-an-inexact-differential

Fluid Dynamics

Fluid dynamics is the study of the flow of liquids. Viscosity is the resistance offered by one part of a fluid to the flow of another part of the fluid. Some aspects of fluid dynamics are explained by viscosity. A fluid can be thought of as an ensemble of superimposed layers of molecules. The area of each of these layers may be defined by the variable A, and the distance between layers by the variable dx. When the layers move between two plates with velocity v1, v2 etc, the process is called *laminar velocity*. In laminar velocity, the force (F) required to maintain a constant velocity difference between two consecutive layers is directly proportional to A and inversely proportional to dx: F = eta x A (dv/dx) eta is the viscosity coefficient. The viscosity coefficient (eta) is the force per unit area required to more a layer of liquid with a velocity difference of 1 cm/s past a parallel layer that is 1 cm away. The viscosity coefficient itself is a physical quantity that is characteristic of each fluid. Fluid viscosity can be measured using the Poisseuille or Stokes equations. At the boundary between the plates and the liquid, the velocity is equal to zero. This is known as the *no-slip condition* The velocities of the fluid layers increase in the middle, and decrease moving outward towards the plates. Each layer has a velocity that differs from the next layer by dv.

First Order Reactions

For a first order reaction of type: aA --> products The rate law expression is: r = -(1/a)(d[A]/dt) = k[A] If we define a rate constant for the rate of change of A such that k_A = a x k, we get either: d[A]/dt = -k_A[A] or d[A]/[A] = - k_Adt To solve this differential equation, we integrate from state one to state 2 and eventually wind up with: ln([A]2/[A]1) = - k_A(t2-t1) If state 1 is the initial state of the reaction where t=0 and [A]_0, the equation becomes: ln([A]/[A]_0) = -k_At or ln[A] = -k_At + ln[A]_0 and so [A] = [A]_0e^(-k_At) [A] decreases exponentially for a first order reaction, and a plot of [A]/[A]_0 versus time gives a straight line with slope -k_A, shown.

Changing Internal Energy of Adiabatic Systems

Heat and work are both means to change the energy of a system, and both are measurable quantities. If a process is adiabatic, meaning that there is no exchange of heat between the system and its surroundings, Q = 0 and the first law of thermodynamics can be written as: deltaU = - W Where U is the total internal energy of the system and W is the work done by the system. U is a system (state) property, and its value depends on the state of the system. Any process that changes the system will change U. This means that the integration of dU gives the difference (the change) in the two values of internal energy. Q and W on the other hand are quantities, and depend only on the path taken by the process. dQ and dW denote infinitesimal quantities, and integration yields the finite (workable?) quantity.

Closed Systems that Exchange Energy with Surroundings

Heat and work are both means to change the energy of a system, and both are measurable quantities. If a closed system is allowed to exchange only heat and work with its surroundings, the first law of thermodynamics can be written: U = Q - W Where U is the total internal energy of the system, Q is the heat added to the system, and W is the work done by the system. U is a system (state) property, and its value depends on the state of the system. Any process that changes the system will change U. Q and W on the other hand are quantities, and depend only on the path taken by the process. dQ and dW denote infinitesimal quantities, and integration yields the finite (workable?) quantity.

Reaction Types

Homogeneous reactions occur in one phase, while heterogeneous reactions occur in more than one phase. Consider the following homogeneous reaction in a closed system: aA + bB --> cC + dD where A, B, C, and D represent different chemical substances, which a, b, c, and d represent the stoichiometric coefficients of the balanced chemical equation.

Blackbody Radiation

If you were to imagine a perfect absorber, it would be one that absorbs all frequencies of light, but emits none. This substance would be black, and it would be called a blackbody. The closest we can get to a true blackbody is a hollow cavity with a very small hole leading to it. Of the light absorbed by the cavity, very little can escape. Previously, the experimentally observed variation of the radiant excitance of a blackbody with wavelength could not be explained by any model, because theory assumed that energy was divided equally between all the vibrations emitting radiation, with the result that the energy had to increase as the wavelength became shorter, since the short wavelengths had more vibrational modes. This is due to the relationship between wavelength and frequency. In 1901 Max Planck proposed that light energy was made of discrete units called quanta, and that the energy of the quantum was directly proportional to the frequency of an oscillator: E = hv, where E is energy, v is frequency, and h is Planck's constant (6.626 x 10^-34 Js^-1). This led to a new treatment of the variation of radiant excitance with wavelength in excellent agreement with experimental results, shown.

The Photoelectric Effect

In 1887 Hertz observed that electrons are emitted from a metal when the metal is irradiated with visible or UV radiation. This effect is known as the photoelectric effect. Three key points are true of the photoelectric effect: (1) Below a given cutoff frequency, no electrons were ejected from the metal surface regardless of radiation intensity (2) Above said cutoff frequency, the number of electrons that were emitted was directly proportional to the radiation intensity (3) As the frequency of the incident radiation was increased, the maximum velocity of ejected electrons increased Classical theory could not explain these results, as it predicted only that the emission of electrons would be directly proportional to the intensity of the radiation

Photons & the Photoelectric Effect

In 1905, Einstein proposed that, not only way the energy of an oscillator quantized, but that light itself consisted of particles or photons with energy = hv He provided a new explanation for the photoelectric effect by proposing that work energy (w) was required to remove and electron of a given energy hv from the surface of the metal according to the equation: hv = 1/2mv^2 + w where 1/2mv^2 is the kinetic energy of the emitted electron. If the energy of the electron (hv) is less than w, the electron cannot be emitted, but it if it is greater than w, the electron will be emitted. There is also a frequency equal to w/h that is just sufficient for the electron to be emitted.

The Uncertainty Principle

In 1926, Heisenberg realized that it was impossible to simultaneously measure the momentum and the position of a particle, because performing one measurement would disturb the particle, thus preventing accurate measurement of the second quantity. This is expressed in the Heisenberg Uncertainty Principle: deltaq x deltap > h/4pi which just stays that the product of the uncertainty of the position of the particle (deltaq) and the uncertainty of the momentum (deltap) is greater than h/4pi.

Quantum Mechanics Operators

In QM, operators are represented with a circumflex accent (basically a half circle, here $) over the symbol representing the variable of interest. Operators are placed to the left of the function they are operating on. For example, the classical variable for kinetic energy is Ek. The corresponding QM operator is $Ek, and the operation that it performs on the function to which it is attached is -(h^2/8pi^2m)V^2, so $Ekphi means the operator Ek is operating on function phi.

Second Order Reaction Type I

One common type of second order rate law comes from reaction type: aA --> products, with rate law: r = k[A] The integrated rate expression comes from: d[A]/dt = - k_A[A]^2, and once again k_A = a x k and so it can be shown that: (1/[A]) - (1/[A]_0) = k_At A plot of 1/[A] versus t gives a straight line with slope k_A.

Reaction Mechanisms

Overall reaction stoichiometry doesn't necessarily yield any information about the reaction mechanism. To be accepted as valid, a reaction mechanism must satisfy two conditions. (A) The sum of its elementary steps must be equal to the balanced equation of the overall reaction (B) The mechanism must be consistent with the experimentally determined rate equation The number of occurrences each step of a reaction undergoes is called the stoichiometric number of the step. Each step on its own is called an elementary reaction. Simple reactions consist of only one elementary reaction, while complex reactions consist of more than one elementary reaction. The rate of each step can be expressed in terms of its molecularity, which refers to the number of molecules that must collide for the reaction to proceed (only one is unimolecular, two is bimolecular etc). The overall rate of the reaction may not exceed the rate of the slowest step. The slowest step is thus referred to as the rate limiting step.

p-p-σ Bonding (MO Theory)

P orbitals are also able to overlap and form molecular orbitals. The +/- signs in the diagram do not represent charges. Instead, they represent the sign of the wave funciton used to mathematically construct the orbitals. An overlap of orbitals is only possible between orbitals that have the same description or sign (+/-). When the + or - halves of two p orbitals overlap, a rotation symmetric molecular orbital is formed, called the p-p-σ bond. The resulting molecular orbital is at a lower energy than the single p orbitals, shown. In the attached energy diagram, two atomic orbitals (AO), both p orbitals, combine to form a p-p-σ bonding MO, which contains the two p electrons that are now shared by the molecule via p-p-σ covalent bond.

Photochemistry Basics

Photochemistry is the study of chemical reactions induced by light. Instead of Ea being provided by intermolecular collisions, energy is supplied by the absorption of light. Usually, there are as many photons absorbed as there are molecules undergoing a transition to the excited state. This is known as the Stark-Einstein Law. During a photochemical reaction, a photon may promote an electron to a higher electronic state, where it is more likely to undergo a chemical reaction than in the ground state. The absorption of energy by a molecule can be described by: A + hv = A* Where A is the molecule in the ground state and Astar the molecule in the excited state. During the course of a photochemical reaction, several processes may occur. Several of them are illustrated by the attached diagram.

Optical Spectra Absorption & Emission

Photon energy (Ep) is given by Ep = hv. A molecule can absorb or emit this energy, thereby altering its rotational, vibrational, or electronic energy by an amount: deltaEm = Ep = hv If delta Em is positive, the photon is absorbed by the molecule in a process called absorption. If deltaEm is negative, the photon is emitted in a process called emission. Electronic absorption takes place when the energy of the absorbed photon is in the UV or visible range, shown by the S0-->S1 arrow on the diagram. If the photon absorbed has energy in the IR region, the absorption is vibrational and takes place between vibrational levels (represented v in the diagram). Absorption is only allowed between states of the same spin multiplicity (ex. singlet to singlet) and is forbidden between states of different multiplicity. However, if two states of different multiplicity lie close to each other, the absorbed energy can be transferred between these states via intersystem crossing. Emission from a triplet state is called phosphorescence, while emission from as singlet state is called fluorescence.

Poisseuille's Equation

Poisseuille's equation is: eta = piPtr^4 / 8LV also shown, where v = volume of the liquid, P = pressure at the start of the process, t= time required for this volume to flow through a capillary tube of radius = r and length = L. The viscosity of a liquid is determined by comparison with a reference liquid, usually water. The ratio of the viscosity coefficients is given by the equations shown, where P1 and P2 are proportional to rho1 and rho2, the densities of the liquids. When rho1, rho2, and eta2 are known, the viscosity coefficient (eta1) of the liquid can be determined.

Units & Constants

Pressure (P): 1 Pa = 1 kg m^-1s^-2 1 atm = 760 mmHg = 101,325 Pa = 1.01325 bars Volume (V): 1 L = 1000 mL = 10^3 mL 1 m^3 = 10^6 cc = 1,000 L Energy (E, U): 1 J = 1 kg m^2s^-2 = 1 N m 1 erg = 10^-7 J 1 cal = 4.184 J Force (F): 1 N = 1 kg m s^-2 Gas Constant: R = 1.987 cal K^-1 mol^-1m = 0.08206 L atm K^-1 mol ^-1

Chemical Kinetics

The activation energy (Ea) is a threshold that exists for every reaction, which must be overcome in order for the reaction to occur. Take for example the reaction: 2 BrNO (g) --> 2 NO (g) + Br2 (g) In this reaction, two Br-N bonds must be broken, which requires energy. According to collision theory, this energy comes from the kinetic energy that molecules have before colliding. This kinetic energy is transformed into potential energy, which is stored in a transition state called the activated complex. Br-N bonds are broken in the activated complex, and Br-Br bonds are formed. In this particular example, the reaction is exothermic. This does not have any bearing on the rate of reaction, since the rate depends of the activation energy, not the amount of energy being liberated. Arrhenius's equation describes the relationship between rate (k) and activation energy (Ea): k = A e^(-Ea/RT) where k is the rate constant, R is the gas constant, T is temp Kelvin and A is a constant known as the preexponential factor. The attached energy profile shows that the colliding molecules must have an amount of kinetic energy that is at least equal to Ea. At a given temperature, only a fraction of the molecules will have this required energy.

What is Heat Capacity?

The effects of temperature on the energy of chemical reactions is treated in terms of heat capacity. Heat capacity is defined as the thermal energy that is required to raise the temperature of a system 1 C under specified conditions. We are able to calculate the heat capacity of a reversible process for which the path is fully specified.

The Zeeman Effect

The environment in which a transition takes place will also effect the spectrum. For example, the presence of a magnetic field will shift the energy levels between which the transition occurs. This is known as the Zeeman effect, shown, which is given by: deltaEz = g x m_j x mu_B x B where g is the Lande factor, m_j is the magnetic quantum number, mu_B is the Bohr magneton, and B is the magnetic field.

The Stark Effect

The environment in which a transition takes place will also effect the spectrum. For example, the presence of an electric field can shift the spectral bands. This is known as the Stark effect, shown. Stark splitting is given by the equation: deltaEs = (3/8π^2)(hm_eZ)(n1-n2)n abs(E)abs where h is Planck's constant, m_e is the mass of the electron, Z is the number of protons in the nucleus, ni is the principle quantum number, and E is the electric field.

Constant Pressure Heat Capacity (C_p)

The equation for heat capacity measured at a constant pressure is shown. In this equation, heat capacity is measured as the amount of heat needed to increase the temperature by dT, when the system is heated in a reversible process at constant pressure. We know that at constant pressure, dH = dU + PdeltaV. We also know that, for a reversible process: dW = PdeltaV and dU = dQ - dW. Thus, dH = dQ, and we get the alternative expression for C_p, shown. This alternative expression also shows that C_p is a state function. Additionally, this equation can be written: dH = C_pdT

Constant Volume Heat Capacity (C_v)

The equation for heat capacity measured at a constant volume is shown. In this equation, heat capacity is measured as the amount of heat needed to increase the temperature by dT, when the system is heated in a reversible process at constant volume. With this restriction, no work can be done by the system. From the mathematical definition of the first law: dU = dQ - dW = dQ - PdeltaV; at constant volume, deltaV=0 and dU = dQ, yielding an alternate definition of C_v, shown. Since U, T and V are system properties, C_v is as well. Additionally, this equation can be written: dU = C_vdT

Reaction Rate Measurement

The most common method for the experimental determination of rate laws is known as the initial rate method. This method works by measuring the initial rate (r_0) for several trials of the reaction, and varying the initial concentration of one reactant at a time. If we consider an example reaction: A + B ---> C + D we know that its rate law is r = k[A]^a[B]^b, where a and b are the partial orders of A and B, respectively. In this example, we will determine the values of a and b using the attached experimental data table. From trial 1 to trial 2, we see [A] is kept constant, but [B] is doubled. As a consequence, the initial rate has doubled. This shows that the reaction is first order in B. Mathematically, we obtain the relationship: (0.005 / 0.010)^b = 1/2 where (0.005/0.010)^b = 1.40/2.80 or (1/2)^b = (1/2)^1, and b=1 From trial 2 to trial 3, [B] is kept constant but [A] is doubled. In this case as well, the initial rate of reaction doubles, which shows that the reaction is also first order in A. Mathematically we obtain the relationship: ([A]2/[A]3)^a = r2/r3 where (0.100/0.20)^a = (2.80/5.60) or (1/2)^a = (1/2)^1 and a=1 Therefore, we know that the overall reaction order is 2, and the experimental rate law is r = k[A][B] (since both exponents are 1). We are able to find the actual value for k using any of the experimental trials, here trial 2: 2.80 x 10^-6 mol x L-1s-1 = k(0.100 mol x L-1)(0.010 mol x L-1) k = 2.80 x 10^-3 L x mol-1s-1 *Note* If initial rate increases by a factor of four rather than doubling upon doubling the reactant, the reaction is second order with respect to that variable rather than first.

Zeroth Order Reactions

The order of most reactions that involve only one reactant aA ---> products is either one or two. A few reactions, however, are zeroth order, and in these cases: r = -(1/a)(d[A]/dt) = k, or d[A]/dt = -k_A Integration gives the expression [A] = -k_At + [A]_0 For a zeroth order reaction, a graph of [A] versus time yields a straight line graph with the slope -k_A, shown.

5 Major Principles of Quantum Mechanics

The postulates of quantum mechanics are as follows: (1) The physical state of a particle can be fully described by a wave function of the type (Ψx,y,z,t) (2) The (Ψx,y,z,t) wave functions are obtained by solving the appropriate Schrodinger equation (time dependent or time independent) (3) Every dynamic variable that correlates with a physically observable property is expressed as a linear operator (4) Operators that represent physical properties are derived from the classical expressions for these properties (5) The eigenvalues obtained by solving the appropriate Schrodinger's equation represent all possible values of an individual measurement of the quantity in question

Reversible First Order Reactions

The reaction A <---> B is first order in both the forward and reverse directions, so that rf = kf[A] and rr = kr[B], where f and r mean forward and reverse, respectively. At equilibrium, the concentration of each species is constant, and it can be shown that: kr[B]_0 = kf[A]_0 = (kf + kr)[A]_eq This equation relates the equilibrium concentration of A to the initial concentration of A and B. It can also be shown that the rate law equation for a reversible first order reaction is given by (shown): [A] - [A]_eq = ([A]_0 - [A]_eq)e^(-(kf+kr)t)

Reaction Orders

The reaction order is the power to which the concentration of a reactant is raised. In this example, x moles of A react with y moles of B to form the product, P: xA + yB ---> P If the rate of formation of P is d[P]/dt, the expression for r can be written as: r = d[P]/dt = k[A]^a[B]^b This final expression is known as a rate law, and is expressed as a function of the reactant concentrations at constant temperature. - The exponents (here a and b) in a rate law are usually integers, and k is known as the rate constant. - The example reaction is of ath order in A, and bth order in B, making the overall reaction order (a+b) - These exponents (a and b) are called partial orders - The sum of the partial orders gives the total order The exponents in the rate law can and often do differ from the values of the stoichiometric coefficients of the balanced reaction equation. This means that rate laws can not be derived from the stoichiometry of the overall equation. They must be determined experimentally.

Entropy and the Flow of Heat Between Hot & Cold Bodies

The second law can be used to show that the flow of heat between two reservoirs, one hot and one cold (Th and Tc), must be from the hotter body to the colder body. When heat is added to or extracted from a system, the system undergoes a finite entropy change at constant temperature, and thus deltaS = dQ/T. The quantity Q is the same for both bodies, but Qh and Qc have opposite signs. This is because the heat added to one body is positive, and the heat extracted from the other body is negative, thus: Qh = -Qc From this, it follows: deltaSh = Qh/Th = - Qc/Tc, and deltaSc = Qc/Tc and so deltaStotal = deltaSh + deltaSc = - Qc/Th + Qc/Tc = = [Qc(Th-Tc)/ThTc] According to the second law, deltaStotal must be positive, and therefore, Qc(Th-Tc) > 0. This means that Qc must be positive and represent the heat added to a cold body, leading us back to the conclusion that heat flows from a hot body to a cold body, now proven by changing entropy. This process as described is spontaneous, and its driving force is the difference in temperature between the two bodies.

The Second Law of Thermodynamics

The second law of thermodynamics states that the entropy change for a system and its surroundings, considered as a whole, is positive and approaches zero for any reversible process. That is to say, the entropy (disorder) of the universe tends to increase.