3.1 Thermodynamics - Chemistry GRE Subject Test
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.
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
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)
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.
Work Given Pressure & Change in Volume
The equation is shown. deltaV = Vf - Vi
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.
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.
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).
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
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
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.
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
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.
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
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.
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
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
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.
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.
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
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.
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.
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
Integrals
Explanation here: http://mathforum.org/library/drmath/view/64571.html
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.)
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
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.
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.
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
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
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)
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.
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
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.
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.
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
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.
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.
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.
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.
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
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.
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)
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.
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.]
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)
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.
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
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.
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.
