Important things to memorize!
thiol
-SH
ester
-oate endings ex: propanoate
millimolar (mM), micromolar (μM), nanomolar (nM)
1 × 10−3 M, 1 × 10−6 M, 1 × 10−9 M
colligative properties
A special set of properties actually depend on the NUMBER OF SOLUTES PRESENT, rather than what kind of solutes they are. These are termed colligative properties: (1) vapor pressure reduction, (2) boiling point elevation, (3) freezing point reduction (4) osmotic pressure.
intensity of wave
I = P/A Intensity = Power/area I = intensity (watts/m^2) P = power (watts) A = area of cross section. (m^2) Intensity is the quantity of energy the wave conveys per unit time across a surface of unit area and it is also equivalent to the energy density multiplied by the wave speed. It is generally measured with units of watts per square meter. Intensity will depend on the strength and amplitude of a wave.
speed of sound in air
340 m/s
1 cal
4.184 J
1 atm
760 mmHg = 760 torr = 101.325 kPA
imine
A double bond between a carbon and a nitrogen C=N
Electrochemical cells
An Ox = Anode oxidation Red Cat = reduction cathode ( more + Ecell is the reduced cathode) Ecell = Ered, cathode - Eoxid, anode current(I) flows from cathode to anode(+ --> -) electron flows from anode (- --> +) electromotive force (emf or Ecell) if Ecell: + than it's spontaneous (Gibbs = -) Standard state delta G = -nF(standard state Ecell) n = number of electrons exchanged A redox reaction is a chemical process in which electrons are transferred between atoms. One atom is reduced, meaning that it gains at least one electron, while another atom is oxidized, meaning that it loses at least one electron. Like all chemical reactions, redox reactions must be balanced, with respect to both charge and the atoms involved. To do this, it is useful to split up redox reactions into half-reactions, focusing on the net ionic products. To balance the charge of a half-reaction, simply add electrons to the more positive side until the charge is the same on both sides. For example, the reaction 2 AgNO3 + Zn → 2 Ag + Zn(NO3)2 can be split into two half-reactions: (1) a reduction half-reaction for silver (Ag + + e− → Ag) and (2) an oxidation half-reaction for zinc (Zn → Zn2+ + 2 e−). The tendency for a species to spontaneously become reduced is measured using a parameter called the standard reduction potential. Reduction potentials (E°) are measured in volts and are defined relative to the standard hydrogen electrode (2 H+ (aq) + 2 e− → H2(g)), which is set at 0 V. Greater (more positive) reduction potentials indicate that a substance 'wants' to be reduced more, while smaller (more negative) reduction potentials indicate that a substance is not prone to reduction. Redox reactions can be carried out in special devices known as electrochemical cells. These cells must have two electrodes, which are where the redox half-reactions occur. The electrode where oxidation happens is known as the anode, while the electrode where reduction happens is known as the cathode. Therefore, a surplus of electrons is generated at the anode (because electrons are lost during oxidation), and they travel to the cathode. In a galvanic (or voltaic) cell, a spontaneous redox reaction is used to generate a positive potential difference that can drive current. The total standard potential generated by a cell, Ecell, can be calculated from the standard reduction potentials of the half-reactions. The simplest way of defining Ecell is presented below: Ecell = E°cathode − E°anode In contrast to a galvanic cell, an electrolytic cell uses a connected power source to conduct a nonspontaneous redox reaction. *galvanic cells have POSITIVE Ecell values (indicating spontaneity), *electrolytic cells are characterized by NEGATIVE Ecell values. Red Cat (Cathode has reduction) An Ox (Anode has oxidation)
Buoyant Force
Archimedes Principle the upward force exerted by a fluid on a submerged object F = force (N = kg*m/s^2) p = density of fluid (kg/L) V = volume of displaced fluid (L) g= acceleration due to gravity (m/s^2)
Ideal gas law
Assumptions 1. no attractions between gas molecules 2. gas molecules do not take up space PV=nRT R = 8.21 x 10^-2 L*atm/(mol*K) 1 mol of any gas = 22.4 L at STP STP: T = 273 K and P = 1 atm Avogadro's Principle n1/v1 = n2/v2 (P and T are constant) Boyle's Law P1V1 = P2V2 (n and T are constant) Charles Law V1/T1 = V2/T2 Gay-Lussac's Law: P1/T1 = P2/T2
boiling point elevation and freezing point depression
Boiling point elevation directly follows from vapor pressure reduction. The lower the vapor pressure, the more energy that will be required to increase that vapor pressure to a level that matches the atmospheric pressure. In other words, the more solute particles present, the lower the vapor pressure, and the higher the boiling point. The extent of this elevation can be calculated using the formula ΔTb = iKbm, where ΔTb = amount by which the boiling point is elevated, i = ionization or van't Hoff factor, which refers to the number of ions each solute molecule dissociates into in solution, m =solute molality. freezing point depression. This property stems from the idea that solute molecules disrupt the lattice structure of the frozen solvent, so more added solute corresponds to more "difficulty" freezing and a lower freezing point. The equation for freezing point depression is ΔTf = iKfm.
acetic acid
CH3COOH
chirality
Chirality is emphasized on the MCAT because it is biologically relevant. Amino acids are chiral = L-stereoisomers comprise the overwhelming majority of amino acids that occur in nature. Carbohydrates = D-isomers occur in nature Many enzymes are stereospecific, meaning that they only function for specific stereoisomers, and the biological functionality of compounds can vary depending on their chirality.
Chromatography
Chromatography is a broad set of separatory techniques based on relative affinity, or tendency for a compound to attract to a certain solvent or structure. Specifically, sample molecules vary in their affinities for a mobile (moving, typically solvent-based) phase versus a stationary (static) phase. In column chromatography, the stationary phase is a vertical column packed with an adsorbent with carefully chosen properties. This adsorbent can attract sample molecules based on charge, size, or affinity for specific ligands.
Zeff (effective nuclear charge)
Due to the presence of protons, the nucleus of an atom is always positive. The attractive force of this positively-charged nucleus on the atom's negatively-charged valence electrons is termed the effective nuclear charge (Zeff). As the number of protons in the nucleus increases from left to right across a period (or row) of the table, Zeff also increases, since each additional proton adds positive charge to the nucleus. However, Zeff is not synonymous with the number of protons held by an atom. Moving down a group, the principal quantum number of the outermost energy level increases, which effectively means that more shells of electrons are added between the nucleus and the outermost, or valence, electrons. These layers of core electrons partially shield the valence electrons from the effects of the positive charge in the nucleus. Thus, Zeff decreases as one moves down a group. Periodic trends in Zeff help explain other trends on the periodic table. Atomic size, or radius, is inversely related to Zeff. Atomic radius decreases from left to right across a period as Zeff increases and the addition of protons pulls the valence electrons closer to the nucleus. In contrast, atomic radius increases down a column as more electron shells are added and electron shielding decreases the attractive force of the nucleus on valence electrons. To summarize this trend, the atoms with the largest radii are found nearest the bottom left of the periodic table.
total mechanical energy
E = U + K - mechanical energy is conserved when the sum of kinetic and potential energies remains constant conservation of mechanical energy E = U + K = 0
Electric field
E = V/m E = electric field magnitude N/C or volts/meter v = voltage (v) 1 v = 1 J/C m = meters
law of conservation of energy
Energy cannot be created or destroyed law of conservation of energy (KE initial + PEinitial = KEfinal + PEfinal), since kinetic energy (KE) = ½mv2. Potential energy (PE) = mgh This approach can be effective, but is only directly applicable when you do not need to include time as a variable.
2nd law of thermodynamics
Every energy transfer or transformation increases the entropy of the universe.
Electric field equation
F = force (N) q = quantity of charge of particle (units = Columbus = C) k = 9.0 x 10^9 N*m^2/C^2 r = separation difference between charges (m) Electric field = force/charge = N/C (Newtons/Coloumbs)
Force
F = ma F = force (Newtons) m = mass (kg) a = acceleration (m/s^2) 1 N = 1 kg*m/s^2.
weight of a volume of fluid
F=ρVg ρ= density (kg/m^3) V=Volume, (L) g = acceleration due to gravity
centripetal force
Fc = mv^2/r v = 2(pi)r/T f = 1/T m = mass (kg) v = velocity (m/s) r = radius (m) pi = 3.14 T = period f = frequency (revolutions/second = Hz or 1/s)
formal charge
Formal charge can be found using the following equation, in which "sticks" refer to bonds and "dots" refer to lone-pair electrons. Formal charge = # of valence electrons - sticks - dots
non-polar hydrophobic amino acids
G,A,V,L,I,M, (F,W,P) (aromatic)
strong acids
HCl, HBr, HI, HNO3, H2SO4, HClO4 completely dissociate into their component ions in aqueous solutions and run is non-reversible strong acid/bases have weak conjugate base/acids
kinetic energy
K.E. = (1/2) mv^2 m = mass (kg) v = velocity m/s units = kg * m^2/s^2 = 1 Joule (J)
conservation of energy
KEinitial = PEfinal mv^2/2 = mgh h = v^2/(2g) initial kinetic energy = final potential energy KE = (1/2)mv^2 PE = mgh
oxyanion prefixes
Oxyanions—polyatomic anions that contain oxygen—use the suffixes "-ite" and "-ate" for compounds with fewer and greater numbers of oxygen atoms, respectively. The prefix "hypo-" is added for oxyanions with one fewer oxygen than "-ite" ions, while the prefix "per-" is added for ions with one more oxygen than "-ate" ions. Examples of this convention include hypochlorite (ClO−), chlorite (ClO2−), chlorate (ClO3−), and perchlorate (ClO4−), as well as nitrite (NO2−) and nitrate (NO3−).
pressure
P = pressure (1 N/m^2 = 1 kg/m*s^2 = 1 J/m^3 = 1 Pascal) F = Force (N) A = area (m^2)
strong bases
NaOH, KOH, other soluble hydroxides of Group IA metals completely dissociate into their component ions in aqueous solutions and run is non-reversible
Newton's three laws
Newton's 1st law: is a statement about inertia. It states that within a reference frame, an object remains at rest or at a constant velocity unless an external force acts upon it. Essentially, this law captures the insight that forces of resistance and friction make moving objects slow down and stop. It can be summarized as Fnet = 0 at equilibrium. Newton's 2nd law defines force. It states that the total sum of forces acting on an object is equivalent to its mass times its acceleration. This is the familiar equation Fnet = ma. Newton's 3rd law is about how forces come in pairs. It states that when body A exerts a force on body B, body B exerts an equal and opposite force on body A: FAB = −FBA. For example, the earth exerts a gravitational force on your body, which is responsible for your weight, but your body also exerts a gravitational force on the earth. In contrast to a common misconception, weight (caused by a gravitational force) and the normal force do not form a Newton's third law pair, because they are caused by different underlying forces and do not always have to be equal.
noncompetitive inhibition
Noncompetitive inhibition is when the inhibitor does not compete with the substrate for the active site, but reduces enzyme activity by binding to another site (the allosteric site) on the enzyme. In noncompetitive inhibition, the inhibitor can combine with either the enzyme or the enzyme-substrate complex. In pure noncompetitive inhibition, the value of Vmax is decreased. Since these inhibitors do not compete with the substrate, their activity is unaffected by substrate concentration. Since the inhibitor always affects a consistent proportion of the available enzyme, Vmax is reduced. However, Km remains the same since if Vmax, is reduced, Vmax/2 is reduced proportionally, and the amount of substrate required to reach this new, reduced Vmax/2 is the same as the original Km. Vmax = decreased Km = unchanged
Gauge pressure (Pg)
Pressure felt at a particular depth = hydrostatic pressure Pg = ρ g D ρ = density of fluid g = accl due to gravity D = depth 1 g/mL = 1,000 kg/m^3
Entropy
Q = heat gained or lost in a reversible process T = temperature in kelvins S = entropy (J/mol*K) Entropy is a state function (which are always path independent)
ether
R-O-R
Recrystallization
Recrystallization is used to purify a solid product that contains impurities. This process involves the dissolution of the solid in a solvent and subsequent heating. The solid then dissolves and is cooled, causing it to solidify (crystallize) again. As the lattice structures of solids tend to exclude impurities, each subsequent recrystallization results in a progressively purer compound.
Polar hydrophilic amino acids
S,T,Y,N,C,Q
Vacuum distillation
Separation techniques are widely used in organic chemistry to prepare purified substances for analysis or reaction. The best technique for a given scenario often depends on the phases of the substances being separated. If all are liquids, one may be able to utilize distillation, which aims to separate liquids by utilizing the difference between their boiling points. The liquids are initially held in the same round-bottom flask, termed the distilling flask. This flask is positioned above a heat source, typically a Bunsen burner. The top of the flask is connected to a column, which leads to a downward-sloping glass condenser over a receiving flask. The condenser is held within a glass casing through which cold water is pumped. As the round-bottom flask is heated, the liquid with the lower boiling point will begin to vaporize, and its vapor will travel up the column and re-condense to fall into the receiving flask. The eventual result is a receiving flask that is rich in the lower-BP liquid, while the distilling flask will still contain the liquid(s) with the higher BP. If boiling points are very high, a vacuum may also be used to lower atmospheric pressure, which lowers the boiling points of all substances involved. put both liquids in distilling flask, heat it up, the liquid with the lower BP will vaporize and go into the column and into the receiving flask. At the end, the receiving flask will have the liquid with the lower BP and the distilling flask will contain the liquid with the higher BP. Boiling chips and vacuum distillation, respectively, are used in distillations to: provide nucleation sites that give the liquid a place to start forming bubbles to prevent superheating; lower the boiling points of the substances to be distilled.
Competitive inhibition
The way to determine the exact type of inhibition is through enzyme kinetic measurements of Vmax and Km. Vmax is the maximum velocity reached by the reaction, and Km is the substrate concentration required to reach Vmax/2. Competitive inhibition is when the inhibitor directly competes with the substrate for the active site on the enzyme. This increases Km since it now takes more substrate to ensure half of the active sites are occupied. If enough substrate is supplied, it will outcompete the inhibitor; therefore, Vmax does not change, although more substrate is needed to reach it. Vmax = unchanged Km = increases
work energy theorem
The work done on an object equals the change in kinetic energy of the object
Solubility Product Constant (Ksp)
Temperature dependent higher Ksp, the more soluble the compound is increase in Ksp with increase temperature = non-gas solute (increase Ksp = increase solubility/dissolution and drive the run forward) complex ions form increase Ksp decrease Ksp with increase temperature = gas solute Ion Product (IP) (analogous to Keq and Q, where IP is the Q) IP = Ksp ---> Saturated (solution is at equilibrium IP< Ksp ----> unsaturated IP > Ksp ----> supersaturated
Electrolytic Cells
driven by external voltage (electrolysis) non spontaneous runs that require an input of energy to proceed (like a battery) Mols of gas that's liberated during electrolysis: M = It/nF M = mols I = current t = time n = number of electron equivalents F = Faraday's Constant ( 1 F = 9.6x10^4 C/mol electron)
functional groups
The functional groups of a molecule predict its physical properties and reactivity, and for the MCAT you are expected to be familiar with several main classes of functional groups. Alkanes and alkenes are hydrocarbons; alkanes only have single C-C bonds, while alkenes have at least one C=C double bond. They interact with each other through weak London dispersion forces, have low melting and boiling points, and do not engage in meaningful acid-base chemistry. There are several important oxygen-containing functional groups: alcohols (RC-OH), aldehydes (RC(=O)H), ketones (RC(=O)R'), and carboxylic acids (R(C=O)OH). Due to hydrogen bonding, alcohols and carboxylic acids have higher melting/boiling points than aldehydes and ketones, and can function as organic weak acids. Carbonyl (C=O) carbons have a significant partial positive charge and therefore often act as electrophiles. The -OH group of carboxylic acids can be replaced by other functional groups to form carboxylic acid derivatives, the most notable are amides (R(C=O)NR'R''), esters (R(C=O)OR'), acid anhydrides (R(C=O)O(C=O)R'), and acid halides (R(C=O)X), in increasing order of reactivity. Amines (R-NH2, R-NHR', or R-NR'R"), imines (R=NH or R=NR'), and enamines (C=C-NH2, C=C-NHR, or C=C-NRR') are nitrogen-containing compounds with medium melting/boiling points that can act as weak bases. Sulfur-containing functional groups contain the root "thio" and generally act similarly to the corresponding oxygen-containing groups. Thiol groups = R-S-H
Coulomb's Law
electric force between charged objects depends on the distance between the objects and the magnitude of the charges. Coulumbs (C ) = Joules/Volume
equation for frequency
f = 1/T T = period
Titration
Titration is the process of finding the concentration of an unknown solution (the analyte) by reacting it with a solution of known concentration (the titrant). The analyte is generally placed in an Erlenmeyer flask, while the titrant is placed in a burette so that the volume of solution added can be monitored. The titrant is added to the analyte until the endpoint is reached. Calculations are then performed to find the unknown concentration of the analyte. Titrations are typically performed for acid/base reactions but are not limited to them. At equivalence points during the titration, the number of acid or base groups added to the solution is equivalent to the number of base/acid groups in the original unknown solution. We can calculate our unknown concentration or volume using the formula NaVa = NbVb, where N and V are the normality (mol/L) and volume of the acidic and basic solutions, respectively. It is important to convert from molarity (M) to normality (N) for polyprotic acids and polyvalent bases. The flat regions of titration curves represent buffering solutions (a roughly equal mix of an acid/base and its conjugate), while the steep, near-vertical sections of the curve contain equivalence point(s), which indicate that enough of the titrant has been added to completely remove one equivalent (acid or base group) from each of the original molecules in the unknown solution. Species with multiple acid or base groups (e.g., H3PO4 or Ca(OH)2) will have multiple equivalence points during the titration. The final key point of any titration is the endpoint. To be successful, there must be some method for observing the endpoint of the reaction. The type of titration reaction that is being used will determine the method used for observing the endpoint. For example, in an acid-base titration, a specific pH value will be the endpoint (monitored by color-changing indicators), while for precipitation reactions, the endpoint is realized by the appearance of a precipitate. Regardless of the details of the reaction involved, the goal of titrations is always to use known volumes/concentrations to determine unknown volumes/concentrations.
uncompetitive inhibition
Uncompetitive inhibition is when the inhibitor binds to only the enzyme-substrate complex, and inactivates it. This causes the number of active enzyme-substrate complexes to decrease, thereby decreasing Vmax. Km also decreases to exactly the same degree as Vmax. Since both kinetic factors decrease to the same extent, the slope of the Lineweaver-Burk plot of enzyme activity will be the same as for the uninhibited enzyme. The effects of an uncompetitive inhibitor cannot be overcome by increasing substrate concentration. Most commonly, the kinetic parameters Vmax and Km are reported via Michaelis-Menten plots. However, Lineweaver-Burk plots may also be used. When the experimental data are examined in the form of double reciprocal Lineweaver-Burk plots, the x- and y-axes are 1/[S] and 1/Vmax, respectively. Since these plots are a straight line, the slope of each line is Km/Vmax. The y-intercept on a Lineweaver-Burk plot represents 1/Vmax, while the x-intercept is −1/Km. These double-reciprocal plots are especially useful for distinguishing between competitive and noncompetitive inhibitors. Vmax = decreases Km = decreases
ohm's law
V=IR v = voltage I = current (amps ) R = resistance (watts)
vapor pressure reduction
Vapor pressure refers to the pressure of the vapor phase that exists (to some degree) immediately above the surface of any liquid. A higher vapor pressure indicates that a larger number of solvent particles were able to escape the liquid and enter the gas phase. When vapor pressure is equal to the atmospheric pressure exerted on the liquid's surface, the liquid will boil. Knowing this, the fact that adding more solute particles causes a reduction in vapor pressure is actually logical. Imagine the surface of an aqueous solution with a lot of solutes in it. Some of the solutes will be at the surface of the solution, so the space they take up is unavailable for the liquid-gas phase transition that is at the core of vapor pressure. This causes vapor pressure reduction. reduction of vapor pressure causes an increase in boiling point and a decrease in melting point vapor pressure: the vapor above a liquid exerts pressure. In a closed container with liquid in it, some of the liquid will vaporize into vapor and then eventually return to liquid state Vapor pressure is only affected with changes in temperature **For vapor pressure to exist, the vapor (gas phase) MUST be in physical contact with the liquid (or solid) it came from. You CAN'T have vapor pressure without two phases being present and in contact!!!!!!
Work equations
W = F*d = Fdcos(theta) work = force * d (units = N * m) Work - energy theorem: W = change in Kinetic energy KE = PE (law of conservation of energy)
isobaric gas-piston system work
W = P* (change in V) = Pressure * Velocity
Pascal's Principle
When force is applied to a confined fluid, the change in pressure, is transmitted equally to all parts of the fluid
Retention factor (Rf)
a ratio used to characterize and compare components of samples in liquid chromatography high Rf = non-polar compounds move up TLC plate rapidly low Rf = polar compounds
Raoult's law (vapor pressure depression)
a solute is added to a solvent, vapor pressure of solvent decreases proportionally
absolute pressure
absolute pressure = Atmospheric pressure (P0) + gauge pressure (Pg) Pg = ρ g D ρ = density of fluid g = accl due to gravity D = depth
frequency of light
f = c/λ f = frequency (Hz) c = speed of light (3.0 x 10^8 m/s^s) λ = wavelength (m)
Centrifugation
centrifugation utilizes a rapidly spinning apparatus to separate particles by density. More dense particles, such as cells, gravitate toward the bottom of the spun tube, while less dense substances remain at the top in a liquid termed the supernatant. This liquid can then be poured off, and further separation or analysis can be conducted.
1st law of thermodynamics
change in internal energy = heat - work
properties of nonmetals
form anions gain electrons (reduced) oxidizing agents
properties of metals
form cations lose an electron (oxidized) reducing agents
Boiling point elevation (BPE)
i = van't Hoff Factor (number of particles compound dissociates into a solution) Kb = proportionality constant (provided) m = molality (= mols solute/kg solvent)
log rules
log (1) = 0 log (10) = 1 log (.5) = -.3
heat transfer during phase change
nHvap. liquid ----> gas or nHfusion solid ----> liquid units for Hfusion/vap. = kJ/mol n = moles
Charged amino acids
negatively charged: D,E Positively charged: K, R, H
density
p = density m = mass (g) v = volume (mL)
pH
pH = -log [H+]
pKa
pKa = -logKa strong acids have low pKa and high Ka pH < pKa, then the compound = protonated. pH > pKa, then the compound = deprotonated.
weak acids and bases
partially dissociate in solution weak base and acids have Kb and Ka <1 have weak conjugates
colligative properties
properties that depend on the concentration of solute particles but not on their identity Vapor pressure depression, boiling point elevation, freezing point depression, osmotic pressure
Heat transfer (during no phase change)
q=mC(change)T q = heat in joules m = mass g C = specific heat of substance (for H20 = 1 Cal/g*K) change T
Acid dissociation constant
smaller Ka = weaker acid, less likely it will dissociate
Gibbs free energy
standard state delta G = -RTlnKeq R = ideal gas constant T = temperature in Kelvins Keq = equilibrium constant ( as this increase, it makes delta G more negative and is more spontaneous) standard state is used for these calculations where: T = 293 K P = 1 atm n = 1 M
frequency shift
the difference between the transmitted and received frequencies upward frequency shift for observers where the sound source is approaching them Doppler effect
elastic potential energy
u = 1/2 kx^2
gravitational potential energy
u = mgh u = potential energy m = mass ( kg) g = acceleration due to gravity (9.8 m/s^2) h = height (m)
definition of power equation
units = kg * m^2/s^2
Power
units: watt, J/s = ft•lb/s = kg•m2/s3. 1 watt = 1 J/s P = Fv Power = Force * velocity P = W/t Power = work/time P = I*V (Power = current x voltage) P = I^2R P = V^2/R. units = watts P = power I = current V = voltage V = IR
speed of wavelength
v = fλ velocity of sound f = frequency (Hz) λ = wavelength (m)
work (one dimensional) - physics
w = F x d work = Force x distance units: W = Joules or N*m F = N (Newtons) d = meters (m)
Henderson-Hasselbalch equation Buffer solutions (weak acid/weak base buffer solutions)
when [conjugate base] = [base] then pH = pKa, this occurs at the half equivalence point and buffering capacity is optimal at this pH
torque
τ = F∙d∙sin(θ) d = distance that the force is applied from the fulcrum F = force applied θ is the angle between the lever arm and the force that is applied. Torque (τ) is the rotational analog of force. Specifically, torque is caused by force applied to a lever arm at a certain distance from an object capable of rotating, known as a fulcrum. Thus, there are three ways to increase the torque applied to an object: (1) increasing the force, (2) increasing the distance at which the force is applied from the fulcrum, (3) adjusting the angle at which the force is applied to make it as close as possible to perpendicular to the lever arm. In equilibrium setups where objects are not moving, it may still be necessary to account for torque, if the potential for rotational motion exists. In such cases, the clockwise and counterclockwise torques will balance out (τCW = −τCCW).
