Physical Chemistry 2 Reviewer PDF

Physical Chemistry 2 Reviewer I. The Perfect Gas 1. The physical state of a sample of a substance, is its form (solid, liquid, or gas) under the current conditions of pressure, volume, and temperature. 2. Mechanical equilibrium is the condition of equality of pressure on eit...

Physical Chemistry 2 Reviewer I. The Perfect Gas 1. The physical state of a sample of a substance, is its form (solid, liquid, or gas) under the current conditions of pressure, volume, and temperature. 2. Mechanical equilibrium is the condition of equality of pressure on either side of a shared movable wall. 3. The thermodynamic and Celsius temperatures are related by the exact expression 4. An equation of state is an equation that interrelates the variables that define the state of a substance. 5. Boyles and Charles’s laws are examples of a limiting law, a law that is strictly true only in a certain limit, in this case p → 0. 6. An isotherm is a line in a graph that corresponds to a single temperature. 7. An isobar is a line in a graph that corresponds to a single pressure. 8. An isochore is a line in a graph that corresponds to a single volume. 9. A perfect gas is a gas that obeys the perfect gas law under all conditions. 10. The expression is the perfect gas law (or perfect gas equation of state). 11. The partial pressure, pJ, of a gas J in a mixture (any gas, not just a perfect gas), is defined as where xJ is the mole fraction of the component J, the amount of J expressed as a fraction of the total amount of molecules, n, in the sample:. 12. Dalton’s law states that the pressure exerted by a mixture of (perfect) gases is the sum of the pressures that each one would exert if it occupied the container alone. II. Real Gases 1. The extent of deviations from perfect behavior is summarized by introducing the compression factor. 2. The compression factor, Z, the ratio of the measured molar volume of a gas, Vm = V/n, to the molar volume of a perfect gas, Vom, at the same pressure and temperature: 3. The virial equation is an empirical extension of the perfect gas equation that summarizes the behavior of real gases over a range of conditions. A more convenient expansion for many applications is 4. The isotherms of a real gas introduce the concept of critical behavior. 5. A gas can be liquefied by pressure alone only if its temperature is at or below its critical temperature. 6. The van der Waals equation is a model equation of state for a real gas expressed in terms of two parameters, one (a) representing molecular attractions and the other (b) representing molecular repulsions. 7. The combined effect of the repulsive and attractive forces gives the van der Waals equation: 8. The van der Waals equation captures the general features of the behavior of real gases, including their critical behavior.  9. The properties of real gases are coordinated by expressing their equations of state in terms of reduced variables. 10. The critical constants are characteristic properties of gases, so it may be that a scale can be set up by using them as yardsticks and to introduce the dimensionless reduced variables of a gas by dividing the actual variable by the corresponding critical constant: III. Internal Energy 1. Work is the process of achieving motion against an opposing force. 2. Energy is the capacity to do work. 3. Heat is the process of transferring energy because of a temperature difference. 4. An exothermic process is a process that releases energy as heat. 5. An endothermic process is a process in which energy is acquired as heat. 6. In molecular terms, work is the transfer of energy that makes use of organized motion of atoms in the surroundings and heat is the transfer of energy that makes use of their disorderly motion. 7. Internal energy, the total energy of a system, is a state function. 8. The internal energy increases as the temperature is raised. 9. The First Law states that the internal energy of an isolated system is constant. 10. If w is the work done on a system, q is the energy transferred as heat to a system, and ΔU is the resulting change in internal energy, then 11. Free expansion (expansion against zero pressure) does no work. 12. The work done when the system expands by dV against a pressure pex is 13. If the change in volume is written as 𝜟V = Vf - Vi, 14. A reversible change is a change that can be reversed by an infinitesimal change in a variable. 15. To achieve reversible expansion, the external pressure is matched at every stage to the pressure of the system. 16. The work of isothermal reversible expansion of a perfect gas from Vi to Vf at a temperature T is 17. The energy transferred as heat at constant volume is equal to the change in internal energy of the system. 18. For a measurable change between states i and f along a path at constant volume, 19. Calorimetry is the measurement of heat transactions. 20. If a constant current, I, from a source of known potential difference, 𝜟𝝓, passes through a heater for a known period, t, the energy supplied as heat is 21. The heat capacity at constant volume is denoted CV and is defined formally as IV. Enthalpy 1. Energy transferred as heat at constant pressure is equal to the change in enthalpy of a system. 2. The enthalpy, H, is defined as 3. The constraint of constant pressure is denoted by a subscript p, so the equation can be written This equation states that, provided there is no additional (no expansion) work done, and the system and surroundings are in mechanical equilibrium, the change in enthalpy is equal to the energy supplied as heat at constant pressure. The result is 4. Enthalpy changes can be measured in an isobaric calorimeter. 5. The change of enthalpy in a reaction that produces or consumes gas under isothermal conditions is 6. The heat capacity at constant pressure (the isobaric heat capacity) is equal to the slope of enthalpy with temperature. Using the partial derivative notation: 7. There is a simple relation between the two heat capacities of a perfect gas: V. Thermochemistry 1. The standard enthalpy of transition is equal to the energy transferred as heat at constant pressure in the transition under standard conditions. 2. The standard state of a substance at a specified temperature is its pure form at 1 bar. 3. A thermochemical equation is a chemical equation and its associated change in enthalpy. 4. Hess’s law states that the standard reaction enthalpy is the sum of the values for the individual reactions into which the overall reaction may be divided. 5. Standard enthalpies of formation are defined in terms of the reference states of elements. 6. The reference state of an element is its most stable state at the specified temperature and 1 bar. 7. The standard reaction enthalpy is expressed as the difference of the standard enthalpies of formation of products and reactants. 8. In the enthalpies of formation of substances, there is enough information to calculate the enthalpy of any reaction by using A more sophisticated way of expressing the same result is to introduce the stoichiometric numbers 𝝂J (as distinct from the stoichiometric coefficients) which are positive for products and negative for reactants. Then 9. The temperature dependence of a reaction enthalpy is expressed by Kirchhoff’s law. 10. The standard reaction enthalpy changes from ΔrH⦵(T1) to 11. Where ΔrCp𝜽 is the difference of the molar heat capacities of products and reactants under standard conditions weighted by the stoichiometric coefficients that appear in the chemical equation:  12. If ΔrCp𝜽 is largely independent of temperature in the range T1 to T2, the integral in Kirchoff’s law evaluates to (T2 – T1) 𝜟rCp𝜽 and that equation becomes VI. Entropy 1. The entropy is a signpost of spontaneous change: the entropy of the universe increases in a spontaneous process. 2. The thermodynamic definition of entropy is based on the expression where qrev is the energy transferred as heat reversibly to the system at the absolute temperature T. 3. A change in entropy is defined in terms of reversible heat transactions. 4. For a measurable change 5. The Boltzmann formula defines entropy in terms of the number of ways that the molecules can be arranged amongst the energy states, subject to the arrangements having the same overall energy. 6. Boltzmann proposed that there is a link between the spread of molecules over the available energy states and the entropy, which he expressed as where k is Boltzmann’s constant (k = 1.381 × 10−23 J/K) and W is the number of microstates, the number of ways in which the molecules of a system can be distributed over the energy states for a specified total energy. 7. The Carnot cycle is used to prove that entropy is a state function. 8. It then follows from |qc|/|qh| = Tc/Th that 9. The efficiency of a heat engine is the basis of the definition of the thermodynamic temperature scale and one realization of such a scale, the Kelvin scale. 10. An engine working reversibly between a hot source at a temperature Th and a cold sink at a temperature T, then it follows that 11. The Clausius inequality is used to show that the entropy of an isolated system increases in a spontaneous change and therefore that the Clausius definition is consistent with the Second Law. 12. From the thermodynamic definition of the entropy (dS = dqrev/T) it then follows that 13. Spontaneous processes are irreversible processes; processes accompanied by no change in entropy are at equilibrium. VII. Entropy changes accompanying specific processes 1. The entropy of a perfect gas increases when it expands isothermally. 2. It is established that the change in entropy of a perfect gas when it expands isothermally from Vi to Vf is 3. The change in entropy of a substance accompanying a change of state at its transition temperature is calculated from its enthalpy of transition. 4. Because at constant pressure q = ΔtrsH, the change in molar entropy of the system is 5. The increase in entropy when a substance is heated is calculated from its heat capacity.  6. When Cp is independent of temperature over the temperature range of interest, it can be taken outside the integral to give with a similar expression for heating at constant volume. VIII. The measurement of entropy 1. Entropies are determined calorimetrically by measuring the heat capacity of a substance from low temperatures up to the temperature of interest and considering any phase transitions in that range. 2. If a substance melts at Tf and boils at Tb, then its molar entropy at a particular temperature T above its boiling temperature is given by 3. The Debye extrapolation (or the Debye T3-law) is used to estimate heat capacities of non-metallic solids close to T = 0. 4. The Nernst heat theorem states that the entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero provided all the substances involved are perfectly ordered. 5. The Third Law of thermodynamics states that the entropy of every perfect crystalline substance is zero at T = 0. 6. The residual entropy of a solid is the entropy arising from disorder that persists at T = 0. 7. Third-Law entropies are entropies based on S(0) = 0. 8. The standard reaction entropy, ΔrS⦵, is defined, like the standard reaction enthalpy, as the difference between the molar entropies of the pure, separated products and the pure, separated reactants, all substances being in their standard states at the specified temperature: A more sophisticated approach is to adopt the notation and to write where the nJ are signed (+ for products, − for reactants) stoichiometric numbers. 9. The standard entropies of ions in solution are based on setting S⦵(H+, aq) = 0 at all temperatures. 10. The standard reaction entropy, ΔrS⦵, is the difference between the molar entropies of the pure, separated products and the pure, separated reactants, all substances being in their standard states. 11. The temperature dependence of the standard reaction entropy, ΔrS⦵, is where ΔrCp𝜽 (T) is the difference of the molar heat capacities of products and reactants under standard conditions at the temperature T weighted by the stoichiometric numbers that appear in the chemical equation. 12. If ΔrCp𝜽 (T) is independent of temperature in the range T1 to T2, the integral (equation in 11) evaluates to ΔrCp𝜃 ln(T2/T1) and IX. Concentrating on the system 1. The Clausius inequality implies several criteria for spontaneous change under a variety of conditions which may be expressed in terms of the properties of the system alone; they are summarized by introducing the Helmholtz and Gibbs energies. 2. This expression states that in a system at constant volume and constant internal energy (such as an isolated system), the entropy increases in a spontaneous change (Second law). 3. That is, in a spontaneous process the entropy of the system at constant pressure must increase if its enthalpy remains constant (under these circumstances there can then be no change in entropy of the surroundings). 4. The Helmholtz energy, A, is defined as 5. A spontaneous process at constant temperature and volume is accompanied by a decrease in the Helmholtz energy. 6. The change in the Helmholtz energy is equal to the maximum work obtainable from a system at constant temperature. 7. Because at constant temperature dA = dU − TdS, it follows that 8. The Gibbs energy, G, defined as 9. A spontaneous process at constant temperature and pressure is accompanied by a decrease in the Gibbs energy. 10. The change in the Gibbs energy is equal to the maximum non-expansion work obtainable from a system at constant temperature and pressure. 11. Because the process is reversible, the work done must have its maximum value, so it follows that 12. Standard Gibbs energies of formation are used to calculate the standard Gibbs energies of reactions. 13. Using the subscript notation to indicate which properties are held constant, the criteria of spontaneous change and equilibrium in terms of dA and dG are 14. Standard enthalpies and entropies of reaction can be combined to obtain the standard Gibbs energy of reaction (or ‘standard reaction Gibbs energy’), ΔrG^⦵: 15. The standard Gibbs energy of a reaction is found by taking the appropriate combination, in the notation introduced where the nJ are the (signed) stoichiometric numbers in the chemical equation. 16. The standard Gibbs energy of formation of H+(aq) is set equal to zero at all temperatures: 17. The standard Gibbs energies of formation of ions may be estimated from a thermodynamic cycle and the Born equation. 18. The molar energy as work is obtained by multiplying by NA, Avogadro’s constant, and the entire expression equated to the standard Gibbs energy of solvation. The result is the Born equation:  19. The chemical potential of a system that consists of a single substance is equal to the molar Gibbs energy of that substance. X. Combining the First and Second Laws 1. From the definition of the entropy change, dS = dqrev / T, it follows that dqrev =TdS and hence the First Law can be written 2. The fundamental equation, a combination of the First and Second Laws, is an expression for the change in internal energy that accompanies changes in the volume and entropy of a system. 3. Relations between thermodynamic properties are generated by combining thermodynamic and mathematical expressions for changes in their values. 4. The Maxwell relations are a series of relations between partial derivatives of thermodynamic properties based on criteria for changes in the properties being exact differentials. 5. The Maxwell relations are used to derive the thermodynamic equation of state and to determine how the internal energy of a substance varies with volume. 6. The variation of the Gibbs energy of a system suggests that it is best regarded as a function of pressure and temperature. 7. For a closed system doing no non-expansion work and at constant composition 8. The Gibbs energy of a substance decreases with temperature and increases with pressure. 9. When applied to the exact differential dG = Vdp − SdT, now gives 10. Alternatively, the above two relations can be expressed in terms of the molar quantities 11. The variation of Gibbs energy with temperature is related to the enthalpy by the Gibbs–Helmholtz equation. 12. The Gibbs energies of solids and liquids are almost independent of pressure; those of gases vary linearly with the logarithm of the pressure. 13. For a perfect gas, substitute Vm = RT/p into the integral, note that T is constant, and find 14. If the final pressure is written as p it therefore follows that 15. The chemical potential, μ, is introduced as another name for the molar Gibbs energy of a pure substance. Expressed in terms of μ, this relation becomes