JEE Chemistry Thermodynamics Notes Set 06

Basic Terms

System is the part of the universe under study; everything else is the surroundings. An open system exchanges matter and energy, a closed system exchanges only energy, and an isolated system exchanges neither.
State functions (P, V, T, E, H, S, G) depend only on the present state; path functions (heat q and work w) depend on how the change is carried out. Extensive properties (mass, volume, E, H, S, G) depend on the amount of matter; intensive properties (T, P, density, molar quantities) do not.
Process types: isothermal (T constant), isobaric (P constant), isochoric (V constant), adiabatic (q = 0), cyclic (initial = final state, so all state functions change by zero).

First Law of Thermodynamics

Energy can be neither created nor destroyed: \( \Delta E = q + w \), where w is work done on the system. With work done by the system written as w, \( \Delta E = q - w \).
Expansion work against constant external pressure: \( w = -P\Delta V \). Free expansion into vacuum: w = 0. Reversible isothermal work: \( w = -2.303\,nRT\log\dfrac{V_2}{V_1} \).
Isothermal ideal gas: \( \Delta E = 0 \) so q = -w. Adiabatic: q = 0 so \( \Delta E = w \). Isochoric: \( \Delta E = q_v \).

Enthalpy and Heat Capacity

\( H = E + PV \). At constant pressure \( q_p = \Delta H \); at constant volume \( q_v = \Delta E \). For reactions involving gases: \( \Delta H = \Delta E + \Delta n_g RT \), where \( \Delta n_g \) is moles of gaseous products minus moles of gaseous reactants.
\( C_p - C_v = R \) for one mole of an ideal gas. \( C_v = \tfrac32R \) for a monatomic gas and \( \tfrac52R \) for a diatomic gas (ignoring vibration). For an adiabatic reversible change \( TV^{\gamma-1} \) is constant, with \( \gamma = C_p/C_v \).

Thermochemistry

Exothermic reactions have \( \Delta H < 0 \); endothermic reactions have \( \Delta H > 0 \). Standard state: pure substance at 1 bar (about 1 atm) and the stated temperature, usually 298 K; the enthalpy of formation of an element in its standard state is zero.
Types: enthalpy of formation, combustion, neutralisation (about -57.3 kJ/mol for a strong acid and strong base; less negative for weak acids or bases because some energy is used in ionisation), fusion, vaporisation, sublimation, atomisation and bond dissociation.
Hess's law: the total enthalpy change of a reaction is the same whether it takes place in one step or several. \( \Delta H_{reaction} = \sum \Delta H_f(products) - \sum \Delta H_f(reactants) \). With bond energies: \( \Delta H = \sum (bonds\ broken) - \sum (bonds\ formed) \).
Kirchhoff's equation: \( \Delta H_2 - \Delta H_1 = \Delta C_p\,(T_2 - T_1) \).

Second and Third Laws, Entropy

Entropy S measures the randomness of a system. For a reversible process \( \Delta S = q_{rev}/T \). At a phase change \( \Delta S = \Delta H/T \). The entropy of a gas is much greater than that of a liquid, which is greater than that of a solid.
Second law: in a spontaneous process the total entropy of the system plus surroundings increases, \( \Delta S_{total} > 0 \); the entropy of the universe tends to a maximum. Third law: the entropy of a perfectly crystalline substance is zero at absolute zero.
Efficiency of a Carnot engine: \( \eta = 1 - T_2/T_1 = (T_1 - T_2)/T_1 \).

Gibbs Free Energy

\( \Delta G = \Delta H - T\Delta S \). A process is spontaneous if \( \Delta G < 0 \), at equilibrium if \( \Delta G = 0 \), and non-spontaneous if \( \Delta G > 0 \).
Cases: \( \Delta H<0, \Delta S>0 \) spontaneous at all temperatures; \( \Delta H>0, \Delta S<0 \) never spontaneous; \( \Delta H<0, \Delta S<0 \) spontaneous at low T; \( \Delta H>0, \Delta S>0 \) spontaneous at high T. The changeover temperature is \( T = \Delta H/\Delta S \).
\( \Delta G^\circ = -RT\ln K = -2.303\,RT\log K \). Maximum non-expansion (useful) work equals \( -\Delta G \).

Worked Examples

Example 1. Find \( \Delta E \) for \( 2NH_3(g)\rightarrow N_2(g)+3H_2(g) \) at 300 K if \( \Delta H = 92.4 \) kJ.
Answer:
\( \Delta n_g = 4-2 = 2 \).
\( \Delta E = \Delta H - \Delta n_gRT = 92.4 - 2(8.314\times10^{-3})(300) = 87.4 \) kJ.

Example 2. Above what temperature is a reaction with \( \Delta H = +30.56 \) kJ and \( \Delta S = 66 \) J/K spontaneous?
Answer:
\( T = \Delta H/\Delta S = 30560/66 = 463 \) K.
Above 463 K, \( \Delta G \) is negative.