Energetics & Equilibria 6: Main recall

Question 1

Expression for weight, W

Answer 1

W = N! / n₀! x n₁! x n₂! x … x n_n!

Question 2

formula for Boltzmann distribution

Answer 2

n_i = n₀e^{−εi /kT}

• n_i is the number of molecules in energy level i
• ε_i is the energy of i
• n₀ is the population of the ground level, which has energy 0
• k is Boltzmann’s constant (k = R/NA, where NA is Avogadro’s constant)

Verbally, as the energy of a level increases, its population decreases. Levels with energies less than or similar to kT have significant populations. So ground level is most populated

Question 3

Entropy is related to the number of ways of achieving a configuration. Give its formula in terms of this.

Answer 3

S = k lnW

Question 4

An object at temperature T reversibly absorbs a small amount of heat. Give the formula for the entropy change.

Answer 4

dS = δq_rev/T

Question 5

Give the formula for the entropy change of the universe

Answer 5

∆S_univ = ∆S_sys + ∆S_surr

= ∆S_sys - q_sys/T_sys

Question 6

equation for the first law of thermodynamics

Answer 6

∆U = q + w

q, heat absorbed by system

w, work done on system

Question 7

ideal gas equation and units used?

Answer 7

pV = nRT

• P in Pa (Nm^-1)
• V in m³
• n in mol
• T in K
• R in J K-1 mol-1

Question 8

A gas is confined inside a cylinder by a piston. Give the expression for the work done by the gas when the piston moves.

Answer 8

δw’ = p_extdV

work done by gas = external pressure x change in volume of gas

Question 9

Integrated relationship for work done by a gas, when the gas expands from volume vi to vf, against constant external pressure.

Answer 9

w’ = p_ext(V_f - V_i)

Question 10

Integrated relationship for work done by an ideal gas, when the gas expands isothermally from volume vi to vi.

Answer 10

w’ = nRTln(V_f/V_i)

Question 11

What expression gives the heat absorbed in a reversible, isothermal expansion of an ideal gas, from volume Vi to Vf?

Answer 11

q_rev = nRTln(V_f/V_i)

isothermal and ideal so w’ = q

Question 12

What expression gives the entropy change of the gas in a reversible, isothermal expansion of an ideal gas, from volume Vi to Vf?

Answer 12

∆S = nRln(V_f/V_i)

Question 13

What is the heat equal to under:

• constant volume conditions?
• constant pressure conditions?

Answer 13

• volume: ∆U, change in internal energy
• pressure: ∆H, enthalpy change

Question 14

Give the definition of the

• constant volume molar heat capacity
• constant pressure molar heat capacity

Answer 14

C_V,m = (∂U_m/∂T)_V

C_p,m = (∂H_m/∂T)_p

Question 15

State the definition of enthalpy (eq)

Answer 15

H = U + pV

Question 16

The molar enthalpy of a substance is known at temperature T1. Give the expression for the molar enthalpy at temperature T2.

Answer 16

H_m(T₂) = H_m(T₁) + C_p,m(T₂ - T₁)

Question 17

How are the molar entropy and enthalpy associated with a phase change related?

Answer 17

ΔS_m,pc = ΔH_m,pc/T_pc

Question 18

Give the relationship for converting molar entropies from one temperature to another.

Answer 18

S_m(T₂) = S_m(T₁) + C_p,mln(T₂/T₁)

Question 19

Definition of Gibbs energy, G (eq)

Answer 19

G = H - TS

ΔG = ΔH - TΔS

Question 20

Give the first, second and third master equations

Answer 20

dU = TdS - pdV

dH = TdS + Vdp

dG = VdP - SdT

Question 21

Give the expression for how molar Gibbs energy varies with pressure, at constant temp, for an ideal gas.

Answer 21

G_m(p) = G^o_m + RT ln(p/p^o) (const. temp, ideal gas)

Question 22

Give the expression for how Gibbs energy varies with volume, at constant temp, for an ideal gas.

Answer 22

G(V₂) = G(V₁) + nRTln(V₁/V₂) (const. temp, ideal gas)

Question 23

Give the expression for how molar Gibbs energy varies with temperature, at constant pressure, for an ideal gas.

Answer 23

Gibbs-Helmholtz equation:

d/dT(G/T) = -H/T² (const. pressure)

Question 24

Give the expressions for partial pressure and mole fraction.

Answer 24

Partial pressure: p_i = x_ip_tot

Mole fraction: x_i = n_i/n_tot

Question 25

Give the expression for how molar Gibbs energy varies with pressure, at constant temp, for an ideal gas in a mixture.

Answer 25

G_m,i(p_i) = G^o_m,i + RT ln(p_i/p^o) (const. temp, ideal gas)

Ideal gases don’t interact, so there is no distinction between a pure gas at pressure p, and that same gas at partial pressure pi in a mixture

Question 26

Give the formula for the Gibbs energy of

• a mixture of ideal gases
• a mixture of solids/liquids/solutions

Answer 26

Ideal gases don’t interact, so just add the molar Gibbs energy of each species, each multiplied by its mole fraction, each at its own partial pressure:

• G = n_AG_m,A(p_A) + n_BG_m,B(p_B​) + …
• G = n_Aμ_A + n_Bμ_B + …

Question 27

Give the expression for how chemical potential of

• gaseous species i varies with pressure, at constant temp, in a mixture
• ideal, dissolved species i varies with pressure, at constant temp, in a mixture

Answer 27

• μ(p_i) = μ^o_i + RT ln(p_i/p^o)
• μ(c_i) = μ^o_i + RT ln(c_i/c^o)

• Both analagous to:*
• G_m,i(p_i) = G^o_m,i + RT ln(p_i/p^o) (const. temp, ideal gas)*

Question 28

Give the expression for the standard Gibbs energy of a reaction.

Answer 28

ΔrG^o = ΔrH^o - TΔrS^o

Question 29

• Δ_rH^o varies with temperature. Give the relationship giving Δ_rH^o at temperature 2, given knowledge of Δ_rH^o at temperature 1.
• Δ_rS^o varies with temperature. State the relationship giving Δ_rS^o at temperature 2, given knowledge of Δ_rS^o at temperature 1.

Answer 29

Δ_rH^o(T2) = Δ_rH^o(T1) + Δ_rC_p^o[T2 - T1]

Δ_rS^o(T2) = Δ_rS^o(T1) + Δ_rC_p^oln(T2/T1)

Question 30

What is true of Gibbs energy at equilibrium?

Answer 30

∆rG = 0

Question 31

Give the expression linking the standard Gibbs energy of reaction and the equilibrium constant.

Answer 31

∆rG^o = -RTlnK

Question 32

Give the van’t Hoff equation.

Answer 32

dlnK/dT = ΔrH^o/RT²

Question 33

Write the equation showing the temperature-dependence of the equilibrium constant

Answer 33

lnK = -(ΔrH^o/R)(1/T) + (ΔrS^o/R)

Question 34

How is the cell potential calculated from the half-cells?

Answer 34

E_cell = E = E_1/2(RHS) - E_1/2(LHS)

E^o = E^o_1/2(RHS) - E^o_1/2(LHS)

Question 35

Give the equation relating cell potential (of the conventional cell reaction) to Gibbs energy.

Answer 35

ΔrG_cell = -nFE (constant temp and pressure)

Question 36

Give the expression for:

• the entropy change of a cell reaction
• the enthalpy change of a cell reaction

Answer 36

ΔrS = nF(∂E)/∂T)_p

ΔrH_cell = TnF(∂E)/∂T)_p - nFE

Question 37

Write the expression linking activity to chemical potential.

Answer 37

µ_i(a_i) = µi^o + RTln(a_i)

Question 38

Give the general form of the Nernst equation.

Answer 38

E = E^o - (RT/nF) ln(products/reactants)

For solids/liquids, chemical potential = standard chemical potential, so E = E^o