## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### Appendix F. Numerical Answers to Selected Problems

3.11 Ar: 129 K, HBr: 467 K.

3.16 Taking *α*_{1}/*α*_{2} = 2, we obtain the following results rendering the Berthelot approximation doubtful.

3.17 *U*_{coh} = 14.3 kJ/mol and *H*_{coh} = *U*_{coh} + *RT* = 16.8 kJ/mol and

*U*_{coh} = 28.9 kJ/mol and *H*_{coh} = *U*_{coh} + *RT* = 31.4 kJ/mol.

3.18 The relative contribution is (11/32)*α*/*σ*^{3} = 0.011.

5.3 *Z* = 1 + exp(−*ε*/*kT*), *p*_{1} = 0.73, *p*_{2} = 0.27, *U* = *ε*/[exp(*ε*/*kT*) + 1],

*C _{V}* = [exp(

*ε*/

*kT*) + 1]

^{−2}(

*ε*

^{2}/

*kT*

^{2})exp(

*ε*/

*kT*).

5.11 *Λ*(He) ≅ 4.3 Å with *ρ*^{−1/3} ≅ 3.7 Å → classical approximation not valid;

*Λ*(Ar) ≅ 0.30 Å with *ρ*^{−1/3} ≅ 4.0 Å → classical approximation valid.

5.21 Assuming deviations less than 5% are allowed, one has 1 + *θ*/3*T* < 1.05 or *T* > 7*θ*.

6.1 *ρ*_{rel} = 0.984·*ρ*_{glass}/*ρ*_{FCC} = 0.984·0.637/0.741 = 0.85. Fig. 6.3 provides 0.9 < *V** < 1.1, hence 0.77 < *ρ*_{rel} < 0.95.

6.3

6.4 *N*^{(1)} = 9.2, *N*^{(2)} = 44.4.

6.6 a) Nearest neighbour Li-Cl ≅ 2.0 Å, next nearest neighbour Li-Cl ≅ 5.3 Å

b) Nearest neighbour Li-Li ≅ Cl-Cl ≅ 3.7 Å.

7.5 *PV*/*KT* = 1 + Σ_{n=1}(*n*^{2} + 3*n*)*η ^{n}*.

7.6 *v* = 1.70; ; ;

8.2 a) *v*_{L}/*v*_{G} = *a*/(*a* − *d*)

b) *V*_{f} = (4π/3)(*a* − *d*)^{3} = (4π*a*^{3}/3)(*v*_{G}/*v*_{L})^{3} = 0.008 (4π*a*^{3}/3)

10.3 H_{2}O: *ε*_{r} = 78.5, CH_{3}OH: *ε*_{r} = 31.6, C_{2}H_{5}OH: *ε*_{r} = 24.3.

10.5 *α*′ = 10.3 Å^{3}, *μ* = 1.57 D.

10.7 *α*′ = 6.4 Å^{3}, *μ* = 2.8 D.

10.10 *α*′ = 13.8 × 10^{−30} m^{3}/mol, *μ* = 0.34 D using the Debye equation. The data point at −110 °C is also at the line for the liquid, indicating that the molecule still rotates in the solid phase. However, *α*′ is large and *μ* is small as compared to independent data (*α*′ = 3.23 × 10^{−30} m^{3}/mol, *μ* = 1.7 D) illustrating the effect of hydrogen bonding.

10.11 *μ**/*μ* = 1.26.

11.5 *V*_{1} = *A* + *B* + *C*[*x*_{1}(2 − *x*_{1})], .

11.6 .

11.10 a) Δ*H* = *zwx*_{1}(1 − *x*_{1}) = 800 J/mol at *x*_{1} = 0.5, hence *zw* = 3200 and *w* = 320 J/mol.

b) or *γ*_{2} = 1.083.

11.12 Since the expressions are symmetric in *x*_{1} and *x*_{2}, yes.

12.2 Δ_{sol}*H* = 4 kJ mol^{−1} > 0, hence Δ*T* > 0. *T*Δ*S* > 4 kJ mol^{−1} or Δ*S* > 13.3 J K^{−1} mol^{−1}.

12.3 .

12.9 Assuming the same structure as in Fig. 12.4, the angle is 40°.

12.11 Δ*H* = *A* − *DT*^{2}, Δ*S* = *C*−2*DT*, Δ*C _{P}* = −2

*DT*.

12.13 *κ*^{−1} ≅ 1.8 nm.

12.17 *t*_{H} = 3 · 10^{−7} s.

12.18 a) Δ*T* = 0.31 °C; b) *Λ*_{m} = 0.0273 Ω^{−1} mol^{−1} m^{2}; c) *I* = 14 mA; d) *I* = 20 mA using *η*(H_{2}O) = 1 · 10^{−3} Pa s, *a*(Ca^{2+}) = 1.0 Å and *a*(I^{−}) = 2.15 Å; e) *t*(Ca^{2+}) = 0.44; f) 2.4 K/hour.

13.1

A: |
B: |

C: |
D: |

13.9 *δ* = 18.2 MPa^{1/2}.

13.10 *δ* = 18.3 MPa^{1/2}.

13.11 *ϕ*_{1} = 0.10, *ϕ*_{2} = 0.41, *ϕ*_{3} = 0.49.

15.1 *u*^{(σ)} = 26.1 mJ m^{−2}.

15.8 *γ* = 0.0264 N/m. The reasonable good agreement for this calculation of cyclohexane with the experimental value 0.0247 N/m is probably fortuitous.