Question G.1: For the binary system diethylamine(1)/n-heptane(2) at 308.15...
For the binary system diethylamine(1)/n-heptane(2) at 308.15 K, find γ_1 and γ_2 when x_1 = 0.4 and x_2 = 0.6.
The subgroups involved are indicated by the chemical formulas:
\mathrm{CH}_3-\mathrm{CH}_2 \mathrm{NH}-\mathrm{CH}_2-\mathrm{CH}_3(1) / \mathrm{CH}_3-\left(\mathrm{CH}_2\right)_5-\mathrm{CH}_3 \text { (2) }
The following table shows the subgroups, their identification numbers k, values of parameters R_k and Q_k (from Table G.1), and the numbers of each subgroup in each molecule:
| k | R_k | Q_k | v_k^{(1)} | v_k^{(2)} | |
| CH_3 | 1 | 0.9011 | 0.848 | 2 | 2 |
| CH_2 | 2 | 0.6744 | 0.54 | 1 | 5 |
| CH_2NH | 33 | 1.207 | 0.936 | 1 | 0 |
By Eq. (G.15),
r_i=\sum_k v_k^{(i)} R_k (G.15)
r_1=(2)(0.9011)+(1)(0.6744)+(1)(1.2070)=3.6836
Similarly,
r_2=(2)(0.9011)+(5)(0.6744)=5.1742
In like manner, by Eq. (G.16),
q_i=\sum_k v_k^{(i)} Q_k (G.16)
q_1=3.1720 \quad \text { and } \quad q_2=4.3960
The r_i and q-i values are molecular properties, independent of composition. Substituting known values into Eq. (G.17) generates the following table for e_{ki} :
e_{k i}=\frac{v_k^{(i)} Q_k}{q_i} (G.17)
| e_{ki} | ||
| k | i = 1 | i = 2 |
| 1 | 0.5347 | 0.3858 |
| 2 | 0.1702 | 0.6142 |
| 33 | 0.2951 | 0.000 |
The following interaction parameters are found from Table G.2:
a_{1,1}=a_{1,2}=a_{2,1}=a_{2,2}=a_{33,33}=0 \mathrm{~K}
a_{1,33}=a_{2,33}=255.7 \mathrm{~K}
a_{33.1}=a_{33.2}=65.33 \mathrm{~K}
Substitution of these values into Eq. (G.21) with T = 308.15 K gives
\tau_{m k}=\exp \frac{-a_{m k}}{T} (G.21)
\tau_{1,1}=\tau_{1,2}=\tau_{2,1}=\tau_{2,2}=\tau_{33,33}=1
\tau_{1,33}=\tau_{2,33}=0.4361
\tau_{33,1}=\tau_{33,2}=0.8090
Application of Eq. (G.18) leads to the values of β_{ik} in the following table:
\beta_{i k}=\sum_m e_{m i} \tau_{m k} (G.18)
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β_{ik}ik = 1k = 2k = 3310.94360.94360.60242110.436
Substitution of these results into Eq. (G.19) yields:
\theta_k=\frac{\sum_i x_i q_i e_{k i}}{\sum_j x_j q_j} (G.19)
\theta_1=0.4342 \quad \theta_2=0.4700 \quad \theta_{33}=0.0958and by Eq. (G.20),
s_k=\sum_m \theta_m \tau_{m k} (G.20)
s_1=0.9817 \quad s_2=0.9817 \quad s_{33}=0.4901
The activity coefficients can now be calculated. By Eq. (G.13),
\ln \gamma_i^C=1-J_i+\ln J_i-5 q_i\left(1-\frac{J_i}{L_i}+\ln \frac{J_i}{L_i}\right) (G.13)
\ln \gamma_1^C=-0.0213 \quad \text { and } \quad \ln \gamma_2^C=-0.0076
and by Eq. (G.14),
\ln \gamma_i^R=q_i\left[1-\sum_k\left(\theta_k \frac{\beta_{i k}}{s_k}-e_{k i} \ln \frac{\beta_{i k}}{s_k}\right)\right] (G.14)
\ln \gamma_1^R=0.1463 \quad \text { and } \quad \ln \gamma_2^R=0.0537
Finally, Eq. (G.7) gives:
\gamma_1=1.133 \quad \text { and } \quad \gamma_2=1.047