For the binary system diethylamine(1)/n-heptane(2) at 308.15 K, find γ1 and γ2 when x1 = 0.4 and x2 = 0.6.

Question

For the binary system diethylamine(1)/n-heptane(2) at 308.15 K, find [latex] γ_1 and γ_2 when x_1 = 0.4 and x_2 = 0.6.[/latex]

Accepted Answer

The subgroups involved are indicated by the chemical formulas:

[latex] \mathrm{CH}_3-\mathrm{CH}_2 \mathrm{NH}-\mathrm{CH}_2-\mathrm{CH}_3(1) / \mathrm{CH}_3-\left(\mathrm{CH}_2 ight)_5-\mathrm{CH}_3 ext { (2) } [/latex]

The following table shows the subgroups, their identification numbers k, values of parameters [latex]R_k and Q_k [/latex] (from Table G.1), and the numbers of each subgroup in each molecule:

	k	[latex]R_k [/latex]	[latex]Q_k [/latex]	[latex] v_k^{(1)} [/latex]	[latex] v_k^{(2)} [/latex]
[latex]CH_3 [/latex]	1	0.9011	0.848	2	2
[latex]CH_2 [/latex]	2	0.6744	0.54	1	5
[latex]CH_2NH [/latex]	33	1.207	0.936	1	0

By Eq. (G.15),

[latex] r_i=\sum_k v_k^{(i)} R_k [/latex] (G.15)

[latex] r_1=(2)(0.9011)+(1)(0.6744)+(1)(1.2070)=3.6836 [/latex]

Similarly,

[latex] r_2=(2)(0.9011)+(5)(0.6744)=5.1742 [/latex]

In like manner, by Eq. (G.16),

[latex] q_i=\sum_k v_k^{(i)} Q_k [/latex] (G.16)

[latex] q_1=3.1720 \quad ext { and } \quad q_2=4.3960 [/latex]

The [latex]r_i and q-i[/latex] values are molecular properties, independent of composition. Substituting known values into Eq. (G.17) generates the following table for [latex]e_{ki} [/latex]:

[latex] e_{k i}=\frac{v_k^{(i)} Q_k}{q_i} [/latex] (G.17)

	[latex]e_{ki} [/latex]
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:

[latex] a_{1,1}=a_{1,2}=a_{2,1}=a_{2,2}=a_{33,33}=0 \mathrm{~K} [/latex]

[latex] a_{1,33}=a_{2,33}=255.7 \mathrm{~K} [/latex]

[latex] a_{33.1}=a_{33.2}=65.33 \mathrm{~K} [/latex]

Substitution of these values into Eq. (G.21) with T = 308.15 K gives

[latex] au_{m k}=\exp \frac{-a_{m k}}{T} [/latex] (G.21)

[latex] au_{1,1}= au_{1,2}= au_{2,1}= au_{2,2}= au_{33,33}=1 [/latex]

[latex] au_{1,33}= au_{2,33}=0.4361 [/latex]

[latex] au_{33,1}= au_{33,2}=0.8090 [/latex]

Application of Eq. (G.18) leads to the values of [latex]β_{ik}[/latex] in the following table:

[latex] \beta_{i k}=\sum_m e_{m i} au_{m k} [/latex] (G.18)

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[latex]β_{ik}[/latex]ik = 1k = 2k = 3310.94360.94360.60242110.436

Substitution of these results into Eq. (G.19) yields:

[latex] heta_k=\frac{\sum_i x_i q_i e_{k i}}{\sum_j x_j q_j} [/latex] (G.19)

[latex] heta_1=0.4342 \quad heta_2=0.4700 \quad heta_{33}=0.0958 [/latex]

and by Eq. (G.20),

[latex] s_k=\sum_m heta_m au_{m k} [/latex] (G.20)

[latex] s_1=0.9817 \quad s_2=0.9817 \quad s_{33}=0.4901 [/latex]

The activity coefficients can now be calculated. By Eq. (G.13),

[latex] \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} ight) [/latex] (G.13)

[latex] \ln \gamma_1^C=-0.0213 \quad ext { and } \quad \ln \gamma_2^C=-0.0076 [/latex]

and by Eq. (G.14),

[latex] \ln \gamma_i^R=q_i\left[1-\sum_k\left( heta_k \frac{\beta_{i k}}{s_k}-e_{k i} \ln \frac{\beta_{i k}}{s_k} ight) ight] [/latex] (G.14)

[latex] \ln \gamma_1^R=0.1463 \quad ext { and } \quad \ln \gamma_2^R=0.0537 [/latex]

Finally, Eq. (G.7) gives:

[latex] \gamma_1=1.133 \quad ext { and } \quad \gamma_2=1.047 [/latex]

Question G.1: For the binary system diethylamine(1)/n-heptane(2) at 308.15...