?Although isothermal VLE data are preferred for extraction of activity coefficients, a

Chapter 13, Problem 13.74

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Although isothermal VLE data are preferred for extraction of activity coefficients, a large body of good isobaric data exists in the literature. For a binary isobaric \(T-x_{1}-y_{1}\) data set, one can extract point values of \(\gamma_{i}\) via Eq. (13.13):

\(\gamma_{i}\left(x, T_{k}\right)=\frac{y_{i} \Phi_{i}\left(T_{k}, P, y\right) P}{x_{i} P_{i}^{\text {sat }}\left(T_{k}\right)}\)

Here, the variable list for \(\gamma_{i}\) recognizes a primary dependence on x and T; pressure dependence is normally negligible. The notation \(T_{k}\) emphasizes that temperature varies with data points across the composition range, and the calculated activity coefficients are at different temperatures. However, the usual goal of VLE data reduction and correlation is to develop an appropriate expression for \(G^{E} / R T\) at a single temperature T. A procedure is needed to correct each activity coefficient to such a T chosen near the average for the data set. If a correlation for \(H^{E}(x)\) is available at or near this T, show that the values of \(\gamma_{i}\) corrected to T can be estimated by the expression:

\(\gamma_{i}(x, T)=\gamma_{i}\left(x, T_{k}\right) \exp \left[\frac{-\bar{H}_{i}^{E}}{R T}\left(\frac{T}{T_{k}}-1\right)\right]\)

Text Transcription:

T-x_1-y_1

gamma_i

gamma_i(x, T_{k)=y_i Phi_i(T_k, P, y)P/x_iP_i^sat (T_k)

T_k

G^E/RT

H^E(x)

gamma_i(x, T)=gamma_i(x, T_k) exp [-bar H_i^ERT(T/T_k-1)]