Question 24.5: Estimate the liquid diffusion coefficient of ethanol, C2H5OH...
Estimate the liquid diffusion coefficient of ethanol, \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}, in a dilute solution of water at 10^{\circ} \mathrm{C}. The molecular volume of ethanol may be evaluated by using values from Table 24.5 as follows:
\begin{aligned} & V_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}}=2 V_{C}+6 V_{H}+V_{O} \\ & V_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}}=2(14.8)+6(3.7)+7.4=59.2 \mathrm{~cm}^{3} / \mathrm{mol} \end{aligned}
Table 24.5 Atomic volumes for complex molecular volumes for simple \text{substances}^{\dagger}
| Element | Atomic volume, in cm3/g mol | Element | Atomic volume, in cm3/g mol |
| Bromine | 27.0 | Oxygen, except as noted below | 7.4 |
| Carbon | 14.8 | Oxygen, in methyl esters | 9.1 |
| Chlorine | 21.6 | Oxygen, in methyl ethers | 9.9 |
| Hydrogen | 3.7 | Oxygen, in higher ethers | |
| Iodine | 37.0 | and other esters | 11.0 |
| Nitrogen, double bond | 15.6 | Oxygen, in acids | 12.0 |
| Nitrogen, in primary amines | 10.5 | Sulfur | 25.6 |
| Nitrogen, in secondary amines | 12.0 |
\text{}^{\dagger} G. Le Bas, The Molecular Volumes of Liquid Chemical Compounds, Longmans, Green & Company, Ltd., London, 1915.
At 10^{\circ} \mathrm{C}, the viscosity of a solution containing 0.05 \mathrm{~mol} of alcohol/liter of water is 1.45 centipoises; the remaining parameters to be used are
\begin{aligned} T & =283 \mathrm{~K} \\ \Phi_{B} \text { for water } & =2.26 \end{aligned}
and
M_{B} \text { for water }=18
Substituting these values into equation (24-52),
{\frac{D_{A B}\mu_{B}}{T}}={\frac{7.4\times10^{-8}(\Phi_{B}M_{B})^{1/2}}{V_{A}^{0.6}}} (24-52)
we obtain
\begin{aligned} D_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}-\mathrm{H}_{2} \mathrm{O}} & =\left(\frac{7.4 \times 10^{-8}(2.26 \times 18)^{1 / 2}}{(59.2)^{0.6}}\right)\left(\frac{283}{1.45}\right) \\ & =7.96 \times 10^{-6} \mathrm{~cm}^{2} / \mathrm{s} \quad\left(7.96 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s}\right) \end{aligned}
This value is in good agreement with the experimental value of 8.3 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s} reported in Appendix J.
Let us compare this value of the liquid diffusivity of ethanol in a dilute solution of water at 10^{\circ} \mathrm{C}, 7.96 \times 10^{-6} \mathrm{~cm}^{2} / \mathrm{s}, with the value of the gas diffusivity of ethanol in air at 10^{\circ} \mathrm{C} and 1 \mathrm{~atm} pressure, 0.118 \mathrm{~cm}^{2} / \mathrm{s}. This emphasizes the order of magnitude difference between the values of the liquid and gas diffusivities.
Performing a similar calculation, the liquid diffusion coefficient of water in an infinite dilute solution of ethanol at the same 10^{\circ} \mathrm{C} temperature predicts that the diffusion coefficient D_{B A} is equal to 1.18 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}. It is important to note that liquid diffusivities D_{A B_{L}} and D_{B A_{L}} are not equal as were the gas diffusivities at the same temperature and pressure.