Estimate the liquid diffusion coefficient of ethanol, C2H5OH, in a dilute solution of water at 10°C. The molecular volume of ethanol may be evaluated by using values from Table 24.5 as follows: VC2H5OH = 2VC + 6VH + VO VC2H5OH = 2(14.8) + 6(3.7) + 7.4 = 59.2 cm³/mol

Question

Estimate the liquid diffusion coefficient of ethanol, [latex]\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}[/latex], in a dilute solution of water at [latex]10^{\circ} \mathrm{C}[/latex]. The molecular volume of ethanol may be evaluated by using values from Table 24.5 as follows:

[latex] \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} [/latex]

Accepted Answer

At [latex]10^{\circ} \mathrm{C}[/latex], the viscosity of a solution containing [latex]0.05 \mathrm{~mol}[/latex] of alcohol/liter of water is 1.45 centipoises; the remaining parameters to be used are

[latex] \begin{aligned} T & =283 \mathrm{~K} \ \Phi_{B} ext { for water } & =2.26 \end{aligned} [/latex]

and

[latex]M_{B} ext { for water }=18[/latex]

Substituting these values into equation (24-52),

[latex]{\frac{D_{A B}\mu_{B}}{T}}={\frac{7.4 imes10^{-8}(\Phi_{B}M_{B})^{1/2}}{V_{A}^{0.6}}}[/latex] (24-52)

we obtain

[latex] \begin{aligned} D_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}-\mathrm{H}_{2} \mathrm{O}} & =\left(\frac{7.4 imes 10^{-8}(2.26 imes 18)^{1 / 2}}{(59.2)^{0.6}} ight)\left(\frac{283}{1.45} ight) \ & =7.96 imes 10^{-6} \mathrm{~cm}^{2} / \mathrm{s} \quad\left(7.96 imes 10^{-10} \mathrm{~m}^{2} / \mathrm{s} ight) \end{aligned} [/latex]

This value is in good agreement with the experimental value of [latex]8.3 imes 10^{-10} \mathrm{~m}^{2} / \mathrm{s}[/latex] reported in Appendix J.

Let us compare this value of the liquid diffusivity of ethanol in a dilute solution of water at [latex]10^{\circ} \mathrm{C}[/latex], [latex]7.96 imes 10^{-6} \mathrm{~cm}^{2} / \mathrm{s}[/latex], with the value of the gas diffusivity of ethanol in air at [latex]10^{\circ} \mathrm{C}[/latex] and [latex]1 \mathrm{~atm}[/latex] pressure, [latex]0.118 \mathrm{~cm}^{2} / \mathrm{s}[/latex]. 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 [latex]10^{\circ} \mathrm{C}[/latex] temperature predicts that the diffusion coefficient [latex]D_{B A}[/latex] is equal to [latex]1.18 imes 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}[/latex]. It is important to note that liquid diffusivities [latex]D_{A B_{L}}[/latex] and [latex]D_{B A_{L}}[/latex] are not equal as were the gas diffusivities at the same temperature and pressure.

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

Question 24.5: Estimate the liquid diffusion coefficient of ethanol, C2H5OH...

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