Evaluate the diffusion coefficient of carbon dioxide in air at 20°C and atmospheric pressure. Compare this value with the experimental value reported in appendix table J.1.

Question

Evaluate the diffusion coefficient of carbon dioxide in air at [latex]20^{\circ} \mathrm{C}[/latex] and atmospheric pressure. Compare this value with the experimental value reported in appendix table J.1.

Accepted Answer

From Appendix Table K.2, the values of [latex]s[/latex] and [latex]e / k[/latex] are obtained

[latex]\begin{array}{lll} & \boldsymbol{\sigma}, extbf{ in } \overset{\circ }{ extbf{A}} & \varepsilon_{A} / \kappa, extbf{ in } \mathbf{K} \ ext { Carbon dioxide } & 3.996 & 190 \ ext { Air } & 3.617 & 97\end{array}[/latex]

The various parameters for equation (24-33)

[latex]D_{A B}=\frac{0.001858F^{3/2}\left[\frac{1}{M_{A}}+\frac{1}{M_{B}} ight]^{1/2}}{P\sigma_{A B}^{2}\Omega_{D}}[/latex] (24-33)

may be evaluated as follows:

[latex] \begin{aligned} \sigma_{A B} & =\frac{\sigma_{A}+\sigma_{B}}{2}=\frac{3.996+3.617}{2}=3.806 \overset{\circ }{ ext{A}} \ \varepsilon_{A B} / \kappa & =\sqrt{\left(\varepsilon_{A} / \kappa ight)\left(\varepsilon_{B} / \kappa ight)}=\sqrt{(190)(97)}=136 \ T & =20+273=293 \mathrm{~K} \ P & =1 \mathrm{~atm} \ \frac{\varepsilon_{A B}}{\kappa T} & =\frac{136}{293}=0.463 \ \frac{\kappa T}{\varepsilon_{A B}} & =2.16 \ \Omega_{D}( ext { Table } \mathrm{K} .1) & =1.047 \ M_{\mathrm{CO}_{2}} & =44 \end{aligned} [/latex]

and

[latex]M_{ ext {Air }}=29[/latex]

Substituting these values into equation (24-33), we obtain

[latex] \begin{aligned} D_{A B} & =\frac{0.001858 T^{3 / 2}\left(1 / M_{A}+1 / M_{B} ight)^{1 / 2}}{P \sigma_{A B}^{2} \Omega_{D}} \ & =\frac{(0.001858)(293)^{3 / 2}(1 / 44+1 / 29)^{1 / 2}}{(1)(3.806)^{2}(1.047)}=0.147 \mathrm{~cm}^{2} / \mathrm{s} \end{aligned} [/latex]

From Appendix Table J.1 for [latex]\mathrm{CO}_{2}[/latex] in air at [latex]273 \mathrm{~K}, 1[/latex] atm, we have

[latex]D_{A B}=0.136 \mathrm{~cm}^{2} / \mathrm{s}[/latex]

Equation (24-41)

[latex]D_{A B_{T_2, P_1}}=D_{A B_{T_1}, p_1}\left(\frac{P_1}{P_2} ight)\left(\frac{T_2}{T_1} ight)^{3 / 2} \frac{\left.\Omega_D ight|_{T_1}}{\left.\Omega_D ight|_{T_2}}[/latex] (24-41)

will be used to correct for the differences in temperature

[latex]\frac{D_{A B, T_{1}}}{D_{A B, T_{2}}}=\left(\frac{T_{1}}{T_{2}} ight)^{3 / 2}\left(\frac{\Omega_{D} \mid T_{2}}{\Omega_{D} \mid T_{1}} ight)[/latex]

Values for [latex]\Omega_{D}[/latex] may be evaluated as follows:

[latex] \begin{array}{ll} ext { at } T_{2}=273 & \varepsilon_{A B} / \kappa T=\frac{136}{273}=\left.0.498 \quad \Omega_{D} ight|_{T_{2}}=1.074 \ ext { at } T_{1}=293 & \left.\Omega_{D} ight|_{T_{1}}=1.074 \quad ext { (previous calculations) } \end{array} [/latex]

The corrected value for the diffusion coefficient at [latex]20^{\circ} \mathrm{C}[/latex] is

[latex] D_{A B, T_{1}}=\left(\frac{293}{273} ight)^{3 / 2}\left(\frac{1.074}{1.047} ight)(0.136)=0.155 \mathrm{~cm}^{2} / \mathrm{s} \quad\left(1.55 imes 10^{-5} \mathrm{~m}^{2} / \mathrm{s} ight) [/latex]

Table J.1 Binary mass diffusivities in gases†
System	T ( K )	$D_{AB}$ P ( cm² atm / s )	$D_{AB}$ P ( m² Pa / s )
Air
Ammonia	273	0.198	2.006
Aniline	298	0.0726	0.735
Benzene	298	0.0962	0.974
Bromine	293	0.091	0.923
Carbon dioxide	273	0.136	1.378
Carbon disulfide	273	0.0883	0.894
Chlorine	273	0.124	1.256
Diphenyl	491	0.160	1.621
Ethyl acetate	273	0.0709	0.718
Ethanol	298	0.132	1.337
Ethyl ether	293	0.0896	0.908
Iodine	298	0.0834	0.845
Methanol	298	0.162	1.641
Mercury	614	0.473	4.791
Naphthalene	298	0.0611	0.619
Nitrobenzene	298	0.0868	0.879
n – Octane	298	0.0602	0.610
Oxygen	273	0.175	1.773
Propyl acetate	315	0.092	0.932
Sulfur dioxide	273	0.122	1.236
Toluene	298	0.0844	0.855
Water	298	0.260	2.634
Ammonia
Ethylene	293	0.177	1.793
Argon
Neon	293	0.329	3.333
Carbon dioxide
Benzene	318	0.0715	0.724
Carbon disulfide	318	0.0715	0.724
Ethyl acetate	319	0.0666	0.675
(continued)

Table J.1 (Continued)
System	T ( K )	$D_{AB}$ P ( cm² atm / s )	$D_{AB}$ P ( m² Pa / s )
Ethanol	273	0.0693	0.702
Ethyl ether	273	0.0541	0.548
Hydrogen	273	0.550	5.572
Methane	273	0.153	1.550
Methanol	298.6	0.105	1.064
Nitrogen	298	0.165	1.672
Nitrous oxide	298	0.117	1.185
Propane	298	0.0863	0.874
Water	298	0.164	1.661
Carbon monoxide
Ethylene	273	0.151	1.530
Hydrogen	273	0.651	6.595
Nitrogen	288	0.192	1.945
Oxygen	273	0.185	1.874
Helium
Argon	273	0.641	6.493
Benzene	298	0.384	3.890
Ethanol	298	0.494	5.004
Hydrogen	293	1.64	16.613
Neon	293	1.23	12.460
Water	298	0.908	9.198
Hydrogen
Ammonia	293	0.849	8.600
Argon	293	0.770	7.800
Benzene	273	0.317	3.211
Ethane	273	0.439	4.447
Methane	273	0.625	6.331
Oxygen	273	0.697	7.061
Water	293	0.850	8.611
Nitrogen
Ammonia	293	0.241	2.441
Ethylene	298	0.163	1.651
Hydrogen	288	0.743	7.527
Iodine	273	0.070	0.709
Oxygen	273	0.181	1.834
Oxygen
Ammonia	293	0.253	2.563
Benzene	296	0.0939	0.951
Ethylene	293	0.182	1.844

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