{"id":4717,"date":"2024-10-05T18:05:46","date_gmt":"2024-10-05T18:05:46","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4717"},"modified":"2024-10-05T18:05:47","modified_gmt":"2024-10-05T18:05:47","slug":"real-gas-mixtures","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/10\/05\/real-gas-mixtures\/","title":{"rendered":"Real Gas Mixtures"},"content":{"rendered":"\n

To this point, we have discussed predicting\u00a0p<\/em>–V<\/em>–T<\/em>\u00a0properties for\u00a0pure<\/em>\u00a0components of real gases. How should you treat mixtures of real gases? The actual critical points of binary mixtures are not linear combinations of the properties of the two components as shown in\u00a0Figure 7.9\u00a0for combinations of CO2<\/sub>\u00a0and SO2<\/sub>. Too many dimensions are involved to draw pictures for three or more components.<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure 7.9. Critical and pseudocritical points for mixtures of CO2<\/sub> and SO2<\/sub><\/p>\n\n\n\n

One way you can make reasonable predictions for\u00a0z<\/em>\u00a0and\u00a0\"Image\"\u00a0for engineering purposes is to use Kay\u2019s method3<\/sup>\u00a0and the compressibility charts. In Kay\u2019s method,\u00a0pseudocritical<\/strong>\u00a0values for mixtures of gases are calculated on the assumption that each component in the mixture contributes to the pseudocritical value in the same proportion as the mole fraction of that component in the gas. Thus, the pseudocritical values are computed as mole averages as follows:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n
\"Image\"\/<\/figure>\n\n\n\n

where\u00a0y<\/em>i<\/em><\/sub>\u00a0is the mole fraction,\u00a0\"Image\"\u00a0is the pseudocritical pressure, and\u00a0\"Image\"\u00a0is the pseudocritical temperature. You can see that these are linearly weighted mole average pseudocritical properties. Look at\u00a0Figure 7.9, which compares the true critical values of a gaseous mixture of CO2<\/sub>\u00a0and SO2<\/sub>\u00a0with the respective pseudocritical values. The respective\u00a0pseudoreduced<\/strong>\u00a0variables are<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Kay\u2019s method\u00a0is known as a two-parameter rule since only\u00a0p<\/em>c<\/em><\/sub>\u00a0and\u00a0T<\/em>c<\/em><\/sub>\u00a0for each component are involved in the calculation of\u00a0z<\/em>. If a third parameter such as\u00a0z<\/em>c<\/em><\/sub>, or the Pitzer acentric factor, or\u00a0\"Image\", is included in the determination of the compressibility factor, you would have a three-parameter rule. Other pseudocritical methods with additional parameters provide better accuracy in predicting\u00a0p<\/em>–V<\/em>–T<\/em>\u00a0properties than Kay\u2019s method, but Kay\u2019s method can suffice for our work, and it is easy to use.<\/p>\n\n\n\n

In instances in which the temperature or pressure of a gas mixture is unknown, to avoid a trial-and-error solution using the generalized compressibility charts, you can compute the pseudocritical ideal volume<\/strong> and a pseudoreduced ideal volume<\/strong> \"Image\" thus<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

\"Image\" can be used in lieu of \"Image\" or \"Image\" in the compressibility charts.<\/p>\n\n\n\n

An enormous literature exists describing proposals for mixing rules<\/em> for equations of state, that is, rules to weight the coefficients or the predictions of each pure component so that the weighted values can be used with the same equations of state as are used for a pure component. Refer to the references at the end of this chapter, or look on the Internet for examples.<\/p>\n\n\n\n

Even if there is only one possible unified theory it is just a set of rules and equations.<\/em><\/p>\n\n\n\n

Stephen Hawking<\/p>\n\n\n\n

\n\n\n\n

<\/a>Example 7.9. Calculation of p-V-T<\/em> Properties for a Real Gas Mixture<\/p>\n\n\n\n

A gaseous mixture has the following composition (in mole percent):<\/p>\n\n\n\n

Methane, CH4<\/sub>                 20<\/p>\n\n\n\n

Ethylene, C2<\/sub>H4<\/sub>               30<\/p>\n\n\n\n

Nitrogen, N2<\/sub>                   50<\/p>\n\n\n\n

at 90 atm pressure and 100\u00b0C. Compare the volume per mole as computed by the methods of (a) the ideal gas law and (b) the pseudoreduced technique (Kay\u2019s method). What other types of averaging might you use?<\/p>\n\n\n\n

Solution<\/p>\n\n\n\n

Basis: 1 g mol of gas mixture<\/p>\n\n\n\n

Additional data needed are<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

The units used are fixed by the units of R. Let R be \"Image\".<\/p>\n\n\n\n

a.<\/strong> Ideal gas law:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

b.<\/strong> According to Kay\u2019s method, you first calculate the pseudocritical values for the mixture:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

T<\/em> \u2032c<\/em><\/sub> = T<\/em>c<\/em>A<\/em><\/sub>y<\/em>A<\/em><\/sub> + T<\/em>c<\/em>B<\/em><\/sub>y<\/em>B<\/em><\/sub> + T<\/em>cC<\/em><\/sub>y<\/em>C<\/em><\/sub> = (191)(0.2) + (283)(0.3) + (126)(0.5) = 186 K<\/p>\n\n\n\n

Then you calculate the pseudoreduced values for the mixture:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

With the aid of these two parameters you can find from Figure 7.8b<\/a> that zT<\/em>\u2032r<\/em><\/sub> = 1.91 and thus z<\/em> = 0.95. Then<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Two of the many possible ways of averaging are to use an equation of state with mole-averaged coefficients, or use the mole-averaged predictions of\u00a0\"Image\"\u00a0obtained from the individual equation of state. If you decided to use Equation (7.9) to calculate\u00a0z<\/em>\u00a0for the mixture, how might you average\u00a0z<\/em>0<\/sup>,\u00a0z<\/em>1<\/sup>, and\u00a0w<\/em>?<\/p>\n","protected":false},"excerpt":{"rendered":"

To this point, we have discussed predicting\u00a0p–V–T\u00a0properties for\u00a0pure\u00a0components of real gases. How should you treat mixtures of real gases? The actual critical points of binary mixtures are not linear combinations of the properties of the two components as shown in\u00a0Figure 7.9\u00a0for combinations of CO2\u00a0and SO2. Too many dimensions are involved to draw pictures for three […]<\/p>\n","protected":false},"author":1,"featured_media":4604,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[570],"tags":[],"class_list":["post-4717","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ideal-and-real-gases"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/gases.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4717","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4717"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4717\/revisions"}],"predecessor-version":[{"id":4718,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4717\/revisions\/4718"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4604"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4717"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4717"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4717"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}