{"id":2195,"date":"2016-03-13T15:40:54","date_gmt":"2016-03-13T15:40:54","guid":{"rendered":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/?p=2195"},"modified":"2016-03-13T15:40:54","modified_gmt":"2016-03-13T15:40:54","slug":"entropythe-entropy-change-of-ideal-gases","status":"publish","type":"post","link":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/entropythe-entropy-change-of-ideal-gases\/","title":{"rendered":"ENTROPY:THE ENTROPY CHANGE OF IDEAL GASES"},"content":{"rendered":"<div class=\"huovf6a0dce4f8534b\" ><script type=\"text\/javascript\">\n\tatOptions = {\n\t\t'key' : '61e5902552e2353963d8d2f1bd1f4a8f',\n\t\t'format' : 'iframe',\n\t\t'height' : 250,\n\t\t'width' : 300,\n\t\t'params' : {}\n\t};\n<\/script>\n<script type=\"text\/javascript\" src=\"\/\/www.highperformanceformat.com\/61e5902552e2353963d8d2f1bd1f4a8f\/invoke.js\"><\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.huovf6a0dce4f8534b {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.huovf6a0dce4f8534b {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.huovf6a0dce4f8534b {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.huovf6a0dce4f8534b {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.huovf6a0dce4f8534b {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n<h3 align=\"justify\"><font size=\"5\">&nbsp;<b>TH<\/b><b>E ENTROPY CHANGE OF IDEAL GASES<\/b><\/font><\/h3>\n<p align=\"justify\"><font size=\"5\">An expression for the entropy change of an ideal gas can be obtained from Eq. 7\u201325 or 7\u201326 by employing the property relations for ideal gases (Fig. 7\u201331). By substituting <i>du <\/i>= <i>C<\/i>u <i>d<\/i><i>T <\/i>and <i>P <\/i>= <i>R<\/i><i>T<\/i>\/u into Eq. 7\u201325, the differential entropy change of an ideal gas becomes<\/font><\/p>\n<p align=\"justify\"><font size=\"5\">A second relation for the entropy change of an ideal gas is obtained in a similar manner by substituting <i>dh <\/i>= <i>C<\/i><i>p <\/i><i>dT <\/i>and u = <i>RT<\/i>\/<i>P <\/i>into Eq. 7\u201326 and integrating. The result is<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0203_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0203_thumb\" border=\"0\" alt=\"ENTROPY-0203_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0203_thumb_thumb.jpg\" width=\"255\" height=\"50\"><\/a><\/font><\/p>\n<p align=\"justify\"><font size=\"5\">The specific heats of ideal gases, with the exception of monatomic gases, de- pend on temperature, and the integrals in Eqs. 7\u201331 and 7\u201332 cannot be per- formed unless the dependence of <i>C<\/i>u and <i>C<\/i><i>p <\/i>on temperature is known. Even when the <i>C<\/i>u(<i>T<\/i>) and <i>C<\/i><i>p<\/i>(<i>T<\/i>) functions are available, performing long integrations every time entropy change is calculated is not practical. Then two reasonable choices are left: either perform these integrations by simply assuming constant specific heats or evaluate those integrals once and tabulate the results. Both approaches are presented next.<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><b>Constan<\/b><b>t Specific Heats (Approximate Analysis)<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><b>&nbsp;<\/b>Assuming constant specific heats for ideal gases is a common approximation, and we used this assumption before on several occasions. It usually simplifies the analysis greatly, and the price we pay for this convenience is some loss in accuracy. The magnitude of the error introduced by this assumption depends on the situation at hand. For example, for monatomic ideal gases such as helium, the specific heats are independent of temperature, and therefore the constant-specific-heat assumption introduces no error. For ideal gases whose specific heats vary almost linearly in the temperature range of interest, the possible error is minimized by using specific heat values evaluated at the av- erage temperature (Fig. 7\u201332). The results obtained in this way usually are sufficiently accurate if the temperature range is not greater than a few hundred degrees.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The entropy-change relations for ideal gases under the constant-specific-<\/font><font size=\"5\">heat assumption are easily obtained by replacing <i>C<\/i>u(<i>T<\/i>) and <i>C<\/i><i>p<\/i>(<i>T<\/i>) in Eqs. 7\u201331 and 7\u201332 by <i>C<\/i>u, av and <i>C<\/i><i>p<\/i>, av, respectively, and performing the integrations. We obtain<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0204_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0204_thumb\" border=\"0\" alt=\"ENTROPY-0204_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0204_thumb_thumb.jpg\" width=\"171\" height=\"221\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0205_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0205_thumb\" border=\"0\" alt=\"ENTROPY-0205_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0205_thumb_thumb.jpg\" width=\"354\" height=\"221\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><strong>Variable Specific Heats (Exact Analysis)<\/strong><\/font>  <\/p>\n<h6 align=\"justify\"><font size=\"5\">When the temperature change during a process is large and the specific heats <\/font><font size=\"5\">of the ideal gas vary nonlinearly within the temperature range, the assumption of constant specific heats may lead to considerable errors in entropy-change calculations. For those cases, the variation of specific heats with temperature should be properly accounted for by utilizing accurate relations for the specific heats as a function of temperature. The entropy change during a process is then determined by substituting these <i>C<\/i>u(<i>T<\/i>) or <i>C<\/i><i>p<\/i>(<i>T<\/i>) relations into Eq. 7\u201331 or 7\u201332 and performing the integrations.<\/font><\/h6>\n<p align=\"justify\"><font size=\"5\">Instead of performing these laborious integrals each time we have a new process, it is convenient to perform these integrals once and tabulate the results. For this purpose, we choose absolute zero as the reference temperature and define a function <i>s<\/i>\u00b0 as<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0206_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0206_thumb\" border=\"0\" alt=\"ENTROPY-0206_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0206_thumb_thumb.jpg\" width=\"349\" height=\"245\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">Note that unlike internal energy and enthalpy, the entropy of an ideal gas varies with specific volume or pressure as well as the temperature. Therefore, entropy cannot be tabulated as a function of temperature alone. The <i>s<\/i>\u00b0 values in the tables account for the temperature dependence of entropy (Fig. 7\u201333).<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The entropy of an ideal gas depends on both <i>T <\/i>and <i>P<\/i><i>. <\/i>The function <i>s<\/i>\u00b0 represents only the temperature- dependent part of entropy.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The variation of entropy with pressure is accounted for by the last term in Eq. 7\u201339. Another relation for entropy change can be developed based on Eq. 7\u201331, but this would require the definition of another function and tabula- tion of its values, which is not practical.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0207_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0207_thumb\" border=\"0\" alt=\"ENTROPY-0207_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0207_thumb_thumb.jpg\" width=\"183\" height=\"198\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0208_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0208_thumb\" border=\"0\" alt=\"ENTROPY-0208_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0208_thumb_thumb.jpg\" width=\"350\" height=\"114\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0209_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0209_thumb\" border=\"0\" alt=\"ENTROPY-0209_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0209_thumb_thumb.jpg\" width=\"488\" height=\"484\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><b>Isentropi<\/b><b>c Processes of Ideal Gases<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">Several relations for the isentropic processes of ideal gases can be obtained by <\/font><font size=\"5\">setting the entropy-change relations developed above equal to zero. Again, this is done first for the case of constant specific heats and then for the case of variable specific heats.<\/font>  <\/p><div class=\"gwhob6a0dce4f855d9\" ><script type=\"text\/javascript\">\n\tatOptions = {\n\t\t'key' : '0c1eb4c533eaedb7b996f49a5a4983a9',\n\t\t'format' : 'iframe',\n\t\t'height' : 300,\n\t\t'width' : 160,\n\t\t'params' : {}\n\t};\n<\/script>\n<script type=\"text\/javascript\" src=\"\/\/www.highperformanceformat.com\/0c1eb4c533eaedb7b996f49a5a4983a9\/invoke.js\"><\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.gwhob6a0dce4f855d9 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.gwhob6a0dce4f855d9 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.gwhob6a0dce4f855d9 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.gwhob6a0dce4f855d9 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.gwhob6a0dce4f855d9 {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n<div class=\"mdrbg6a0dce4f85483\" ><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-0778475562755157\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- 300x600 hydraulics-and-pneumatics -->\n<ins class=\"adsbygoogle\"\n     style=\"display:inline-block;width:300px;height:600px\"\n     data-ad-client=\"ca-pub-0778475562755157\"\n     data-ad-slot=\"3735577695\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.mdrbg6a0dce4f85483 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.mdrbg6a0dce4f85483 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.mdrbg6a0dce4f85483 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.mdrbg6a0dce4f85483 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.mdrbg6a0dce4f85483 {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n\n<p align=\"justify\"><b><font size=\"5\"><\/font><\/b> <\/p>\n<p align=\"justify\"><font size=\"5\"><strong>Constant Specific Heats (Approximate Analysis)<\/strong><\/font><\/p>\n<h6 align=\"justify\"><font size=\"5\">When the constant-specific-heat assumption is valid, the isentropic relations <\/font><font size=\"5\">for ideal gases are obtained by setting Eqs. 7\u201333 and 7\u201334 equal to zero. From Eq. 7\u201333,<\/font><\/h6>\n<p align=\"justify\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0210_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0210_thumb\" border=\"0\" alt=\"ENTROPY-0210_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0210_thumb_thumb.jpg\" width=\"357\" height=\"400\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0211_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0211_thumb\" border=\"0\" alt=\"ENTROPY-0211_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0211_thumb_thumb.jpg\" width=\"183\" height=\"238\"><\/a><\/p>\n<p align=\"justify\"><font size=\"5\">The specific heat ratio <i>k<\/i>, in general, varies with temperature, and thus an average <i>k <\/i>value for the given temperature range should be used.<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><\/font> <\/p>\n<p align=\"justify\"><font size=\"5\">Note that the ideal-gas isentropic relations above, as the name implies, are strictly valid for isentropic processes only when the constant-specific-heat assumption is appropriate (Fig. 7\u201336).<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><b>V<\/b><b>ariabl<\/b><b>e Specific Heats (Exact Analysis)<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">When the constant-specific-heat assumption is not appropriate, the isentropic <\/font><font size=\"5\">relations developed above will yield results that are not quite accurate. For such cases, we should use an isentropic relation obtained from Eq. 7\u201339 that accounts for the variation of specific heats with temperature. Setting this equation equal to zero gives<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0212_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0212_thumb\" border=\"0\" alt=\"ENTROPY-0212_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0212_thumb_thumb.jpg\" width=\"365\" height=\"125\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><b>Relativ<\/b><b>e Pressure and Relative Specific Volume<\/b><b> <\/b>Equation 7\u201348 provides an accurate way of evaluating property changes of ideal gases during isentropic processes since it accounts for the variation of specific heats with temperature. However, it involves tedious iterations when the volume ratio is given instead of the pressure ratio. This is quite an inconvenience in optimization studies, which usually require numerous repetitive calculations. To remedy this deficiency, we define two new dimensionless quantities associated with isentropic processes.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The definition of the first is based on Eq. 7\u201348, which can be rearranged as<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0213_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0213_thumb\" border=\"0\" alt=\"ENTROPY-0213_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0213_thumb_thumb.jpg\" width=\"167\" height=\"223\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0214_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0214_thumb\" border=\"0\" alt=\"ENTROPY-0214_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0214_thumb_thumb.jpg\" width=\"360\" height=\"176\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">Note that the relative pressure <i>P<\/i><i>r <\/i>is a <i>dimensionles<\/i><i>s <\/i>quantity that is a function of temperature only since <i>s<\/i>\u00b0 depends on temperature alone. Therefore, values of <i>P<\/i><i>r <\/i>can be tabulated against temperature. This is done for air in Table A\u201321. The use of <i>P<\/i><i>r <\/i>data is illustrated in Fig. 7\u201337.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">Process: isentropic Given: <i>P<\/i>1, <i>T<\/i>1, and <i>P<\/i>2 Find: <i>T<\/i>2<\/font>  <\/p>\n<h6 align=\"justify\"><font size=\"5\">Sometimes specific volume ratios are given instead of pressure ratios. This is particularly the case when automotive engines are analyzed. In such cases, one needs to work with volume ratios. Therefore, we define another quantity related to specific volume ratios for isentropic processes. This is done by utilizing the ideal-gas relation and Eq. 7\u201349:<\/font><\/h6>\n<p align=\"justify\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0215_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0215_thumb\" border=\"0\" alt=\"ENTROPY-0215_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0215_thumb_thumb.jpg\" width=\"357\" height=\"116\"><\/a><\/p>\n<p align=\"justify\"><font size=\"5\">Equations 7\u201349 and 7\u201350 are strictly valid for isentropic processes of ideal gases only. They account for the variation of specific heats with temperature and therefore give more accurate results than Eqs. 7\u201342 through 7\u201347. The values of <i>P<\/i><i>r <\/i>and u<i>r <\/i>are listed for air in Table A\u201321.<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0216_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0216_thumb\" border=\"0\" alt=\"ENTROPY-0216_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0216_thumb_thumb.jpg\" width=\"449\" height=\"484\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0217_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0217_thumb\" border=\"0\" alt=\"ENTROPY-0217_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0217_thumb_thumb.jpg\" width=\"399\" height=\"484\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0218_thumb.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-bottom: 0px; border-left: 0px; padding-left: 0px; padding-right: 0px; display: block; float: none; margin-left: auto; border-top: 0px; margin-right: auto; border-right: 0px; padding-top: 0px\" title=\"ENTROPY-0218_thumb\" border=\"0\" alt=\"ENTROPY-0218_thumb\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/ENTROPY-0218_thumb_thumb.jpg\" width=\"357\" height=\"109\"><\/a><\/font><\/p>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp;THE ENTROPY CHANGE OF IDEAL GASES An expression for the entropy change of an ideal gas can be obtained from Eq. 7\u201325 or 7\u201326 by employing the property relations for ideal gases (Fig. 7\u201331). By substituting du = Cu dT and P = RT\/u into Eq. 7\u201325, the differential entropy change of an ideal gas [&hellip;]<br \/><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/entropythe-entropy-change-of-ideal-gases\/\" class=\"more-link\" >Continue reading&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/2195"}],"collection":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/comments?post=2195"}],"version-history":[{"count":1,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/2195\/revisions"}],"predecessor-version":[{"id":2196,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/2195\/revisions\/2196"}],"wp:attachment":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/media?parent=2195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/categories?post=2195"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/tags?post=2195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}