{"id":1325,"date":"2021-07-09T20:26:51","date_gmt":"2021-07-10T00:26:51","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/3-1-ideal-gas-and-ideal-gas-equation-of-state\/"},"modified":"2022-08-09T19:14:30","modified_gmt":"2022-08-09T23:14:30","slug":"3-1-ideal-gas-and-ideal-gas-equation-of-state","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/3-1-ideal-gas-and-ideal-gas-equation-of-state\/","title":{"raw":"3.1 Ideal gas and ideal gas equation of state","rendered":"3.1 Ideal gas and ideal gas equation of state"},"content":{"raw":"
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Consider <\/span>a container of fixed volume <\/span>filled with <\/span>a gas<\/span>. When the container <\/span>is heated<\/span>, <\/span>the gas<\/span> temperature <\/span>will increase<\/span>,<\/span> causing the gas<\/span> pressure<\/span> to increase<\/span>.\u00a0<\/span>The variations of<\/span> gas pressure and temperature are<\/span> governed by the equations of state. An<\/span> [pb_glossary id=\"1375\"]equation of state[\/pb_glossary]<\/span> (EOS) is an <\/span>expression<\/span> that relates pressure, temperature, and specific volume <\/span>of a gas<\/span>.<\/span><\/p>\r\n \r\n

The <\/span>simplest equation of state is the <\/span>ideal gas equation of state<\/em><\/strong>, which<\/span> is <\/span>expressed<\/span> as<\/span><\/p>\r\n

\u00a0[latex]Pv=RT [\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0 or \u00a0 \u00a0 [latex]P\\mathbb{V}=mRT [\/latex]<\/span><\/p>\r\n

where\r\n<\/span><\/p>\r\n

[latex]m[\/latex]<\/span>: mass, in kg <\/span><\/p>\r\n

[latex]\\mathbb{V}[\/latex]<\/span>: volume, in m3<\/sup><\/p>\r\n

[latex]v[\/latex]<\/span>: specific volume, in m3<\/sup>\/kg<\/p>\r\n

[latex]T[\/latex]<\/span>: absolute temperature, in K<\/p>\r\n

[latex]P[\/latex]<\/span>: pressure, in kPa or Pa<\/p>\r\n

[latex]R[\/latex]<\/span>: gas constant in kJ\/kgK or J\/kgK<\/span><\/p>\r\n \r\n

A gas which obeys the ideal gas EOS is called an <\/span>[pb_glossary id=\"1379\"]ideal gas[\/pb_glossary]<\/strong>. <\/span>The i<\/span>deal gas <\/span>model <\/span>is a hypothetical model. <\/span>It<\/span> approximates the [latex]P-v-T[\/latex]<\/span> behaviour of a gas <\/span>at <\/span>high <\/span>temperatures and low <\/span>pressures in the superheated <\/span>vapour region.<\/span><\/p>\r\n \r\n

When a gas is at a state near the saturation region or its critical point, the gas behaviour<\/span> deviate<\/span>s<\/span> from the ideal gas <\/span>model significantly<\/span>.<\/span> For example, Figure 3.1.1<\/a> shows <\/span>the [latex]T-v[\/latex] diagram for water.\u00a0 Steam in the shaded region is either at a high temperature or a low pressure. The ideal gas model is valid in this region with a relative error of less than 1%.\u00a0 Moving out of the shaded region and towards the saturated vapour line or the critical point, the relative error increases significantly because the ideal gas EOS can no longer represent the gas behaviour in these regions.\r\n<\/span><\/p>\r\n\r\n<\/div>\r\nA common mistake that students tend to make is to use the ideal gas EOS <\/span>in all calculations without evaluating its suitability for the given conditions. It is important to note that, although many gasses may be treated as ideal gases in a certain range of pressures and temperatures,\u00a0 the ideal gas EOS is NOT valid for gases in all conditions.<\/strong><\/em> Therefore, it cannot be used without verification. T<\/span>he compressibility factor in Section 3.2 explains how to verify <\/a>if a gas is an \"ideal\" or real gas.<\/span>\r\n

\r\n\r\n \r\n\r\n[caption id=\"attachment_855\" align=\"aligncenter\" width=\"489\"]\"T-v<\/a> Figure 3.1.1<\/strong> T-v diagram for water<\/em>[\/caption]\r\n\r\n
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Example 1<\/p>\r\n\r\n<\/header>\r\n

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Two tanks contain methane. For the given conditions, methane can be treated as an ideal gas.<\/p>\r\n\r\n