{"id":338,"date":"2021-05-29T17:24:05","date_gmt":"2021-05-29T21:24:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/?post_type=chapter&p=338"},"modified":"2022-09-02T15:20:45","modified_gmt":"2022-09-02T19:20:45","slug":"key-equations-4","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/key-equations-4\/","title":{"raw":"4.6 Key equations","rendered":"4.6 Key equations"},"content":{"raw":"\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Constant-volume specific heat<\/td>\r\n[latex]C_v=\\left(\\displaystyle\\frac{\\partial u}{\\partial T}\\right)_v[\/latex]<\/td>\r\n<\/tr>\r\n
Change in specific internal energy for a<\/strong>ll fluids<\/strong><\/td>\r\n[latex]\\Delta u = u_2-u_1[\/latex]<\/td>\r\n<\/tr>\r\n
Change in specific internal energy for i<\/strong>deal gases<\/strong><\/td>\r\n[latex]\\Delta u = C_v\\left(T_2-T_1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
Specific heat transfer<\/td>\r\n[latex]q=\\displaystyle\\frac{Q}{m}[\/latex]<\/td>\r\n<\/tr>\r\n
Boundary work<\/td>\r\n[latex]{}_{1}W_{2}=\\displaystyle\\int_{1}^{2}{Pd\\mathbb{V}\\ } [\/latex]<\/td>\r\n<\/tr>\r\n
Specific boundary work<\/td>\r\n[latex]{}_{1}w_{2}=\\displaystyle\\int_{1}^{2}{Pdv\\ }[\/latex]<\/td>\r\n<\/tr>\r\n
Spring force<\/td>\r\n[latex]F=Kx[\/latex]<\/td>\r\n<\/tr>\r\n
Spring work<\/td>\r\n[latex]W_{spring}=\\displaystyle\\int_{1}^{2}{Fdx=}\\displaystyle\\frac{1}{2}K\\left(x_2^2-x_1^2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
The first law of thermodynamics for closed systems<\/strong>\r\n<\/strong><\/td>\r\n[latex]\\Delta U = U_2-U_1 = {}_{1}Q_{2} - {}_{1}W_{2}[\/latex], assuming\u00a0 [latex]\\Delta KE = \\Delta PE = 0 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nEquations for polytropic Processes<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Process function<\/td>\r\n[latex]{P}{v}^{n}= \\rm{constant}[\/latex]<\/td>\r\n<\/tr>\r\n
Boundary work for real gases <\/strong><\/td>\r\nIf\u00a0 [latex] n \\neq 1[\/latex],\r\n

[latex]{}_{1}W_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}-{P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\r\nIf\u00a0 [latex]\\ n=1,[\/latex]\r\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}\\\\\u00a0 [\/latex]<\/p>\r\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0 [\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Specific boundary work for real gases<\/strong><\/td>\r\nIf\u00a0 [latex] n \\neq 1[\/latex]\r\n

[latex]\\ {}_{1}w_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}{v}_\\mathbf{2}-{P}_\\mathbf{1}{v}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\r\nIf\u00a0 [latex]\\ n=1,[\/latex]\r\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}\\\\ [\/latex]<\/p>\r\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\ [\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Boundary work for ideal gases<\/strong><\/td>\r\nIf\u00a0 [latex] n \\neq 1[\/latex]\r\n

[latex]{}_{1}W_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}-{P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\r\nIf\u00a0 [latex]\\ n=1,[\/latex]\r\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}\\\\\u00a0 [\/latex]<\/p>\r\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0 [\/latex]<\/p>\r\n

[latex]\\ {}_{1}W_{2}={{mRT}}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={{mRT}}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0 [\/latex]<\/p>\r\n

([latex]T[\/latex] in Kelvin)<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Specific boundary work for ideal gases<\/strong>\r\n<\/em><\/strong><\/td>\r\nIf\u00a0 [latex] n \\neq 1[\/latex]\r\n

[latex]\\ {}_{1}w_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}{v}_\\mathbf{2}-{P}_\\mathbf{1}{v}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\r\nIf\u00a0 [latex]\\ n=1,[\/latex]\r\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}\\\\ [\/latex]<\/p>\r\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\ [\/latex]<\/p>\r\n

[latex]\\ {}_{1}w_{2}={{RT}}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={{RT}}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\ [\/latex]<\/p>\r\n

([latex]T[\/latex] in Kelvin)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"\n\n\n\n\n\n\n\n\n\n\n
Constant-volume specific heat<\/td>\n[latex]C_v=\\left(\\displaystyle\\frac{\\partial u}{\\partial T}\\right)_v[\/latex]<\/td>\n<\/tr>\n
Change in specific internal energy for a<\/strong>ll fluids<\/strong><\/td>\n[latex]\\Delta u = u_2-u_1[\/latex]<\/td>\n<\/tr>\n
Change in specific internal energy for i<\/strong>deal gases<\/strong><\/td>\n[latex]\\Delta u = C_v\\left(T_2-T_1\\right)[\/latex]<\/td>\n<\/tr>\n
Specific heat transfer<\/td>\n[latex]q=\\displaystyle\\frac{Q}{m}[\/latex]<\/td>\n<\/tr>\n
Boundary work<\/td>\n[latex]{}_{1}W_{2}=\\displaystyle\\int_{1}^{2}{Pd\\mathbb{V}\\ }[\/latex]<\/td>\n<\/tr>\n
Specific boundary work<\/td>\n[latex]{}_{1}w_{2}=\\displaystyle\\int_{1}^{2}{Pdv\\ }[\/latex]<\/td>\n<\/tr>\n
Spring force<\/td>\n[latex]F=Kx[\/latex]<\/td>\n<\/tr>\n
Spring work<\/td>\n[latex]W_{spring}=\\displaystyle\\int_{1}^{2}{Fdx=}\\displaystyle\\frac{1}{2}K\\left(x_2^2-x_1^2\\right)[\/latex]<\/td>\n<\/tr>\n
The first law of thermodynamics for closed systems<\/strong>
\n<\/strong><\/td>\n		[latex]\\Delta U = U_2-U_1 = {}_{1}Q_{2} - {}_{1}W_{2}[\/latex], assuming\u00a0 [latex]\\Delta KE = \\Delta PE = 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Equations for polytropic Processes<\/strong><\/p>\n\n\n\n\n\n\n\n
Process function<\/td>\n[latex]{P}{v}^{n}= \\rm{constant}[\/latex]<\/td>\n<\/tr>\n
Boundary work for real gases <\/strong><\/td>\nIf\u00a0 [latex]n \\neq 1[\/latex],<\/p>\n

[latex]{}_{1}W_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}-{P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\n

If\u00a0 [latex]\\ n=1,[\/latex]<\/p>\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}\\\\\u00a0[\/latex]<\/p>\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0[\/latex]<\/p>\n<\/td>\n<\/tr>\n

Specific boundary work for real gases<\/strong><\/td>\nIf\u00a0 [latex]n \\neq 1[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}{v}_\\mathbf{2}-{P}_\\mathbf{1}{v}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\n

If\u00a0 [latex]\\ n=1,[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}\\\\[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\[\/latex]<\/p>\n<\/td>\n<\/tr>\n

Boundary work for ideal gases<\/strong><\/td>\nIf\u00a0 [latex]n \\neq 1[\/latex]<\/p>\n

[latex]{}_{1}W_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}-{P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\n

If\u00a0 [latex]\\ n=1,[\/latex]<\/p>\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}\\\\\u00a0[\/latex]<\/p>\n

[latex]{}_{1}W_{2}={P}_\\mathbf{1}\\mathbb{V}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}\\mathbb{V}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0[\/latex]<\/p>\n

[latex]\\ {}_{1}W_{2}={{mRT}}{ln}{\\displaystyle\\frac{\\mathbb{V}_\\mathbf{2}}{\\mathbb{V}_\\mathbf{1}}}={{mRT}}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\\u00a0[\/latex]<\/p>\n

([latex]T[\/latex] in Kelvin)<\/p>\n<\/td>\n<\/tr>\n

Specific boundary work for ideal gases<\/strong>
\n<\/em><\/strong><\/td>\n		If\u00a0 [latex]n \\neq 1[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}=\\displaystyle\\frac{{P}_\\mathbf{2}{v}_\\mathbf{2}-{P}_\\mathbf{1}{v}_\\mathbf{1}}{1-n}\\\\[\/latex]<\/p>\n

If\u00a0 [latex]\\ n=1,[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}\\\\[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}={P}_\\mathbf{1}{v}_\\mathbf{1}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}={P}_\\mathbf{2}{v}_\\mathbf{2}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\[\/latex]<\/p>\n

[latex]\\ {}_{1}w_{2}={{RT}}{ln}{\\displaystyle\\frac{{v}_\\mathbf{2}}{{v}_\\mathbf{1}}}={{RT}}{ln}{\\displaystyle\\frac{{P}_\\mathbf{1}}{{P}_\\mathbf{2}}}\\\\[\/latex]<\/p>\n

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