4. The First Law of Thermodynamics for Closed Systems
4.6 Key equations
| Constant-volume specific heat | [latex]C_v=\left(\displaystyle\frac{\partial u}{\partial T}\right)_v[/latex] |
| Change in specific internal energy for all fluids | [latex]\Delta u = u_2-u_1[/latex] |
| Change in specific internal energy for ideal gases | [latex]\Delta u = C_v\left(T_2-T_1\right)[/latex] |
| Specific heat transfer | [latex]q=\displaystyle\frac{Q}{m}[/latex] |
| Boundary work | [latex]{}_{1}W_{2}=\displaystyle\int_{1}^{2}{Pd\mathbb{V}\ }[/latex] |
| Specific boundary work | [latex]{}_{1}w_{2}=\displaystyle\int_{1}^{2}{Pdv\ }[/latex] |
| Spring force | [latex]F=Kx[/latex] |
| Spring work | [latex]W_{spring}=\displaystyle\int_{1}^{2}{Fdx=}\displaystyle\frac{1}{2}K\left(x_2^2-x_1^2\right)[/latex] |
| The first law of thermodynamics for closed systems |
[latex]\Delta U = U_2-U_1 = {}_{1}Q_{2} - {}_{1}W_{2}[/latex], assuming [latex]\Delta KE = \Delta PE = 0[/latex] |
Equations for polytropic Processes
| Process function | [latex]{P}{v}^{n}= \rm{constant}[/latex] |
| Boundary work for real gases | If [latex]n \neq 1[/latex],
[latex]{}_{1}W_{2}=\displaystyle\frac{{P}_\mathbf{2}\mathbb{V}_\mathbf{2}-{P}_\mathbf{1}\mathbb{V}_\mathbf{1}}{1-n}\\[/latex] If [latex]\ n=1,[/latex] [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}}}\\ [/latex] [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}}}\\ [/latex] |
| Specific boundary work for real gases | If [latex]n \neq 1[/latex]
[latex]\ {}_{1}w_{2}=\displaystyle\frac{{P}_\mathbf{2}{v}_\mathbf{2}-{P}_\mathbf{1}{v}_\mathbf{1}}{1-n}\\[/latex] If [latex]\ n=1,[/latex] [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] [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] |
| Boundary work for ideal gases | If [latex]n \neq 1[/latex]
[latex]{}_{1}W_{2}=\displaystyle\frac{{P}_\mathbf{2}\mathbb{V}_\mathbf{2}-{P}_\mathbf{1}\mathbb{V}_\mathbf{1}}{1-n}\\[/latex] If [latex]\ n=1,[/latex] [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}}}\\ [/latex] [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}}}\\ [/latex] [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}}}\\ [/latex] ([latex]T[/latex] in Kelvin) |
| Specific boundary work for ideal gases |
If [latex]n \neq 1[/latex]
[latex]\ {}_{1}w_{2}=\displaystyle\frac{{P}_\mathbf{2}{v}_\mathbf{2}-{P}_\mathbf{1}{v}_\mathbf{1}}{1-n}\\[/latex] If [latex]\ n=1,[/latex] [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] [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] [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] ([latex]T[/latex] in Kelvin) |
