4. The First Law of Thermodynamics for Closed Systems

4.6 Key equations

 Constant-volume specific heat $C_v=\left(\displaystyle\frac{\partial u}{\partial T}\right)_v$ Change in specific internal energy for all fluids $\Delta u = u_2-u_1$ Change in specific internal energy for ideal gases $\Delta u = C_v\left(T_2-T_1\right)$ Specific heat transfer $q=\displaystyle\frac{Q}{m}$ Boundary work ${}_{1}W_{2}=\displaystyle\int_{1}^{2}{Pd\mathbb{V}\ }$ Specific boundary work ${}_{1}w_{2}=\displaystyle\int_{1}^{2}{Pdv\ }$ Spring force $F=Kx$ Spring work $W_{spring}=\displaystyle\int_{1}^{2}{Fdx=}\displaystyle\frac{1}{2}K\left(x_2^2-x_1^2\right)$ The first law of thermodynamics for closed systems $\Delta U = U_2-U_1 = {}_{1}Q_{2} - {}_{1}W_{2}$, assuming  $\Delta KE = \Delta PE = 0$

Equations for polytropic Processes

 Process function ${P}{v}^{n}= \rm{constant}$ Boundary work for real gases If  $n \neq 1$, ${}_{1}W_{2}=\displaystyle\frac{{P}_\mathbf{2}\mathbb{V}_\mathbf{2}-{P}_\mathbf{1}\mathbb{V}_\mathbf{1}}{1-n}\\$ If  $\ n=1,$ ${}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}\\$ ${}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ Specific boundary work for real gases If  $n \neq 1$ $\ {}_{1}w_{2}=\displaystyle\frac{{P}_\mathbf{2}{v}_\mathbf{2}-{P}_\mathbf{1}{v}_\mathbf{1}}{1-n}\\$ If  $\ n=1,$ $\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac{{v}_\mathbf{2}}{{v}_\mathbf{1}}}={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac{{v}_\mathbf{2}}{{v}_\mathbf{1}}}\\$ $\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ Boundary work for ideal gases If  $n \neq 1$ ${}_{1}W_{2}=\displaystyle\frac{{P}_\mathbf{2}\mathbb{V}_\mathbf{2}-{P}_\mathbf{1}\mathbb{V}_\mathbf{1}}{1-n}\\$ If  $\ n=1,$ ${}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}\\$ ${}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ $\ {}_{1}W_{2}={{mRT}}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}={{mRT}}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ ($T$ in Kelvin) Specific boundary work for ideal gases If  $n \neq 1$ $\ {}_{1}w_{2}=\displaystyle\frac{{P}_\mathbf{2}{v}_\mathbf{2}-{P}_\mathbf{1}{v}_\mathbf{1}}{1-n}\\$ If  $\ n=1,$ $\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac{{v}_\mathbf{2}}{{v}_\mathbf{1}}}={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac{{v}_\mathbf{2}}{{v}_\mathbf{1}}}\\$ $\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ $\ {}_{1}w_{2}={{RT}}{ln}{\displaystyle\frac{{v}_\mathbf{2}}{{v}_\mathbf{1}}}={{RT}}{ln}{\displaystyle\frac{{P}_\mathbf{1}}{{P}_\mathbf{2}}}\\$ ($T$ in Kelvin)