4.6 Key equations

Claire Yu Yan

4. The First Law of Thermodynamics for Closed Systems

4.6 Key equations

Constant-volume specific heat

C_{v} = {(\frac{\partial u}{\partial T})}_{v}

$C_v=\left(\displaystyle\frac{\partial u}{\partial T}\right)_v$

Change in specific internal energy for all fluids

Δ u = u_{2} - u_{1}

$\Delta u = u_2-u_1$

Change in specific internal energy for ideal gases

Δ u = C_{v} (T_{2} - T_{1})

$\Delta u = C_v\left(T_2-T_1\right)$

Specific heat transfer

q = \frac{Q}{m}

$q=\displaystyle\frac{Q}{m}$

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)

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Introduction to Engineering Thermodynamics Copyright © 2022 by Claire Yu Yan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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