03 Heat and Work

Last Updated: 1 year ago2024-01-24 ; Contributors: AhmedThahir

Addable Quantities¶

note that specific quanties like $h, u$ can not be added

\begin{aligned} F &= kx \\ W &= \frac{1}{2} k x^2 \\ &= \frac12 k ({x_2} ^2 - {x_1}^2) \\ \end{aligned}

\begin{aligned} \dot W &= VI \\ W &= VI \Delta t \end{aligned}

Note that temperature should be in $K$ (Kelvin)

W_\text{out, b} = \int \limits_{v_1}^{v_2} P \cdot dv

Type	Condition(s)	$W_b$
Isochoric	$V = c$	$0$
Isobaric	$P = c$	$P_1(V_2 - V_1)$	$mP_1(\nu_2 - \nu_1)$
Isothermal	$\begin{aligned} T &= c \\ PV &= mRT \\ P_1 V_1 &= P_2 V_2 \end{aligned}$	$P_i V_i \ \ln \vert \frac{V_2}{V_1} \vert \\ P_i V_i \ \ln \vert \frac{P_1}{P_2} \vert$	$mRT \ \ln \vert \frac{V_2}{V_1} \vert$ $mRT \ \ln \vert \frac{P_1}{P_2} \vert$
Polytropic	$\begin{aligned} P V^n &= c \\ P_1 (V_1)^n &= P_2 (V_2)^n \\ \frac{P_1}{P_2} &= \left( \frac{V_2}{V_1} \right)^n \end{aligned}$	$\frac{P_2 V_2 - P_1 V_1}{1-n}$	$\frac{mR(T_2 - T_1)}{1-n}$

Type	Condition(s)	\(W_b\)
Isochoric	\(V = c\)	\(0\)
Isobaric	\(P = c\)	\(P_1(V_2 - V_1)\)	\(mP_1(\nu_2 - \nu_1)\)
Isothermal	\(\begin{aligned} T &= c \\ PV &= mRT \\ P_1 V_1 &= P_2 V_2 \end{aligned}\)	\(P_i V_i \ \ln \vert \frac{V_2}{V_1} \vert \\ P_i V_i \ \ln \vert \frac{P_1}{P_2} \vert\)	\(mRT \ \ln \vert \frac{V_2}{V_1} \vert\) \(mRT \ \ln \vert \frac{P_1}{P_2} \vert\)
Polytropic	\(\begin{aligned} P V^n &= c \\ P_1 (V_1)^n &= P_2 (V_2)^n \\ \frac{P_1}{P_2} &= \left( \frac{V_2}{V_1} \right)^n \end{aligned}\)	\(\frac{P_2 V_2 - P_1 V_1}{1-n}\)	\(\frac{mR(T_2 - T_1)}{1-n}\)