03 Heat and Work
Addable Quantities¶
- mass
- volume
- U
- H
- \(u\) for closed system
note that specific quanties like \(h, u\) can not be added
Work¶
Spring¶
\[ \begin{aligned} F &= kx \\ W &= \frac{1}{2} k x^2 \\ &= \frac12 k ({x_2} ^2 - {x_1}^2) \\ \end{aligned} \]
Electric¶
\[ \begin{aligned} \dot W &= VI \\ W &= VI \Delta t \end{aligned} \]
Boundary Work¶
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 | \(\begin{aligned} P V^n &= c \\ P_1 (V_1)^n &= P_2 (V_2)^n \\ \frac{P_1}{P_2} &= \left( \frac{V_2}{V_1} \right)^n \end{aligned}\) | \(\frac{P_2 V_2 - P_1 V_1}{1-n}\) | \(\frac{mR(T_2 - T_1)}{1-n}\) |
Sign Convention¶
| Quantity | Sign |
|---|---|
| \(Q_\text{in}\) | + |
| \(Q_\text{out}\) | - |
| \(W_\text{in}\) | - |
| \(W_\text{out}\) | + |
| expansion | + |
| compression | - |