热力学第一定律

1. 准静态过程

中间态视为平衡态

无耗散的准静态过程是可逆的.

较快的过程近似为绝热过程.

1.1. 等体过程

吸收热量等于内能增量 \[C_{V, m} = \left( \frac{\mathrm{d}U}{\mathrm{d}T} \right)_V\]

1.2. 等压过程

吸收热量等于焓的增量 \[C_{p, m} = \left( \frac{\partial H}{\partial T} \right)_p, \ H = U + pV\]

1.3. 等体过程

过程中理想气体不做功 \(\delta Q = \nu C_{V, m}\mathrm{d}T\) \[\Rightarrow Q = \Delta U = \int_{T_1}^{T_2}\nu C_{V, m}\mathrm{d} T\]

1.4. 等压过程

\(\mathrm{d}W = -p \mathrm{d}V = \nu R\mathrm{d}T\)
\(\mathrm{d}U = \nu C_{V, m}\mathrm{d}T\)

\[\Rightarrow \nu C_{V, m}\mathrm{d}T + \nu R \mathrm{d}T\] \[\Rightarrow C_{p, m} - C_{V, m} = R\]

上式对理想气体成立.

1.5. 等温过程

\(PV = \text{const}\)
\(\mathrm{d} Q = p \mathrm{d}V = \nu (RT/V)\mathrm{d}V\)

\[\Rightarrow Q = \nu RT \ln \frac{V_2}{V_1}\]

1.6. 绝热过程

\(\mathrm{d} U = - p\mathrm{d}V = \nu C_{V, m}\mathrm{d}T\)
\(\mathrm{d}(pV) = \nu R\mathrm{d}T\)

\[\Rightarrow pV^{\gamma} = \text{const}\] 其中 \[\gamma = \frac{C_p}{C_V} > 1\]

走在绝热线上方的是吸热过程, vice versa.

\[W_{\text{adiabatic}} = \nu C_{V, m}(T_2 - T_1)\] 另外由 \(pV = \nu RT\) 可得: \[TV^{\gamma - 1} = c,\ \frac{p^{\gamma -1}}{T^{\gamma}} = c\] \[\Rightarrow W = \frac{p_2 V_2 - p_1 V_1}{\gamma - 1}\]

1.7. 多方过程

\(P V^n = c\)
\(TV^{n-1} = c\)
\(\frac{p^{n-1}}{T^n} = c\)

易得

等容: \(n = \infty\)
等压: \(n = 0\)
等温: \(n = 1\)
绝热: \(n = \gamma\)

\[C_n = \frac{n - \gamma}{n - 1} C_V\]

2. 热机与循环

热机效率: \[\eta = 1 - \frac{|Q_2|}{|Q_1|}\]
卡诺循环: 两个等温过程和两个绝热过程. \[\eta = 1- \frac{T_2}{T_1}\]
内燃机循环: \[\eta = 1 - \frac{\rho^{\gamma} - 1}{\gamma(\rho - 1)K^{\gamma - 1}}\] \[K = \left( \frac{V_1}{V_2} \right)\] \[\frac{T_3}{T_2} = \frac{V_3}{V_2} = \rho\]
制冷系数: \[\eta = \frac{|Q_2|}{W} = \frac{|Q_2|}{|Q_1| - |Q_2|}\]