POINT
【関連記事】
- 音速を熱力学の観点から見てみる．
- 音速を等温変化で表す．
断熱圧縮率と等温圧縮率
断熱圧縮率と等温圧縮率
\begin{aligned}
-\frac{1}{v} \biggl(\frac{\partial v}{\partial p} \biggr)_{\mathrm{ad}}
&=\frac{c_{p}}{c_{v}}
\biggl[ -\frac{1}{v} \biggl(\frac{\partial v}{\partial p} \biggr)_{T} \biggr]
\end{aligned}
-\frac{1}{v} \biggl(\frac{\partial v}{\partial p} \biggr)_{\mathrm{ad}}
&=\frac{c_{p}}{c_{v}}
\biggl[ -\frac{1}{v} \biggl(\frac{\partial v}{\partial p} \biggr)_{T} \biggr]
\end{aligned}
$v = 1 / \rho$を比容（単位質量あたりの体積），$u$を単位質量あたりの内部エネルギーとすると
\begin{aligned}
\mathrm{d}^{\prime} q
&= \mathrm{d}u + p \,\mathrm{d}v
\end{aligned}
\mathrm{d}^{\prime} q
&= \mathrm{d}u + p \,\mathrm{d}v
\end{aligned}
これより，定積比熱と定圧比熱はそれぞれ
\begin{aligned}
c_{v}
&= \biggl(\frac{\mathrm{d}^{\prime} q}{\mathrm{d}T}\biggr)_{v}
= \biggl(\frac{\partial u}{\partial T} \biggr)_{v} \\
c_{p}
&= \biggl(\frac{\mathrm{d}^{\prime} q}{\mathrm{d}T}\biggr)_{p}
= \biggl(\frac{\partial u}{\partial T} \biggr)_{p}
+ p \biggl(\frac{\partial v}{\partial T} \biggr)_{p}
\end{aligned}
となる．c_{v}
&= \biggl(\frac{\mathrm{d}^{\prime} q}{\mathrm{d}T}\biggr)_{v}
= \biggl(\frac{\partial u}{\partial T} \biggr)_{v} \\
c_{p}
&= \biggl(\frac{\mathrm{d}^{\prime} q}{\mathrm{d}T}\biggr)_{p}
= \biggl(\frac{\partial u}{\partial T} \biggr)_{p}
+ p \biggl(\frac{\partial v}{\partial T} \biggr)_{p}
\end{aligned}
また，$u = u(p, v)$とすると
\begin{aligned}
\mathrm{d}^{\prime} q
&= \biggl(\frac{\partial u}{\partial p} \biggr)_{v} \,\mathrm{d}p
+ \biggl[p + \biggl(\frac{\partial u}{\partial v} \biggr)_{p} \biggr]
\,\mathrm{d}v
\end{aligned}
断熱過程では$\mathrm{d}^{\prime} q = 0$だから\mathrm{d}^{\prime} q
&= \biggl(\frac{\partial u}{\partial p} \biggr)_{v} \,\mathrm{d}p
+ \biggl[p + \biggl(\frac{\partial u}{\partial v} \biggr)_{p} \biggr]
\,\mathrm{d}v
\end{aligned}
\begin{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
&= - \biggl[p + \biggl(\frac{\partial u}{\partial v} \biggr)_{p} \biggr]
\biggl(\frac{\partial u}{\partial p} \biggr)_{v}^{-1} \\
&=-\biggl[p \biggl(\frac{\partial v}{\partial T} \biggr)_{p}
+ \biggl(\frac{\partial u}{\partial T} \biggr)_{p} \biggr]
\biggl(\frac{\partial T}{\partial v} \biggr)_{p} \\
&\qquad \times
\biggl[\biggl(\frac{\partial u}{\partial T} \biggr)_{v}
\biggl(\frac{\partial T}{\partial p} \biggr)_{v} \biggr]^{-1} \\
&=-c_{p} \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
\biggl / c_{v} \biggl(\frac{\partial T}{\partial p} \biggr)_{v}
\tag{1}
\end{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
&= - \biggl[p + \biggl(\frac{\partial u}{\partial v} \biggr)_{p} \biggr]
\biggl(\frac{\partial u}{\partial p} \biggr)_{v}^{-1} \\
&=-\biggl[p \biggl(\frac{\partial v}{\partial T} \biggr)_{p}
+ \biggl(\frac{\partial u}{\partial T} \biggr)_{p} \biggr]
\biggl(\frac{\partial T}{\partial v} \biggr)_{p} \\
&\qquad \times
\biggl[\biggl(\frac{\partial u}{\partial T} \biggr)_{v}
\biggl(\frac{\partial T}{\partial p} \biggr)_{v} \biggr]^{-1} \\
&=-c_{p} \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
\biggl / c_{v} \biggl(\frac{\partial T}{\partial p} \biggr)_{v}
\tag{1}
\end{aligned}
ここで
\begin{aligned}
\mathrm{d}T
&= \biggl(\frac{\partial T}{\partial v} \biggr)_{p} \,\mathrm{d}v
+\biggl(\frac{\partial T}{\partial p} \biggr)_{v} \,\mathrm{d}p
\end{aligned}
から\mathrm{d}T
&= \biggl(\frac{\partial T}{\partial v} \biggr)_{p} \,\mathrm{d}v
+\biggl(\frac{\partial T}{\partial p} \biggr)_{v} \,\mathrm{d}p
\end{aligned}
\begin{aligned}
0
&= \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
+\biggl(\frac{\partial T}{\partial p} \biggr)_{v}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
0
&= \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
+\biggl(\frac{\partial T}{\partial p} \biggr)_{v}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
\begin{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
&= - \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
\biggl(\frac{\partial T}{\partial p} \biggr)_{v}^{-1}
\tag{2}
\end{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
&= - \biggl(\frac{\partial T}{\partial v} \biggr)_{p}
\biggl(\frac{\partial T}{\partial p} \biggr)_{v}^{-1}
\tag{2}
\end{aligned}
式(1), (2)から
\begin{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
&=\frac{c_{p}}{c_{v}} \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
&=\frac{c_{p}}{c_{v}} \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
音速と等温変化
音速を$c$，$\gamma = c_{p} / c_{v} $とすると\begin{aligned}
c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
\biggl(\frac{\partial v}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\gamma \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\biggl(\frac{\partial v}{\partial \rho} \biggr)_{T}
=\gamma \biggl(\frac{\partial p}{\partial \rho} \biggr)_{T}
\end{aligned}
c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\biggl(\frac{\partial p}{\partial v} \biggr)_{\mathrm{ad}}
\biggl(\frac{\partial v}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\gamma \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\biggl(\frac{\partial v}{\partial \rho} \biggr)_{T}
=\gamma \biggl(\frac{\partial p}{\partial \rho} \biggr)_{T}
\end{aligned}
$v = 1 / \rho$だから
\begin{aligned}
c^{2}
&=\gamma \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\biggl(-\frac{1}{\rho^{2}}\biggr) \\
&=-\frac{\gamma}{\rho^{2}}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
となる．c^{2}
&=\gamma \biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\biggl(-\frac{1}{\rho^{2}}\biggr) \\
&=-\frac{\gamma}{\rho^{2}}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
\end{aligned}
