664. Under the conditions of the previous problem, determine the work of adiabatic compression of a gas, in which its volume $v$ and pressure $p$ are related by the equation $p v^{k}=c=$ const (Poisson's law), where $k$ is a constant for the given gas, greater than one. (For air, $k \approx 1.4$.)
Solution. Repeating the same reasoning and using the same notation as in the solution of the previous problem, we find the following expression for the differential of work:

$$
d q(x)=\frac{c d x}{S^{k-1}(H-x)^{k}}
$$

Integrating from $x=0$ to $x=h$, we obtain the entire sought work

$$
\begin{aligned}
& Q=\frac{c}{S^{k-1}} \int_{0}^{n} \frac{d x}{(H-x)^{k}}=\frac{c}{S^{k-1}} \int_{t}^{n}(H-x)^{-k} d(H-x)= \\
& =\left.\frac{c}{S^{k-1}} \frac{(H-x)^{1-k}}{1-k}\right|_{11} ^{0}=\frac{p_{0} v_{0}^{k}}{S^{k-1}(k-1)}\left[\frac{1}{(H-h)^{k-1}}-\frac{1}{H^{k-1}}\right]= \\
& =\frac{\rho_{0} v_{0}}{k-1}\left[\left(\frac{H}{H-h}\right)^{k-1}-1\right] .
\end{aligned}
$$

Assuming $H=1.5 \, \text{m}, k=1.4$, we find

$$
Q \approx \frac{2479.2 \pi}{0.4}\left[\left(\frac{1.5}{0.3}\right)^{0.4}-1\right] \approx 17593.4 \kappa \Gamma \mathcal{M} \approx 172591.3 \partial \varkappa
$$

Comparing this result with the previous ones shows that the work spent on the adiabatic compression of the gas is greater,

![](https://cdn.mathpix.com/cropped/2024_05_22_8a069bcd8684ea7ea3dbg-215.jpg?height=309&width=270&top_left_y=1094&top_left_x=105)

than during isothermal compression.