Example 7. In a cylindrical vessel with a volume of $V_{0}$, atmospheric air is adiabatically (without heat exchange with the environment) compressed to a volume of $V_{1}$. Calculate the work of compression.
Solution. It is known that the adiabatic process is characterized by the Poisson equation

$$
\frac{p}{p_{0}}=\left(\frac{V_{0}}{V}\right)^{k}
$$

where $V_{0}$ is the initial volume of the gas, $p_{0}$ is the initial pressure of the gas, and $k$ is a constant for the given gas. Let $V$ and $p$ denote the volume and pressure of the gas at the moment when the piston is at height $h$, and let $S$ be the area of the piston. Then, when the piston is lowered by an amount $d h$, the volume of the gas decreases by an amount $d V = S d h$. The work done in this process is

$$
d W=-p S d h, \quad \text { or } \quad d W=-p d V
$$

Finding $p$ from (3) and substituting it into (4), we obtain the differential equation of the process

$$
d W=-\frac{p_{0} V_{0}^{k}}{V^{k}} d V
$$

Integrating this equation, we get

$$
W=-p_{0} V_{0}^{k} \int \frac{d V}{V^{k}}=\frac{p_{0} V_{0}^{k}}{(k-1) V^{k-1}}+C, \quad k \neq 1
$$

According to the initial condition $\left.W\right|_{V=V_{0}}=0$, from (5) we obtain

$$
C=-\frac{p_{0} V_{0}}{(k-1)}
$$

Thus, the work of adiabatic compression (from $V_{0}$ to $V$) will be

$$
W=\frac{p_{0} V_{0}}{k-1}\left[\left(\frac{V_{0}}{V}\right)^{k-1}-1\right]
$$

For $V=V_{1}$, we get

$$
W_{1}=\frac{p_{0} V_{0}}{k-1}\left[\left(\frac{V_{0}}{V_{1}}\right)^{k-1}-1\right]
$$