663. A cylinder with a height of $H=1.5$ m and a radius of $R=0.4$ m, filled with gas at atmospheric pressure ( $10330 \kappa \Gamma / \boldsymbol{M}^{2}$ ), is closed with a piston. Determine the work spent on isothermal compression of the gas when the piston is moved a distance of $h=1.2$ m inside the cylinder.
Solution. During an isothermal change of state of a gas, when its temperature remains constant, the dependence between the volume $v$ and the pressure $p$ of the gas is expressed by the formula $p v = c = \text{const}$. (Law of

![](https://cdn.mathpix.com/cropped/2024_05_22_8a069bcd8684ea7ea3dbg-214.jpg?height=231&width=381&top_left_y=537&top_left_x=807)

Fig. 123 Boyle-Mariotte.)

Therefore, if the piston is pushed in by $x$ m into the cylinder (Fig. 123), the pressure $p(x)$ of the gas per unit area of the piston will be $p(x) = \frac{c}{v(x)} = \frac{c}{S(H-x)}$, and the pressure on the entire area $S$ of the piston will be $P(x) = S p(x) = \frac{c}{H-x}$.

Assuming that the work spent on pushing the piston in by $x \mu$ is some function $q(x)$, and allowing that during further pushing of the piston by a small distance $dx$ the pressure $P(x)$ experienced by it remains unchanged, we find the approximate value (differential) of the increment of the function $q(x)$:

$$
\Delta q \approx P(x) dx = \frac{c}{H-x} dx = dq
$$

The entire sought work $Q$ corresponds to the change of $x$ from 0 to $h$, therefore

$$
Q = c \int_{0}^{h} \frac{dx}{H-x} = -\left. c \ln (H-x) \right|_{0}^{h} = c \ln \frac{H}{H-h}
$$

For $H = 1.5 \text{ m}, R = 0.4 \text{ m}, h = 1.2 \text{ m}, p_{0} = 10330 \kappa \Gamma / \mu^{2}$, we find $v_{0} = \pi R^{2} H = 0.24 \pi \text{ m}^{3}; c = p_{0} v_{0} = 2479.2 \pi; Q \approx 12533.3 \kappa \Gamma \mathcal{M} \approx 122951.7$ J.