4. In two identical cylinders closed with light pistons, there is the same amount of gases: in one - nitrogen, in the other - water vapor. Both cylinders are maintained at a constant temperature of $100^{\circ} \mathrm{C}$. At the initial moment, the pressure of the gas in both cylinders is 0.5 atm, the volume is 2 liters, and the distance from the bottom to the piston is 1 m. The pistons are pushed into the cylinders at a constant speed of 10 cm/min for 7.5 minutes. Determine the ratio of the powers of the external forces applied to the pistons at the end of the movement. Was there a time interval of $30 \mathrm{s}$ during which at least one external force did work greater than 15 J?
Answer: $2 ;$ Yes.

Solution. Consider the isothermal compression of nitrogen. Since the ratio of the volumes of the gas at the beginning and end of the motion $l /(l-u t)=\alpha=4$, the pressure of the nitrogen will be $p=\alpha p_{0}=2$ atm. Water vapor cannot exist at such a pressure at the given temperature, so condensation will begin when the pressure reaches the saturation vapor pressure, and the pressure in the cylinder with water vapor will be constant at $p_{v}=1$ atm.

The forces applied to the pistons are proportional to the pressures, the speeds of the pistons are the same, so the ratio of the powers is $p / p_{v}=2$.

In $\tau=30$ s, the volume changes by $10^{-4} \mathrm{~m}^{3}$. The pressure of the water vapor does not exceed $10^{5}$ Pa, so the work done by the force pushing the piston in the vapor does not exceed 10 J.

Let's calculate the average pressure required to perform the given work of compressing the nitrogen:

$$
A=p_{a} S v \tau=p_{a} \frac{V}{l_{0}} u \tau, \quad p_{a}=\frac{A l_{0}}{V u \tau}=1.5 \text { atm }
$$

During the time interval $\tau$, the pressure in the cylinder until the end of the compression is $p_{1}=\frac{p_{0} l_{0}}{l_{0}-u(t-\operatorname{tau})}=\frac{5}{3}$ atm. Since the pressure was almost always greater than $p_{1}$ during the last time interval $\tau$, the work done is greater than the specified work.

Answer to variant 172: 3 ; No.

Answer to variant 173: 2; Yes.

Answer to variant $174: 4 ;$ No.