There is a heat-insulating walled container filled with a normal-state gas of volume $V$. The container has a small hole through which the gas can flow unimpeded. How much heat must be transferred to increase the temperature of the gas in the container to $T_{1}$? The molar mass of the gas is $M$, and its specific heat at constant pressure is $c_{p}$. The external pressure is $1 \mathrm{~atm}$. (The walls of the container do not absorb heat.)
Assuming that the gas is in equilibrium with itself during heating, the state equation for the gas is:

$$
p V=(m / M) R T
$$

where $m$ is the mass of the gas, and $M$ is the mass of a molar quantity of the gas. During the process, the pressure and volume of the gas remain constant. The mass, due to the above relationship, is inversely proportional to the absolute temperature

$$
m=(p V M / R)(1 / T)
$$

The heat absorbed by the gas is proportional to the mass and the change in temperature

$$
\Delta Q=m c_{p} \Delta T=c_{p} \cdot(p V M / R T) \Delta T
$$

This relationship is valid for a small change in temperature, which causes only a negligible change in the mass of the gas. The heat absorbed from $T_{0}$ to $T_{1}$ can be obtained as follows. Divide the interval $\left(T_{0}, T_{1}\right)$ into small parts using division points $\tau_{1}, \tau_{2}, \ldots, \tau_{n-1}$, where

$$
T_{0}=\tau_{0}<\tau_{1}<\tau_{2}<\ldots<\tau_{n-1}<\tau_{n}=T_{1}
$$

Then the heat absorbed between $T_{0}$ and $T_{1}$ is the sum of the heat absorbed in each $\left(\tau_{j-1}, \tau_{j}\right)$ interval, which is approximately

$$
\sum_{j=1}^{n} c_{p}\left(p V M / R \tau_{j}\right)\left(\tau_{j}-\tau_{j-1}\right)
$$

(The approximation is more accurate the finer the division of the $\left(T_{0}, T_{1}\right)$ interval.) This sum approaches the

$$
\int_{T_{0}}^{T_{1}} c_{p}(p V M / R T) \mathrm{d} T
$$

integral as the division of the $\left(T_{0}, T_{1}\right)$ interval is refined beyond any limit. Therefore, the sought heat is

$$
Q=c_{p} \frac{p V M}{R} \int_{T_{0}}^{T_{1}} \frac{\mathrm{d} T}{T}=c_{p} \cdot \frac{p V M}{R} \ln \frac{T_{1}}{T_{0}}
$$

Attila Virosztek (Szolnok, F. Verseghy Gymnasium, 4th grade)