Solution

Problem

$5.5.1.$ The volume of a gas was halved, and its temperature was increased by a factor of $1.5$. By what factor did the gas pressure increase?

Solution

The amount of gas in this process remains constant, so for the two states (index 1 for the initial state, index 2 for the final state), we can write the ideal gas equation of state:

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

According to the problem statement, we are given: $V_2 = \frac{1}{2}V_1$ and $T_2 = \frac{3}{2}T_1$.

Substituting these into the equation:

$$\frac{P_1V_1}{T_1} = \frac{P_2 \cdot \frac{1}{2}V_1}{\frac{3}{2}T_1}$$

Hence:

$$P_2 = 3P_1$$