An ideal gas expands according to the law ${\mathrm{P}}^2\mathrm{V}= \mathrm{constant}$  On expansion, the temperature

An ideal gas expands according to the law P2V = constant, which implies that the product of the square of the pressure (P²) and the volume (V) of the gas remains constant throughout the expansion. This relationship suggests that, as the gas expands, the volume of the gas must increase. The pressure, therefore, adjusts in such a way that this condition is maintained.

In the process of expansion, an ideal gas expands according to the law P2V, meaning the gas must either cool down or warm up depending on the specific thermodynamic conditions and the nature of the expansion. For an ideal gas, the temperature and volume are related through Charles's Law, which states that for a fixed amount of gas, the volume is directly proportional to the temperature when the pressure is constant.

Therefore, when an ideal gas expands according to the law P2V, the volume increases, and as per Charles’s Law, the temperature of the gas also increases. This is because the increased volume requires the gas particles to spread out more, which, in turn, requires more energy in the form of heat to maintain the kinetic energy of the particles, leading to a temperature rise.

To summarize, in the case where an ideal gas expands according to the law P2V, both the volume and the temperature of the gas increase during expansion, assuming the process happens in a manner where heat is not lost to the surroundings.

This analysis shows that an ideal gas expands according to the law P2V, and it results in a temperature increase due to the direct relationship between volume and temperature as described by Charles's Law.