The relation ΔU = CV,avg ΔT, where ΔU is the change in internal energy, CV,avg is the average molar heat capacity at constant volume, and ΔT is the change in temperature, is specifically valid for constant volume processes of an ideal gas.
In thermodynamics, the heat capacity is defined as the amount of heat energy required to raise the temperature of a substance by a certain amount. The specific heat capacity (per unit mass) or molar heat capacity (per mole) depends on the conditions under which the heating occurs.
For an ideal gas, the heat capacity at constant volume (CV) is defined as the amount of heat required to raise the temperature of a given amount of gas by one degree Celsius (or one Kelvin) at constant volume. It is a characteristic property of the gas.
The relation ΔU = CV,avg ΔT is derived specifically for constant volume processes, where the volume remains constant throughout the process. In such a scenario, no work is done by the gas (W = 0), and therefore, the change in internal energy (ΔU) is equal to the heat energy transferred to the gas (Q).
For processes involving changes in volume (such as isobaric, isothermal, or adiabatic processes), the relationship between internal energy and temperature change is different. In those cases, one needs to consider the work done by or on the gas in addition to the heat transfer.
Therefore, while ΔU = CV,avg ΔT is applicable only to constant volume processes of an ideal gas, for other processes, different relationships and equations need to be used, taking into account the specific conditions and the work done.