To solve this problem, we need to understand the relationship between $C_P$ and $C_V$ for one mole of an ideal gas.$\\ \\$For an ideal gas, the specific heat capacities at constant pressure $(C_P)$ and constant volume $(C_V)$ are related by the equation:$\\ C_P - C_V = R \\ \\$where $R$ is the universal gas constant.$\\ \\$Let's break down why this relationship holds:$\\ •$The specific heat capacity at constant volume, $C_V,$ is defined as the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius at constant volume.$\\ •$The specific heat capacity at constant pressure, $C_P,$ is defined as the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius at constant pressure.$\\ •$When heating a gas at constant pressure, work is done by the gas as it expands. This requires additional energy compared to heating at constant volume.$\\ •$The additional energy required is precisely the work done by the gas, which is equal to $R$ for one mole of an ideal gas.$\\ \\$Therefore, the correct relationship is:$\\ C_P - C_V = R \\ \\$This corresponds to Option 2.