Reversible Adiabatic Expansion of a Mono-atomic Gas

Introduction:
In thermodynamics, adiabatic processes are those in which no heat exchange takes place between the system and its surroundings. Reversible adiabatic expansion is a type of adiabatic process in which the system expands slowly enough that it can be considered to be in equilibrium at all times. In this process, the internal energy of the system changes due to the work done by the system on its surroundings.

Relationship between Volume and Internal Energy:
The relationship between the volume V and the internal energy U of a perfect mono-atomic gas undergoing reversible adiabatic expansion can be expressed mathematically as follows:

U = CV(T^(γ-1))

where U is the internal energy of the gas, V is the volume of the gas, T is its temperature, C is the specific heat capacity of the gas at constant volume, and γ is the ratio of the specific heat capacities at constant pressure and constant volume.

Explanation:
The internal energy of a gas is a measure of the kinetic energy of its molecules. During reversible adiabatic expansion, the gas expands against a piston or other external force, doing work on its surroundings. As a result, the kinetic energy of the gas molecules decreases, and the temperature of the gas also decreases. However, since the process is adiabatic, there is no heat exchange between the gas and its surroundings, so the total internal energy of the gas remains constant.

The specific heat capacity of a gas is a measure of the amount of energy required to raise its temperature by a certain amount. Since the gas is undergoing reversible adiabatic expansion, its volume is changing slowly enough that it can be considered to be in equilibrium at all times. Therefore, the specific heat capacity of the gas at constant volume can be used to calculate the change in internal energy during the expansion process.

The relationship between volume and internal energy during reversible adiabatic expansion is given by the equation U = CV(T^(γ-1)). This equation shows that the internal energy of the gas is directly proportional to its volume, and also depends on the temperature and specific heat capacity of the gas. As the gas expands during the adiabatic process, its volume increases, and its internal energy decreases. However, the decrease in internal energy is proportional to the increase in volume, so the relationship between volume and internal energy remains constant throughout the process.

Conclusion:
In conclusion, the relationship between the volume V and the internal energy U of a perfect mono-atomic gas undergoing reversible adiabatic expansion can be expressed mathematically as U = CV(T^(γ-1)). This equation shows that the internal energy of the gas is directly proportional to its volume, and also depends on the temperature and specific heat capacity of the gas. During reversible adiabatic expansion, the gas expands slowly enough that it can be considered to be in equilibrium at all times, and the total internal energy of the gas remains constant.