Free Expansion of an Ideal Gas under Adiabatic Conditions

In a free expansion of an ideal gas, the gas expands into a larger volume without any external work being done on it and without any heat exchange with the surroundings. This process is also known as an irreversible expansion. The adiabatic condition implies that there is no heat transfer between the gas and its surroundings during the expansion.

Effect on Internal Energy

During free expansion under adiabatic conditions, the internal energy of the gas remains constant. This means that the internal energy before and after the expansion is the same.

The internal energy of an ideal gas depends only on its temperature. Since there is no heat transfer during free expansion, the temperature of the gas remains constant. Therefore, the internal energy also remains constant, as it is directly proportional to the temperature.

Explanation

When a gas expands into a larger volume without any external work being done on it, the gas molecules move farther apart, resulting in a decrease in the density of the gas. As a result, the gas molecules have more space to move around and their average kinetic energy remains the same.

In an adiabatic process, the equation of state for an ideal gas can be given by:
P1V1^γ = P2V2^γ

where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and γ is the adiabatic index (specific heat ratio).

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As a result, the ratio of specific heat capacities (γ) remains constant during the expansion. This implies that the pressure and volume are inversely proportional to each other:
P1V1^γ = P2V2^γ

Therefore, the decrease in pressure is compensated by the increase in volume, resulting in the constant internal energy of the gas.

Conclusion

In conclusion, during free expansion of an ideal gas under adiabatic conditions, the internal energy of the gas remains constant. This is because there is no heat transfer during the expansion and the temperature of the gas remains constant. The decrease in pressure is compensated by the increase in volume, resulting in no change in the internal energy of the gas.