Calculate the internal energy change of one mole of an ideal gas during an adiabatic free exmapnsion process from a volume of 2 litre to 5 litre.

To determine the internal energy change of one mole of an ideal gas during an adiabatic free expansion from a volume of 2 liters to 5 liters, we need to consider the nature of the process. In an adiabatic free expansion, the gas expands into a vacuum without doing any work and without heat exchange with the surroundings. This unique scenario leads us to a straightforward conclusion regarding the internal energy change.

Understanding Internal Energy in Ideal Gases

The internal energy of an ideal gas is primarily a function of its temperature. For an ideal gas, the internal energy (U) can be expressed as:

• U = n * C_v * T

Here, n is the number of moles, C_v is the molar heat capacity at constant volume, and T is the temperature in Kelvin. Since we are dealing with an ideal gas and the process is adiabatic, we need to focus on the implications of the free expansion.

Characteristics of Adiabatic Free Expansion

In an adiabatic free expansion:

• No heat is exchanged with the surroundings (Q = 0).
• No work is done on or by the gas (W = 0).

According to the first law of thermodynamics, which states:

• ΔU = Q - W

Substituting the values for our adiabatic free expansion process:

• ΔU = 0 - 0 = 0

Conclusion on Internal Energy Change

Since there is no heat transfer and no work done, the internal energy change (ΔU) of the gas during this process is zero. This means that even though the gas expands from 2 liters to 5 liters, its internal energy remains unchanged, provided that the temperature does not change during the process. In essence, the internal energy of one mole of an ideal gas undergoing adiabatic free expansion remains constant.

Final Thoughts

This concept highlights the unique characteristics of ideal gases and adiabatic processes. In real-world applications, while ideal gas behavior can be a useful approximation, it’s essential to remember that deviations can occur under certain conditions, particularly at high pressures or low temperatures. However, for our ideal gas in this scenario, the internal energy change is definitively zero.