A metallic vessel of volume V with a narrow outlet tube is immersed in a water bath.The vessel contains n moles of a gas at a high pressure P.The gas is allowed to leak out slowly to the atmosphere(pressure P') via the tube.The vesel and bath are maintained at const temperature equal to the atmospheric temperature throughout the process by dissipating heat through an electical resistor placed inside the bath. The resistance of the resistor is r and it carries a current I.Equilibrium is attained in a time T.

Obtain an expression for the change in internal energy of the n moles of the gas from the initial to final equilibrium state.Take the molar volume of the gas at atmospheric pressure to be V'.

To derive the expression for the change in internal energy of the gas as it leaks out of the vessel, we need to consider several key concepts from thermodynamics. The process involves a gas at high pressure leaking into an atmosphere at a lower pressure while maintaining a constant temperature. This scenario allows us to apply the principles of the first law of thermodynamics and the ideal gas law.

Understanding the System

We start with a metallic vessel containing n moles of gas at an initial pressure P and volume V. The gas is allowed to escape through a narrow outlet tube into the atmosphere, which has a pressure P'. Throughout this process, the temperature is kept constant, which indicates that we are dealing with an isothermal process.

First Law of Thermodynamics

The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W):

• ΔU = Q - W

Isothermal Process and Internal Energy

For an ideal gas, the internal energy is a function of temperature only. Since the temperature remains constant throughout the process, the change in internal energy for the gas can be expressed as:

• ΔU = 0

This means that there is no change in internal energy as the gas escapes, provided that the gas behaves ideally and the temperature remains constant.

Heat Transfer and Work Done

Even though the internal energy does not change, we need to consider the heat transfer and work done during the process. The gas does work as it expands against the atmospheric pressure when it leaks out. The work done by the gas can be calculated using the formula:

• W = P' * ΔV

Here, ΔV is the change in volume of the gas as it escapes. Since the gas is leaking out, we can express this in terms of the moles of gas that have escaped.

Calculating the Work Done

Let’s denote the final number of moles of gas remaining in the vessel as n_f. The initial volume of the gas is V = nRT/P, where R is the ideal gas constant. The final volume can be expressed as:

• V' = n_fRT/P'

The change in volume due to the gas escaping is:

• ΔV = V - V' = (n - n_f)RT/P'

Substituting this into the work done equation gives us:

• W = P' * (n - n_f)RT/P'

Thus, the work done simplifies to:

• W = (n - n_f)RT

Heat Transfer Calculation

Since the process is isothermal, the heat added to the system must equal the work done by the gas to maintain the temperature. Therefore:

• Q = W = (n - n_f)RT

Final Expression for Change in Internal Energy

Now, substituting back into the first law of thermodynamics, we find:

• ΔU = Q - W = 0

This confirms that the change in internal energy remains zero throughout the process, as expected for an ideal gas undergoing an isothermal expansion.

Summary

In conclusion, for the scenario described, the change in internal energy of the gas as it leaks out of the vessel is:

• ΔU = 0

This result is consistent with the principles of thermodynamics, particularly for an ideal gas undergoing an isothermal process. The heat added to the system equals the work done by the gas, resulting in no net change in internal energy.