To determine the change in internal energy when the volume of a gas changes from V to 2V at constant pressure, we need to consider the relationship between internal energy, volume, and temperature in the context of the first law of thermodynamics. The internal energy of an ideal gas is primarily a function of its temperature, and for an ideal gas, the change in internal energy (ΔU) can be expressed as:

Understanding Internal Energy Changes

For an ideal gas, the change in internal energy is given by the equation:

ΔU = nC_vΔT

where:

• n is the number of moles of the gas.
• C_v is the molar heat capacity at constant volume.
• ΔT is the change in temperature.

Applying the Ideal Gas Law

At constant pressure, we can use the ideal gas law, which states:

PV = nRT

Here, R is the universal gas constant. If the volume changes from V to 2V while keeping the pressure constant, we can analyze how this affects the temperature:

Calculating the Temperature Change

Initially, we have:

P * V = n * R * T_initial

When the volume doubles:

P * (2V) = n * R * T_final

From these two equations, we can express the temperatures:

• T_initial = PV / (nR)
• T_final = 2PV / (nR)

Finding the Change in Temperature

The change in temperature (ΔT) is then:

ΔT = T_final - T_initial = (2PV / (nR)) - (PV / (nR)) = PV / (nR)

Calculating the Change in Internal Energy

Now, substituting ΔT back into the equation for ΔU:

ΔU = nC_v * (PV / (nR))

Since we are looking for a general expression, we can simplify this further. For a monatomic ideal gas, C_v = (3/2)R, and for a diatomic gas, C_v = (5/2)R. However, without loss of generality, we can express the change in internal energy in terms of pressure and volume:

Final Expression for Change in Internal Energy

Thus, we can express the change in internal energy as:

ΔU = (PV / R) * C_v

For a monatomic gas, this results in:

ΔU = (3/2)PV

For a diatomic gas, it would be:

ΔU = (5/2)PV

Identifying the Correct Option

Given the options provided:

• (a) PV
• (b) 3PV
• (c) PV / -1
• (d) RV / -1

The most appropriate answer, assuming we are dealing with a monatomic ideal gas, is:

(b) 3PV

In summary, when the volume of an ideal gas doubles at constant pressure, the change in internal energy is directly related to the pressure and the initial volume, leading us to conclude that the correct answer is 3PV for a monatomic gas. This illustrates the fundamental principles of thermodynamics and how they apply to real-world scenarios involving gases.