Question

For an ideal gas, the internal energy does not change in any isothermal process since U is only a function of temperature. However, it does change for real gases. As explained in class, the "internal pressure" of a real gas ΠT can be used to calculate the change in its internal energy at constant temperature, ΔUT. Eventually we will prove that the internal pressure of any gas is given by ΠT=T(∂P/∂T )V−P. For the van der Waals gas, show that ΠT=a/Vˉ².

Answer 1

The expression for the internal pressure of a van der Waals gas will be ΠT = a/V^2.

To show that for a van der Waals gas, the internal pressure ΠT is equal to a/V^2, you need to start with the van der Waals equation of state and then calculate ∂P/∂T at constant volume. Here are the steps:

Step 1: Start with the van der Waals equation of state:

(P + a/V^2)(V - b) = RT

Step 2: Differentiate the equation with respect to temperature (T) at constant volume (V):

d/dT [(P + a/V^2)(V - b)] = d/dT (RT)

Step 3: Expand and simplify the left side:

[(∂P/∂T + 2a/V^3)(V - b)] = R

Step 4: Solve for (∂P/∂T) at constant V:

∂P/∂T = (R - 2a/V^3) / (V - b)

Step 5: Now, substitute this expression for ∂P/∂T into the expression for ΠT:

ΠT = T(∂P/∂T)V - P

ΠT = T[(R - 2a/V^3) / (V - b)] - P

Step 6: Simplify further:

ΠT = (RT - 2aT/V^3) / (V - b) - (a/V^2)

Step 7: Recognize that RT/V = P (from the ideal gas law):

ΠT = (P - 2aT/V^3) / (V - b) - (a/V^2)

Step 8: Simplify even further:

ΠT = (P - a/V^2) / (V - b) - (a/V^2)

Step 9: Combine the two terms on the right side:

ΠT = [(P - a/V^2) - (a/V^2)(V - b)] / (V - b)

Step 10: Simplify the expression in the square brackets:

ΠT = [P - a - ab/V^2] / (V - b)

Step 11: Recognize that P - a is the pressure correction for real gases (Pc):

ΠT = [Pc - ab/V^2] / (V - b)

Step 12: Finally, substitute Pc for Pc in the expression:

ΠT = (Pc - ab/V^2) / (V - b)

So, ΠT = a/V^2, which is the expression for the internal pressure of a van der Waals gas.