Although the ideal gas is useful, other, more realistic descriptions of gases ex

Question

Although the ideal gas is useful, other, more realistic descriptions of gases exist. A good example is the van der Waals gas, which for a monatomic gas is defined by the following equations P = NkT/V - Nb - aN^2/V^2, U = 3/2NkT - aN^2/V The quantities a and b are constants which depend on the particular gas. Below we consider thermodynamic processes where N is always fixed. (a) Consider isothermal expansion of the van der Waals gas from volume V_a to V_b, at temperature T_0. What is the work done on the gas? (b) Determine the change in internal energy for the isothermal expansion described above, and hence determine the heat added to the gas. (c) Consider expansion from volume V_a to V_b along the line T^3/2(V - Nb) = C, where C is a constant. What is the work done on the gas, and the change in internal energy? What type of process is this?

Explanation / Answer

a)

Work done in an isothermal expansion = PdV (We will simply use integration)

Using the given expression for pressure,

NkT dV/(V-Nb) -aN^2 dV/V^2

= NkT ln( Vb/ Va) + aN^2[ 1/Vb - 1/Va] { T is the value given in the question, and it's value isn't changing as it is isothermal. }

b)

Using the given expression for U, the term 3 NkT/ 2 will cancel out (isothermal process).

We will be left with: aN^2 ( 1/Va - 1/Vb)

Heat added:

Q = U + W

We get NkT ln( Vb/ Va). (The second term from work done cancels out from internal energy term)

c)

This is an adiabatic process.

dQ = 0, for this process.

So, dU = -dW

Work done = PdV

It can be written as P^3/2 V^3/2 (V-Nb) = C

So, P = C / V (V-Nb)^2/3

Again, Change in internal energy will be negative of work done.

Since the question asks us to write P as a function of V only, so this should be the final answer.

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