Answer van der Waals' Eqn State for Adiabatic Expansion Temperature

[sty2004]

Homework Statement

The van der Waals’ equation of state for one mole of gas is given by
(P+a/V²)(V-b)=RT
where V is the molar volume at temperature T, and a and b are constants. The internal energy of the gas is given by
E=(3/2)RT-a/V
Initially, one mole of the gas is at a temperature T1 and occupies a volume V1. The gas is allowed to expand adiabatically into a vacuum so that it occupies a total volume V2. What is the final temperature of the gas?

Homework Equations

The Attempt at a Solution

I guess the energy of the gas is constant as it is an adiabatic expansion(right?..).
So E₁=E₁ and so T₂=T₁+2a/(3R)(1/V₂-1/V₁)
Do I need the first equation provided?

 

[hikaru1221]

I guess the energy of the gas is constant as it is an adiabatic expansion(right?..).

Not really. Adiabatic only means no HEAT exchange.

The question is not very clear. I think the situation is like this: You have a fixed, closed and heat-insulating chamber of volume V1 containing the gas of temperature T1, and another fixed, closed and heat-insulating chamber of volume V2 containing chamber V1. The space between the two chambers is vacuum. You somehow open chamber V1 and wait until the gas gets back to equilibrium state. At that time, you are to find temperature T2 of the gas.

In that situation, the gas neither exchanges heat with the chambers (because the chambers are heat-insulating; this means the expansion is adiabatic) nor does work on the chambers (because the chambers are fixed; this is one point the question didn't mention). Therefore:
Internal energy of the gas before release = Internal energy of the gas when coming to equilibrium at T2.
If that's so, you won't need the first equation.

However, that doesn't mean internal energy of the gas remains the same throughout the expansion.