# The work done by a perfect gas

**1. The problem statement, all variables and given/known data**

An ideal gas at initial temperature T

_{1}and pressure P

_{1}is compressed by a piston to half its original volume. The temperature is varied so that the relation P=AV always holds and A is a constant. What is the work done n the gas in terms of n (moles of gas) R (gas constant) and T

_{1}?

**2. Relevant equations**

Using ideal gas law, PV= nRT

and assuming the work done is [itex]W = \int^{V_b}_{V_a} P dV[/itex]

**3. The attempt at a solution**

OK, simple enough, right?

V_{b} = (1/2) V_{a}

So V = nRT/P and [itex]V_a = \frac{nRT}{P_1}[/itex] and [itex]V_b = \frac{nRT}{2P_1}[/itex]

Plug this into the integral above. Since P=AV (in the givens) it should be

[itex]W = \int^{V_b}_{V_a} (AV) dV = A \int^{V_b}_{V_a} V dV = A\frac{V^2}{2}|^{V_a}_{V_b}[/itex]

Which leaves me with

[itex]=\frac{A}{2} \left[\left(\frac{nRT}{2P_1}\right)^2 – \left(\frac{nRT}{P_1}\right)^2\right] = \frac{A}{2}\left(\frac{nRT}{P_1}\right)^2(-3/4) = \frac{-3A}{8}\left(\frac{nRT}{P_1}\right)^2[/itex]

Yet the answer is listed as [itex]\frac{-3A}{8}\left(\frac{nRT}{P_1}\right)[/itex]

So I am trying to figure out how I got the extra factor in there. I think I did everything right, but is there some stupid mathematical error I made?

Anyhow, it’s possible there’s a typo in the book’s answer too. But…

any help is appreciated, even though I anticipate it will be trivial…

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