Thermodynamics: Calculating the work done

1. Jun 16, 2013

Saitama

1. The problem statement, all variables and given/known data
One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a temperature $T$. The space over the piston opens into the atmosphere. What work has to be performed in order to increase isothermally the gas volume under the piston $n$ times by slowly raising the piston? The friction of piston against the cylinder walls is negligibly small.

2. Relevant equations

3. The attempt at a solution
Work done in isothermal process is
$$nRT\ln\frac{V_2}{V_1}$$
In the given question, $n=1 \text{mol}$ and $V_2=nV_1$. Hence work done is:
$$RT\ln n$$
but this is wrong.

2. Jun 16, 2013

I like Serena

Hey Pranav!

You have calculated the work done on the gas, but you have omitted the minus sign. :surprised
So if you (slowly) extract an amount of energy equal to $RT\ln n$ from the gas, the process will be isothermal.
This is not the total amount of work.

Note that it is a bit hard to force a change and still getting energy back instead of putting it into it.

3. Jun 16, 2013

voko

Where is the atmosphere in your equations?

4. Jun 16, 2013

Saitama

Thanks voko and ILS for the replies!

Okay, so there will be some work done on the atmosphere too. The change in volume of atmosphere is $(n-1)V_1$. The work done on the atmosphere is $PV_1(n-1)$ where P is the initial pressure. Also $PV_1=RT$, hence net work done is $(n-1)RT-RT\ln n=RT(n-1-\ln n)$, correct?

5. Jun 16, 2013

voko

This looks good to me.

6. Jun 16, 2013

I like Serena

Yep. Looks good.

7. Jun 16, 2013

Saitama

Thanks!

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