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SoggyBottoms

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## Homework Statement

We have an ideal gas of N particles with mass m and temperature T and volume V.

a) Calculate [itex]\langle E_{kin} \rangle[/itex]

We now reversibly compress the gas from volume V to V/2. During this compression heat Q is added, work W is done on the gas and the energy of the gas changes by [itex]\Delta E[/itex].

b) Calculate Q, W and [itex]\Delta E[/itex] in case the compression is isobaric.

## The Attempt at a Solution

a) This one I know how to do, the answer is [itex]\langle E_{kin} \rangle = \frac{3}{2}N k_B T[/itex]

b) The change is isobaric and the work done on the gas is positive, so [itex]W = p \Delta V = p(V - \frac{V}{2}) = \frac{N k_B \Delta T}{2}[/itex].

We also have that [itex]\Delta E = \Delta U = C_V \Delta T = \left(\frac{\partial \langle E_{kin} \rangle}{\partial T}\right)_V \Delta T = \frac{N k_B \Delta T}{2}[/itex].

Now: [tex]\Delta Q = \Delta U + \Delta W = \Delta T (\frac{N k_B}{2} + \frac{N k_B}{2}) \\

= N k_B \Delta T[/tex]

So [itex]Q = N k_B T[/itex]

Is this correct?