# Work definition in thermodynamics

1. May 25, 2014

### ricard.py

Hello,
I have been self-learning Thermodynamics and I am having a bit of trouble with calculating the work in different circumstances.

Along the lectures we have come up with three different equations for work
1) W = pΔV
2) W = nRTln(V2/V1)
3) W = CvΔT

So my questions are:
1) which ones must be used in which type of thermodynamic process? For instance, the third is used in adiabatic processes, but the second
2) If using the second formula in a reaction that changes the temperature along it, we have to take as T the initial temperature, the last temperature, the difference..?
2) Accoding to the first equation, if V is constant, then W=0. However, according to the last formula the work only depends on T and we can get work done without modifying the volume. Why this is not contradictory?

Thanks!

2. May 25, 2014

### dauto

Formula 1) should be used for processes at constant pressure. Formula 2) is used for processes at constant temperature. Formula 3) is wrong. It should read Q = CvΔT, where Q is the heat exchanged in a constant volume process.

Last edited: May 25, 2014
3. May 25, 2014

### ricard.py

Ok thanks!
Concerning the third equation I forgot to say that it is in a context of a diabatic expansion (q=0). Therefore, ΔU=CvΔT=W.

Then in a diabatic expansion, we can have work only dependent on the T and not on the V. How does this not contradict the "classical" definition of W=pΔV?

4. May 25, 2014

### Andrew Mason

There is no sense in memorizing formulae. These all derive from the first law: Q = ΔU + W (where Q is the heat flow into the system, ΔU is the change in internal energy of the system and W is the work done BY the system that undergoes a change in thermodynamic states).

The first law applies between any two thermodynamic equilibrium states regardless of the process followed in moving between those two states. However it can be rather difficult to calculate these quantities if the thermodynamic properties are undefined during the process.

In the case of an expansion at constant pressure - where, for example, the work is done against constant atmospheric pressure - the work done BY the system is just W = ∫PdV = P∫V = PΔV. So Q = nCPΔT = ΔU + PΔV

In the case of an adiabatic expansion, Q = 0 so ΔU + W = 0 which means W = -ΔU. If you are dealing with an ideal gas where ΔU = nCVΔT then W = -nCVΔT

In the case of an isothermal compression of an ideal gas where P = nRT/V, the work done in compressing the gas ( -W = work done ON the system) is:

-W = - ∫PdV = - ∫(nRT/V)dV = -nRT∫dV/V = nRTln(V1/V2)

AM

Last edited: May 25, 2014
5. May 25, 2014

### dauto

duplicate post

Last edited: May 25, 2014
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