Specific heat capacity at constant pressure

[v_pino]

I've read that:

specific heat capacity at constant pressure = dU-W / m. dT

dU = change in internal energy
W = work done
m = mass of gas
dT = change in temperature

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However, shouldn't the right hand side equate to specific heat capacity at constant VOLUME?

By saying that it is specific heat capacity at constant pressure, I thought we have already taken into account the energy used to expand the gas.

 

[Mapes]

Specific heat capacity at constant pressure, by definition, is

$$c_P=\frac{1}{m}\left(\frac{\partial H}{\partial T}\right)_P=\frac{1}{m}\left(\frac{\partial (U+PV)}{\partial T}\right)_P=\frac{1}{m}\left(\frac{\partial U}{\partial T}\right)_P+\frac{P}{m}\left(\frac{\partial V}{\partial T}\right)_P$$

If $c_P$ is constant, we can integrate and get

$$c_P=\frac{\Delta U+W}{m\Delta T}$$

In contrast,

$$c_V=\frac{1}{m}\left(\frac{\partial U}{\partial T}\right)_V$$

and, if $c_V$ is constant,

$$c_V=\frac{\Delta U}{m\Delta T}$$

Does this answer your question?