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?

 

[astrokreng]

Your understanding is correct. The specific heat capacity at constant pressure is indeed related to the specific heat capacity at constant volume. However, the difference lies in the fact that at constant pressure, work is being done on the system to maintain the pressure, whereas at constant volume, no work is being done. Therefore, the specific heat capacity at constant pressure takes into account the energy used to expand the gas, whereas the specific heat capacity at constant volume does not. Both values are important in understanding the thermodynamic properties of a system.