# When does the molar heat capacity equation hold good?

What are the conditions for the molar heat capacity equation viz. $$ΔQ=nCΔT$$ to be true?

For example, if the process is isothermal, then change is temperature change is zero and so should be the change in heat i.e. $$ΔQ=0$$ (according to the equation). But that is not true. I think that the condition should be that work done by gas should be zero. And if that's the case then can we apply this equation to a problem where work done by gas is not zero?

What are the conditions for the molar heat capacity equation viz. ΔQ=nCΔT to be true.

First of all it is usually written as $$Q$$ not $$\Delta Q$$. Secondly, you generally need to know the substance involved (liquid, solid, gas), and whether or not a phase change is involved. For gases, the specific heat is a function of the process (e.g., constant pressure process, constant volume process). For liquids and solids it is generally process independent (one value for a limited range of temperatures)

For example if the process is isothermal, then change is temperature change is zero and so should be the change in heat i.e. ΔQ=0 (according to the equation).

That's not correct.

For example, in a reversible isothermal expansion process involving an ideal gas, internal energy depends only on temperature and is zero. Thus from the first law, the heat transfer to the gas equals the work done by the gas, or $$Q=W$$, so $$Q$$ is not zero.

For an isothermal phase change process, the change in temperature is zero but the heat transfer is not zero. It is called the latent heat of fusion or vaporization which causes a phase change at constant temperature.

You need to look at the whole picture in the context of the first law which, for a closed system, is

$$\Delta U=Q-W$$

And you also need to consider whether or not a phase change is involved.

Hope this helps.

• Can you please explain why the equation Q=nCΔT for isothermal expansion of ideal gas (no phase change) is inconsistent? I get that heat change won't be zero according to the first law, but molar heat capacity equation says otherwise, why is that? Commented May 21, 2020 at 18:40
• I'm saying a phase change occurs at constant temperature, so $\Delta T=0$. b\But $Q$ is not zero. For this case you can't use your equation. The proper equation for $Q$ is $Q=nh$ where $h$ is the molar enthalpy kJ/mole, or $Q=mh$ where $h$ is the specific enthalpy in kJ/kg. Commented May 21, 2020 at 18:50

You're confused because your original equation is incorrect (in Thermodynamics). In thermodynamics, the heat capacity(its) are defined in terms of the internal energy U and the enthalpy H, not in terms of the heat flow Q. For an ideal gas, ALWAYS $$C_v=\frac{\Delta U}{n\Delta T}$$ and $$C_p=\frac{\Delta H}{n\Delta T}$$For an arbitrary substance (including real gases), ALWAYS $$C_v=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V$$and$$C_p=\frac{1}{n}\left(\frac{\partial H}{\partial T}\right)_P$$ With those definitions, you can never go wrong.