Is the heat capacity infinite for a process at constant temperature?


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> The heat capacity of $\ce{H2}$ at constant $P$ and temperature is \_\_\_\_\_\_.
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It was explained to me that the answer is infinity with the following reasoning:



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> Since heat capacity = $\mathrm{d}H/\mathrm{d}T$ and $\mathrm{d}T$ is zero since constant temperature is part of the question, the answer is infinity.
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**My attempt:** Since $\ce{H2}$ is a diatomic gas, its $C\_p$ is $7R/2$ which gives $7~\mathrm{cal \cdot mol^{-1} \cdot K^{-1}}$.


Which method is correct?




If you assume ideal behaviour and a closed system, then since the ideal gas law holds, with pressure and temperature remaining constant volume must also be constant.So that means that nothing is happening to the system, everything is as it is with $dq=0$ and $dw=0$.Since there is no process involved there is no meaning for the term heat capacity.


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In my humble opinion, your answer is correct and your teacher's answer is incorrect. Heat capacity is an intensive property of a material, defined as you expressed it. If the gas is ideal, then, if dT is equal to zero, dH is also equal to zero, so the ratio dH/dT appears to be indeterminate. However the limit of $\frac{\Delta H}{\Delta T}$ as $\Delta T$ approaches zero is the heat capacity, and that is *not indeterminate*. 


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**I think the answer infinity is correct.** 


We know that $c=\frac{\partial H}{\partial T}$. But, in your case, $\partial T$ is not $\to0$, in fact, $\partial T$ is said to be **exactly** zero. You might be tempted now to say that $c=\frac{\partial H}{\partial T}=\frac{\partial H}0=\infty$ but this is NOT the correct way to interpret this question (and neither is it mathematically valid). The correct way to interpret this question is that "a very large change in enthalpy causes a negligible change in temperature". Hence, $c\to\infty$ (a mathematical way to express "enormously large").


**PS:** This process is commonly *taken* to be the phase change process at the boiling point or the melting point. However, it is more complicated at that point than it seems, because the *mass* of the system also changes. At this point, we may have to thus use the specific heat capacity, $c=\frac 1m \frac{\partial H}{\partial T}$ which is defined per unit mass. But again, the specific heat isn't defined for processes in which mass *changes*. So, it is really tough to do any reasonable calorimetry at this point.


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Heat capacity $C$ is not well-defined for a closed system undergoing constant temperature processes, at least with a reversible process. For a reversible process in a closed system, 


$$\delta q\_\pu{rev} = C\pu{d}T.$$


If temperature is constant, then $\mathrm{d}T=0$ gives $q\_\pu{rev} = 0$ for *any finite value* of $C$. Other formulae for heat capacity, expressed through internal energy or enthalpy, require additional constraints such as constant volume or pressure, and only expansion work is allowed. Neither of these alternative depictions change the outcome.


This is not a problem because heat capacity does not make sense anyway if we assume *a priori* that temperature cannot change at all.



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> **My attempt:** Since $\ce{H2}$ is a diatomic gas, its $C\_P$ is $7R/2$ which gives $\pu{7 cal mol-1 K^-1}$.
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I would be fine with this answer as well. If heat is supplied at constant pressure to a closed non-reactive system, then the temperature must change. The alternative would be an change in volume to keep temperature constant but this would alter the pressure of the system.


In other words, in the net-constant pressure and net-constant temperature process dihydrogen had to warm up (according to its heat capacity at constant pressure), and then cool off again. It is not a good idea to equate the heat capacity here to infinity; that would make little physical and mathematical sense in my opinion.


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