# Why is the “total heat capacity” an intensive path function?

My textbook states the following:

"The total heat capacity, $C$ (Heat required to raise the temperature of the system by 1°C) is an intensive path function. On the other hand, $C_V$(Molar heat capacity at constant volume) and $C_P$ (Molar heat capacity at constant pressure) are intensive but state functions."

Firstly, I don't understand why $C$ is an intensive property, especially because it does depend on the mass of the system. It does, however, make sense to say that $C_V$ and $C_P$ are intensive properties, since the heat considered in the calculations involving these two, is the heat per mole of the substance and this quantity will remain constant for any amount of the same substance.

Secondly, I am very confused as to why $C$ is a path function where as $C_V$ and $C_P$ are not. I am not able to understand whether they ought to be path/state functions, because on the one hand, temperature is a state function, where as heat is only defined for a process.

In freshman physics, we learned that, when heat is added to a constant volume system, we can write Q = mCΔT, where C is called the specific heat capacity. However, when we got more deeply into the basics and learned thermodynamics, we found that this elementary approach is no longer adequate (or precise). We found that Q depends on process path and that, if work W is occurring, this changes things. However, we still wanted C to continue to represent a physical property of the material being processed, and not to depend on process path or whether work is occurring. This is dealt with in thermodynamics by changing the definition of C a little. Rather than associating C with the path dependent heat Q, in thermodynamics, we associate C with parameters relating to the state of the material being processed, in particular specific internal energy U and specific enthalpy H. We define the specific heat capacity at constant volume $C_v$ in terms of the derivative of the specific internal energy U with respect to temperature at constant volume: $$C_v=\left(\frac{\partial U}{\partial T}\right)_v\tag{1}$$ We also found that we could define a specific heat capacity at constant pressure $C_p$ as the derivative of the specific enthalpy H with respect to temperature at constant pressure:$$C_p=\left(\frac{\partial H}{\partial T}\right)_p\tag{2}$$ The question is, "do either of these definitions reduce to the more elementary version from freshman physics under any circumstances." The answer is "yes." From the first law of thermodynamics, we find that, for a closed system of constant volume (no work being done), $Q=m\Delta U=mC_v\Delta T$, and, for a closed system experiencing a constant pressure change (with $W=p\Delta v$), $Q=m\Delta H=mC_p\Delta T$. Of course, Eqns. 1 and 2 are much more generally applicable than this.

For a given process, the heat added divided by the temperature change of the system (I am assuming they are calling this C) varies with the amount of work that is done. Like you, I can't see why they would possibly call this an intensive property, although it is certainly a path function. Maybe intensive is a typo, and they meant extensive.

The specific heat capacities Cv and Cp are intensive state functions, because they are defined as the partial derivatives of the specific internal energy and the specific enthalpy, respectively, with respect to temperature (the former at constant volume and the latter at constant pressure), and the specific internal energy and specific enthalpy are state functions.

• Thanks Chester, that extensive/ intensive thing ( that might be a typo), was giving me a headache, my source is p163 of Schroeder thermal physics. – user108787 Sep 14 '16 at 16:11
• Oh, but $C_V$ and $C_P$ haven't been defined like that in my textbook. They are defined simply as "heat capacity divided by number of moles at unit volume/pressure", which is why I wasn't getting a clear picture as to why they are state functions, where as $C$ is not. I'll look this up some more. Thanks anyway :-) – user106570 Sep 14 '16 at 23:37

If you divide one extensive property by another extensive property, you arrive at an intensive property.

If you multiply one extensive property by an intensive property, you arrive at an extensive property. So volume by density is mass.

From State Functions, which gives a good introduction.

If certain property is a state function , keep this rule in mind: is this property or value affected by the path or way taken to establish it? If the answer is no, then it is a state function, if is yes, then it is not a state function.

In thermodynamics, a quantity that is well defined so as to describe the path of a process through the equilibrium state space of a thermodynamic system is termed a process function, or, alternatively, a process quantity, or a path function. As an example, mechanical work and heat are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.

Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include work, heat and arc length. In contrast to path functions, state functions are independent of the path taken. Thermodynamic state variables are point functions, differing from path functions. For a given state, considered as a point, there is a definite value for each state variable and state function.

• No problem, there are so many thermodynamic equations and rules, I keep a separate notebook for them. – user108787 Sep 14 '16 at 23:50