# Expression of the specific heat capacity in statistical mechanics from the thermodynamical definition

The specific heat capacity as I recall from thermodynamics(and as Wikipedia defines it) is defined as

$$C_v= \frac{1}{M}\frac{dQ}{dT}$$

$$C_v$$ is specific heat capacity at constant volume, M is the mass

In statistical mechanics, after deriving the Boltzmann-Gibbs probability measure, the canonical partition function Z and defining the Helmholtz free energy, they start proving the thermodynamic relations from these equations. At some point the following equation is used, which I like to know how to obtain, it must be a thermodynamical thing, since they are not proving it

$$C_v= \left.\frac{\partial \langle H \rangle}{\partial T}\right|_{V,N}= \left.\frac{\partial U}{\partial T}\right|_{V,N}$$

H is the Hamiltonian of the system, and usually they put =E, where E is the energy, I think that they are not using U, to distinguish it from the thermodynamical internal energy, since when constructing the theory we don't know beforehand if they coincide, then we prove they do

I am not familiar with the second one, can it be proved using thermodynamics? How does one get from the first one to the second one?

• Please define your symbols. For example, is $C_v$ specific heat at constant volume and is $H$ enthalpy? Who is the "they", and what additional information do "they" provide between the step of deriving the canonical partition function and presenting the equation that you show? It seems that you are asking for us to fill in those gaps, but perhaps "they" have done so already. What don't you understand about the missing steps? Sep 10 '20 at 13:37
• Did you have a look at the wikipedia article for Specific heat capacity: en.wikipedia.org/wiki/…, which I think fills in the gaps you are missing. Sep 10 '20 at 13:41
• The second one is the correct one if you replace the H with a U. The first one is not the thermodynamic definition of heat capacity. Sep 10 '20 at 13:53
• @ Jeffrey J Weime I have updated my question Sep 10 '20 at 15:39
• @Chet Miller The first one is given in wikipedia as definition of specific heat capacity and is the one I am familiar with, but its the second expression that they use in my notes to calculate $C_v$ from U, so it should somehow derive from the first one. You are saying the second one is the definition?, then it should equivalent to the first one Sep 10 '20 at 15:44

$$c_v=\frac{1}{M}\frac{dQ}{dT}$$ is an informal definition of heat capacity per unit of mass called specific heat capacity. Your statistic mechanical text defines heat capacity for the whole object, not per unit of mass. Usually "specific" version are written with lower case. I'll do this.

$$c_v=\frac{1}{M}\frac{dQ}{dT}$$

$$Q$$ is not a state function, $$dQ$$ is not a differential, you can't write something like $$dQ$$. You can only talk of $$\delta Q$$ for a certain transformation. Transformations that matter to define heat capacity are transformations at constant volume.

The first principle says : $$dU=\delta Q+\delta W$$. Strictly speaking, a transformation at constant volume cannot do work on a system: you can't do work on a gas without touching the piston. More generally, if no external variable the system's Hamiltonian depends on (here it is $$V$$ but it could be other variables) are allowed to change, the work is $$\delta W=0$$ by definition. Hence, any transformation at constant volume is such that $$dU=\delta Q$$.

Finally, you can define heat capacity as (here, you have state functions only and you don't need to say for what transformation.):

$$C_v=\frac{\partial U}{\partial T}|V$$

And the "per unit of mass" specific version:

$$c_v=\frac{1}{M}\frac{\partial U}{\partial T}|V$$

$$\langle H\rangle=U$$ by definition. The Hamiltonian is the energy (it's named "Hamiltonian" to say it's a function depending on the micro-state, while energy usually refers to a single numerical value). For example, you can say the micro-canonical ensemble is the set of micro states whose Hamiltonian (=energy) is equal to a certain value $$U$$. $$U$$ is the average if the energy is not precesily fixed (as in the canonical ensemble for example).

• I am not very confortable with just dropping the mass, this "informal definition" is connected to the equation $Q=C_v m \Delta T$, which we know is correct. Sep 10 '20 at 17:16
• Your statistical mechanics book gives a definition that is not per unit of mass. You have to reconcile them in one way or another: either you drop the mass from Wikipedia, or add the mass in the your textbook. They clearly don't talk about same $C_v$. Sep 10 '20 at 17:24
• uhm, I didn't know the definition wasn't standard. Isn't the term heat capacity used when the formula is $Q=C \Delta T$ , and defined intensively as the amound of heat a certain mass must gain to raise its temperature by $1 K$? Sep 10 '20 at 17:29
• $Q=C\Delta T$ defines an "extensive" version called "heat capacity". It's the amount of heat required to raise the temperature of $1K$ for the whose object. $Q=mc\Delta T$ defines an intensive version called "specific heat capacity", It's the amount of heat required to raise the temperature of $1K$ for one kilogram of it. Most often people use the letter case the way I just did. That's the case here :en.wikipedia.org/wiki/Heat_capacity and here: en.wikipedia.org/wiki/Specific_heat_capacity Sep 10 '20 at 18:02
• I see your title mentionned "specific". I can correct my answer accordingly if you want. Sep 10 '20 at 18:04