We have already established a definition of temperature from earlier in the course:

$$T=\frac{2{K}_{tot}}{3N{k}_{B}}$$

If we rearrange this definition for the totalkinetic energy of molecules, we have a first view at what an expression for internal energy is going to look like:

$${K}_{tot}=\frac{3}{2}N{k}_{B}T$$

This would approximate the internal energy of a *monoatomic* ideal gas, because the amount of ways you can store energy in a gas like helium or neon is so limited; those “molecules” (which, of course, are mere single atoms) can’t rotate or vibrate like even hydrogen gas can.

*k _{B}*, you’ll remember, is the constant equivalent to 1.381 × 10

^{-23}J K

^{-1}, the Boltzmann constant (to use the common value and not a value with significant figure overkill). This turns out to be a reference energy

*per molecule*; if you multiply this constant by Avogadro’s number, you get the ideal gas constant

*R*, which is 8.3145 J mol

^{-1}K

^{-1}.

If we transform the number of molecules times a molecular reference energy to a number of *moles* times a *molar* reference energy - which is what the ideal gas constant turns out to be! - and we presume that this kinetic energy is equal to internal energy for an ideal gas of single atoms, this is what results:

$$U=\frac{3}{2}nRT$$

Other texts use this as the foundational definition of internal energy for a monoatomic system. They even ascribe a meaning to the “3” in the numerator - the three dimensions, *x, y, *and *z*, that an ideal gas atom can move about in.

We’ve already argued that if there’s no volume change in a gas, there’s no work done on or by that gas. One of the key outcomes of that statement is that, under circumstances of constant volume, the change in internal energy is exactly equivalent to the heat absorbed.

That statement makes *both* of these statements true:

$${q}_{V}=\mathrm{\Delta}U=\frac{3}{2}nR\mathrm{\Delta}T\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

$${q}_{V}=\mathrm{\Delta}U=n\overline{{C}_{V}}\mathrm{\Delta}T\text{}\text{}\text{}\text{}\text{}\text{(true in general)}$$

The equivalence of these two statements can be simplified into a single term, which we’re going to use to define a brand-new specific heat:

$$\overline{{C}_{V}}=\frac{3}{2}R=12.47{\text{J mol}}^{-1}{\text{K}}^{-1}\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

The monoatomic ideal gas constant-volume specific heat $\overline{{C}_{V}}$ is one of the more remarkable theoretical results - the first four periodic gases in the periodic table all have molar specific heats of 12.5 J mol^{-1} K^{-1} under conditions of constant volume, and deviations for the larger ideal gases are minor and only in the third significant figure (xenon’s value for the molar constant-volume specific heat is12.7 J mol^{-1} K^{-1}; it’s larger owing to size and intermolecular attraction).

Even more remarkable are the patterns this reveals about gas behavior.

Let’s think about reality for a bit.

We need to study gases in containers of constant volume. When we receive gases to study in a lab, they’re typically shipped in steel tanks, and the volume of those tanks isn’t going to shift. Rigid containers are a pretty important part of the story of our study of gases.

They’re also not the conditions we normally experience on a day-to-day basis. The rooms we study in are open, and matter can move in those rooms freely. To the extent that we might generate gas in the lab, that gas typically is collected over water - hardly a constant-volume situation, given the capacity we have to displace water.

Behavior we want to model that will have real, practical utility for us isn’t behavior at constant volume, but at constant *pressure.* Pressure is far less likely to change than volume. The atmosphere keeps the pressure we experience in a lab setting, if not perfectly constant, then reasonably close to constant.

The bad news is that we have to deal with work in that circumstance. The good news is that the work is taking place (by definition!) at constant pressure, so that we don’t have to deal with an integral variety of work; we can simply make the statement that *w* = －*P *Δ*V*.

What this will result in is a new, *different* specific heat. Now, internal energy change and heat transfer are *not* equal; the presence of work changes one with respect to the other. What is most interesting to do in this case is to actually solve for a heat transfer at constant pressure - call this *q** _{P}*, as a counter to

*q*

*- by rearranging the first law:*

_{V}$${q}_{P}=\mathrm{\Delta}U-w$$

Note that subtracting work flips the sign on *P *Δ*V*:

$${q}_{P}=\frac{3}{2}nR\mathrm{\Delta}T+P\mathrm{\Delta}V\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

It should be somewhat obvious here that applying the ideal gas law to *P *Δ*V* can make this relationship easily simplified if we work on the assumption that this is a closed container where the count of particles in the container cannot change, and that changing the volume of this matter (likely a gas) at constant pressure will only serve to change the temperature:

$$P\mathrm{\Delta}V=nR\mathrm{\Delta}T\text{}\text{}\text{}\text{}\text{}\text{(assuming constant pressure and moles)}$$

$${q}_{P}=\frac{3}{2}nR\mathrm{\Delta}T+nR\mathrm{\Delta}T\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

$${q}_{P}=\frac{5}{2}nR\mathrm{\Delta}T$$

Once again, we can recognize that there is a molar specific heat embedded within this expression, and we can separate that specific heat out to identify that this is different than what holds for matter at constant volume:

$$\overline{{C}_{P}}=\frac{5}{2}R=20.79{\text{J mol}}^{-1}{\text{K}}^{-1}\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

Again, for most noble gases, there's spectacular agreement with this ideal value ofconstant-pressure specific heat $\overline{{C}_{P}}$ to three significant figures.

We therefore have the key difference in the molar specific heats of gases. Gases at constant volume have a lower specific heat than gases at constant pressure. What’s more…

$$\overline{{C}_{P}}-\overline{{C}_{V}}=\frac{5}{2}R-\frac{3}{2}R\text{}\text{}\text{}\text{}\text{}\text{(presuming monoatomic ideal gas)}$$

$$\overline{{C}_{P}}-\overline{{C}_{V}}=R\text{}\text{}\text{}\text{}\text{}\text{(true in general)}$$

...that difference is *exactly* the ideal gas constant. And while we’ve done all of this derivation for monoatomic ideal gases, because that difference emerges from the addition of the work term in pursuing $\overline{{C}_{P}}$, *the difference between molar specific heats at constant pressure and at constant volume will always be equal to the ideal gas constant.*

Diatomic ideal gases, with rotational and vibrational degrees of freedom to store internal energy (in addition to translational degrees of freedom), have higher values of the constant-pressure and constant-volume specific heats:

$$\overline{{C}_{P}}=\frac{7}{2}R\text{}\text{}\text{}\text{}\text{}\text{(presuming diatomic ideal gas)}$$

$$\overline{{C}_{V}}=\frac{5}{2}R\text{}\text{}\text{}\text{}\text{}\text{(presuming diatomic ideal gas)}$$

The difference between those two values for diatomic gases - and for any ideal gases, for that matter - will remain *R.*

(A quick footnote: The textbook I was using to teach physical chemistry when I started working on these notes, which will remain nameless to protect the guilty,*consistently*switched the *V*and*P*subscripts when identifying the constant-pressure and constant-volume specific heats, in 14 different instances throughout the second chapter of the text, despite the ease with which it can be demonstrated that the constant-pressure specific heat should*always*be greater than the constant-volume specific heat for a gas. As you get more advanced in your chemical education, these types of typographical errors*will* turn up in textbooks, and you need to be aware of the possibility of them and work through the theory for yourself to demonstrate that the text you're studying from is accurate.)