The Physics Philes, lesson 129: Return of Heat Capacities!
For the past few episodes of The Physics Philes, we’ve been trying to understand the kinetic-molecular model of idea gases. Now we need to rewind a little big and go back thinking about heat capacities.
Remember way back in the day we talked about molar heat capacities, or a quantity that determines the heat required for some number n of moles to change temperature. We can express this relationship as Q = nCΔT, where Q is the heat, C is the molar specific heat capacity, and ΔT is the change in temperature.
This is fine for as far as it goes, but what if we want to predict molar specific heat capacities on a purely theoretical basis? We can do this, but we need our old friend the kinetic-molecular model of ideal gases.
First I should note that we’re only going to worry about the constant volume – or isochoric – situation. This way we don’t have to deal with the energy transfer that comes with mechanical work. If we let a gas move the walls of its container, it would do work. Don’t worry; we’ll get to that later. Just not right now.
OK, so where was I? Oh right. Kinetic-molecular model of ideal gases.
We know from this simple model that the translational kinetic energy is the only molecular energy of point-like molecules. The energy is directly proportional to the absolute temperature T. We found an equation for the translational kinetic energy:
If the temperature only changes a very small amount, we can write this using differentials:
All this means is that a very small change in temperature corresponds to a very small change in translational kinetic energy.
Now, let’s take a look again at the definition for molar heat capacity at constant volume, denoted with Cv. If the temperature changes only a very little, we can expect the heat Q to change only a very little:
Remember that heat is nothing more than the transfer of energy. We’ve assumed that the translational kinetic energy represents all of the molecular energy. That means that dK and dQ must be the same! Setting those equations equal to each other and doing a little algebra, we get
This says that the molar heat capacity at constant volume of every gas that has molecules that can be represented as points is equal to 3R/2. And we know what R is, so Cv has a definite value.
Except it doesn’t. Well, not for every gas. This equation works well for monatomic gas, but if we have a gas with molecules any bigger than that, it doesn’t work very well at all. Why should this be?
Diatomic and polyatomic molecules can’t be modeled in this way as a point particle. It needs something more complicated. For illustration, let’s think about a diatomic molecule. That’s a molecule with two masses with an interaction force between them. These types of molecules have translational kinetic energy, but they can also have kinetic energy associated with rotation through its center of mass. In addition to vibrating motion along the line joining the two masses together that gives yet more kinetic and potential energies.
All of that means that, unlike a monatomic gas, when the temperature is increased in a diatomic or polyatomic gas energy needs to go to translational kinetic energy, rotational kinetic energy, and vibrational energies. So to increase the temperature of a monatomic and diatomic or polyatomic gas the same amount, additional heat needs to fee the additional energies. It’s for that reason that diatomic and polyatomic gases have larger molar specific heat capacities.
You may be wondering just how we know how much energy is associated with what type of energy. Good question, and it’s a question that is too complex for me to answer in full. But it has to do with the principle of equipartition of energy. This principle states that, on average, each component of velocity has an associated kinetic energy per molecule of one half the product of the Boltzmann constant k and the absolute temperature. The number of velocity components needed to fully describe the motion of the molecule is called degrees of freedom. For example, a monatomic gas has three degrees of freedom because it has three velocity components: the x, y, and z. So the average kinetic energy per molecule is (3/2)kT, which is consistent with what we’ve found previously.
For a diatomic molecule, on the other hand, there are two possible axes of rotation: perpendicular to each other and to the molecule’s axis. A diatomic molecule has 5 degrees of freedom, so the average total kinetic energy per molecule is (5/2)kT. The total kinetic energy of n moles is
Thus the molar heat capacity at a constant volume is
It turns out that this agrees remarkably well with the values of Cv we’ve measured for diatomic gases. Neat!
Vibrational motion can contribute to the heat capacities of gases. Molecular bonds stretch and bend, which adds degrees of freedom and some additional energies. However, for most diatomic molecules, the vibrational energy doesn’t matter. To understand why we need to use some concepts from quantum mechanics. See, vibrational energy can only change in finite steps. If the change in energy of the first step is much larger than the energy possessed by most molecules, then nearly all the molecules will stay in its minimum energy state of motion. The temperature change doesn’t change the average vibrational energy too much and the degrees of freedom that may have resulted are said to be “frozen out.” In more complex molecules the gaps between energy levels might be smaller and the vibration might contribute to the heat capacity. Rotational energy also might contribute to the heat capacity. The rotational energy also only changed by finite steps, but are usually much smaller. “Freezing out” of the degrees of freedom only rarely occurs.
Aren’t you glad we spent so much time on the kinetic-molecular model of ideal gases? Otherwise, none of this would make sense! Next week, however, we’re moving on to the last section in the chapter on thermal properties of matter, and that is phases of matter.
Featured image credit: Siyavula Education via Flickr