 So, the internal energy, the molar internal energy for a diatomic molecule that we're treating as a harmonic oscillator, we've seen has this rather complicated form. So let's take a little closer look at this and how this function depends on temperature. Clearly, there's some temperature dependence here. We have at least briefly talked about the cold temperature limit. When the temperature gets very cold, temperature goes to zero Kelvin. This ratio of the vibrational temperature, property of that molecule over the temperature, that's going to become very large. So the e to the minus theta over t, it shows up in two different places, e to the minus infinity gets very small, that goes to zero. So this fraction looks like a zero in the numerator, one minus a zero in the denominator. So this entire fraction is zero, so the entire second term is zero. So the internal energy approaches just one half h nu, and then we add to that zero times h nu. So what that means is the number of excitations is zero at zero Kelvin, and that makes sense. Very cold temperatures, few of the molecules are excited into excited energy levels, and the only energy the system has is the zero point energy, the heat capacity, the temperature derivative of that internal energy. There's no temperatures here, all the temperatures have gone away. So in this cold temperature limit, the heat capacity goes to zero. That's what happens at cold temperatures. It's also interesting to think about what would happen as we approach high temperatures. And in some degrees, just the opposite happens, but the limits get a little more complicated when we look at this exponential. So when the temperature is approaching very large values, this ratio of theta vibrational over temperature, that's of course going to become very small. e to the negative theta over t, so that now looks like e to the zero, e to the negative zero in particular, e to the zero is one. If we just said e to the theta over t is approaching one, that's true, but it's a little inconvenient because the one for the exponential in the denominator one minus one in this denominator makes this fraction blow up, and so we have infinity would be our answer. It's a little more informative if we remember that e to the x, when x is a small number, whether it's zero or a very small number, e to the x can be approximated by its Taylor series, one plus x plus a half of an x squared plus and so on. So the smaller x is the fewer of these terms we have to include in the Taylor series. This approximation is equivalent to just saying forget about the x and the x squared and all the terms just say it's equal to one. That's only true in the limit where the temperature has become infinite. At very high temperatures, if we want to include one more term in this series and say that e to the x is one plus x, ignoring the x squared and the higher terms, then we could say that e to the minus theta over t is approaching one, but more specifically it's equal to one minus theta vibrational over t, and then we've ignored the terms beyond that. So with that approximation, then the internal energy is going to approach, we've got a one half h nu plus this h nu, and now what we're multiplying by in the numerator, e to the minus theta over t is one minus theta over t. That ratio is small, so in the numerator it's fine to replace that with a one. In the denominator, we couldn't replace the exponential by a one. Now what we're going to replace it with is one minus e to the minus, sorry, one minus theta over t. So we've used this extended approximation, this Taylor series, because this one is now going to cancel this one, and when we do that one minus one cancels negative, negative theta over t in the denominator makes this look like, so I've got an h nu out front, negative theta over t in the denominator, looks like t over theta, and we can simplify a little further if we remember that the, let's put that up here, h nu is equal to k times theta, so this h nu I can rewrite as k theta, when I do that the theta from the h nu cancels the theta in the denominator, and I can rewrite this as one half h nu plus k theta divided by theta just gives me k, and I still have a t left over from the numerator. So that's a much simpler expression, still blows up, still becomes infinitely large as the temperature approaches infinity, but now we have a little information, more information about specifically how it's increasing and how it's becoming infinite as the temperature gets large, it's increasing linearly with the temperature. So at high temperatures the energy is not just the zero point energy, the energy of the ground state, one half h nu, but also a term that is linearly proportional to the temperature with this factor of r or k out front, and we can also write the heat capacity. If I take the temperature derivative of this term, the constant volume heat capacity is going to be, the derivative of kt is just k, or if we prefer r. So notice that we have yet another heat capacity that looks like some multiples of one half r, as we've seen for the particle in the box and the rigid rotor. So we have a different answer, the heat capacity is zero in the cold temperature limit, the heat capacity looks like r in the high temperature limit, and in between there's some temperature dependence. If we take a look even more closely at how the energy and the heat capacity depend on the temperature. So if I have temperature on this axis, if I plot how the internal energy due to vibrations, if I plot this function as a function of temperature, what it's going to look like is it's going to start out at one half h nu at cold temperatures. We know that from the cold temperature limit. It's going to look like a function that depends linearly on temperature in the high temperature limit and in between it's just going to make a transition between those two things. So when temperature is low, it has this value one half h nu, when the temperature is large, it just looks like a straight line that's heading upwards with a slope of r. So that's how the internal energy depends on temperature. The other property that's interesting to look at is how the constant volume heat capacity depends on temperature, and that's just the temperature derivative of this curve. At low temperatures it's flat, the temperature derivative is zero as we've seen here. At high temperatures the slope is approaching constant and that constant is the gas constant. So what the graph of the heat capacity looks like is this. At low temperatures the heat capacity is zero, at high temperatures it's approaching this value of r, and in between there's some temperature dependence to what the heat capacity is. Again this is the energy that's due to the vibration of a harmonic oscillator and the heat capacity that's due only to the vibrations of the harmonic oscillator. So if I were to add this constant, the vibrational temperature, that's a constant that has units of temperature. On this graph the theta vibrational might be roughly somewhere around here, and what we see is that when the temperature is quite cold compared to the vibrational temperature we have zero heat capacity and just the zero point energy. When the temperature becomes as large as or even larger than the vibrational temperature that's when we begin to see the constant heat capacity of r or a linearly increasing internal energy.