In the last lecture we spent some time discussing Molecular energy without getting a very clear or concise picture of it. Today we'll be hoping to achieve just that, and we'll do it by looking at a few concepts familiar to most of us. Included in this discussion will be conversations on translation, rotation, and vibration, as well as the degrees of freedom that are represented in each. We'll also take our first look at how these notions apply to both linear and non linear molecules.
But before we do that, let's take a look at an old chemical friend: heat capacity. Using C as the variable for heat capacity, E for energy, and delta T for temperature, we can quickly discover through either mathematical manipulation or physical experimentation that
E = C * delta T
Let's look at this for a little bit. Suppose we have a container with one mole of a monatomic gas at 293 Kelvins. How much energy do you think it would take to raise the temperature to 303 Kelvins? Well, obviously, we could just look up the heat capacity for the molecule in question, but that's too easy. Instead let us have some fun.
To begin with, let's use Helium in this example. There isn't any special reason, it's just easier writing "He" than "monotomic gas" all of the time. And let's think about what happens when we put energy into a monatomic gas. Where does that energy go? I'll give you a hint: it ain't potential. That's right. All of the energy going into the gas goes in as Kinetic Energy. And it just so happens that we have a nifty little equation,
Kavg = 3/2 * R * T
to help us solve this little problem. For this of course R must be in units of Joules per Kelvin * mol, in which case the constant is 8.314 units. When we plug this in, along with a delta T of 10, we determine that the energy needed is right around 120 Joules.
Let's look briefly now then at what we have accomplished. We've taken our ability to combine the micro and macroscopic properties of gasses and used it to determine the energy necessary to raise the temperature of gas, which is related to a seemingly completely different equation. So congratulations to us.
But there are questions that arise. For instance, why does the gas have to be monatomic in the example above? Well, to put it simply, it is because we can't do the above with any other type of gas...yet. Shortly we will be able to.
With a monatomic gas, we can ignore pesky little problems like intramolecular kinetic energy. Things like rotational, vibrational, and translational energy. These things, remember, have been discussed in the previous section, and if you need a refresher on them, well, you are not alone. That section will be up once I refresh myself.
Now, let's get back to work. Consider a molecule with n atoms. Let's start with a linear atom. We know from previous discussions that the number of possible degrees of freedom for a molecule is always 3n. With a linear molecule, three degrees are translational, with the molecule being able to move in the X, Y, and Z directions. How many rotational degrees are there? Well, use a model. Hold the molecule horizontal to the ground along an imaginary X axis. Rotation through this axis does not produce noticable results, but rotation through the center of mass from the Y and Z directions yields results, therefore yielding two degrees of freedom. But how many vibrational degrees are you? This we can't determine through logic like with the translational and rotational kinetic energies. But we can use the 3n idea.
Remember, every molecule has 3n degrees of freedom. Let's pick our molecule as diatomic Nitrogen. Here we have two atoms so n = 2. Therefore we should see six degrees of freedom. But look! Five of them are already accounted for, three translational and two rotational. That leaves just one degree left for vibrational energy. If we look at another linear molecule, say Carbon dioxide, we have n = 3, but still three translational and two rotational degrees of freedom. Using 3n, we see we should have nine degrees of freedom and thus must have four vibrational degrees.
Nonlinear molecules turn out to be much the same. They too have three translational degrees of freedom (the maximum possible number for what we're doing), but they have three rotational degrees of freedom as well (again the maximum possible number for what we're doing). Using the 3n rule once more, the number of vibrational degrees of freedom comes to 3n - 6. An example is water, H2O, which has 3n of nine, a trans degree of 3, a rotational degree of 3, and a vibrational degree of 3.
So what is it vitally important to memorize from this? Not much. Most of what is above can be figured out simply by looking at the numbers and analyzing them properly. The one thing that can't be forgotten is 3n. That is a very important number that is difficult to derive. Remember that, and you'll be okay.