If the temperature of an object is simply a measurement of the average kinetic energy of the molecules within it; is it possible to convert the temperature of an object, given it's volume, into either a value for the average speed of movement of each molecule or the average kinetic energy of each molecule?

I understand I might be completely barking up the wrong tree here, but I'm studying A Level physics and am just a bit curious.

Thanks in advance!


Yes, more or less. You do not need the volume of the object, because temperature is what's called an intensive quantity--it doesn't matter how much "stuff" there is. What you need to know is what kind of molecules you're dealing with. According to the equipartition theorem each degree of freedom gets $\frac{1}{2} k_B T$ energy, where $k_B$ is Boltzmann's constant and $T$ is measured in kelvins. A "degree of freedom" is a way the system can move, so it's a place for the system to store energy. Let me demonstrate for an ideal gas: an ideal gas has no interactions, so there is no potential energy. However, there is kinetic energy in three dimensions, $x, y,$ and $z$. So there are three degrees of freedom here, and from the equipartition theorem we get: $$ \langle E \rangle = \langle \frac{1}{2} m v^2 \rangle = \frac {3}{2} k_B T $$ The angle brackets mean "average." Do some algebra and you get: $$ \langle v^2 \rangle = \frac{3 k_B T}{m} $$ You'll sometime see people take the square root at this point, but strictly speaking you can't always do that. There is a slight difference between the "square of the average" and the "average of the squares."

Now, what if this wasn't an ideal gas? What if it was a gas made of diatomic molecules, like $O_2$? We can imagine that we have two atoms on a spring in a classical model. In that case, we have far more degrees of freedom. Each molecule can move in three directions (3 d.o.f.), it can vibrate back and forth (2 d.o.f. -- one for potential and one for kinetic energy along that imaginary spring), and it can rotate (2 d.o.f., since it can rotate in multiple planes). Equivalently, each atom has three d.o.f., and then we have one more for the "spring" between them. So for this case, the three in our above formula is now a seven, and we get: $$ \langle v^2 \rangle = \frac{7 k T}{m} $$ One of the most important driving forces in thermodynamics was the measurement of heat capacities, $\Delta E/ \Delta T$, and the discovery of places where these formulas do or do not hold. For instance, the diatomic gas limit turns out to not work at low temperatures, because quantum effects suppress the influence of the rotations.

Your comment about oscillations confuses me a bit because in a standard gas the particles aren't "oscillating" which would imply they return to old positions/velocities. However, you can also apply this thinking to harmonic oscillators (masses on springs) and get what is known as the "Einstein model of a solid". That, too, drove a lot of research in what is today solid-state physics.

  • $\begingroup$ Ahh okay, that makes a lot of sense, thanks! The oscillation comment was a poor choice of words and has since been edited :) $\endgroup$ – Persistence Jun 5 '15 at 19:28

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