BOLTZMAN'S cONSTANT and Thermal laws.

[notsam]

Homework Statement

A cylinder contains a mixture of helium and
argon gas in equilibrium at a temperature of
121 C.
Boltzmann’s constant is 1.38066 ×
10−23 J/K, and Avogadro’s number is 6.02 ×
1023 mol−1. What is the average kinetic energy of each
type of molecule?
Answer in units of J.

Homework Equations

The Attempt at a Solution

I think that the awnser is just simply taking the temperature and multiplying it my Boltzmann constant, but that seems to simple.

 

[cepheid]

The Attempt at a Solution

I think that the awnser is just simply taking the temperature and multiplying it my Boltzmann constant, but that seems to simple.

You're close -- the average energy per particle will be proportional to k_BT, but not exactly equal to it. I think that you're just missing an important concept. Here is that concept: in a system of particles, there are many degrees of freedom, meaning that there are many different modes that can contribute to the internal kinetic energy of the system. There is translational kinetic energy (i.e. energy of particles moving from one place to another). In the case of a monatomic gas, that's it. Since each particle in the gas consists of single atom, all the particles can do is move around. There are three translational degrees of freedom corresponding to the three independent directions through space in which particles can move. In the case of a gas in which each particle of the gas is a molecule consisting of more than one atom, those molecules can also vibrate and rotate. So, you have additional modes or degrees of freedom in which kinetic energy can be present (in this case kinetic energy due to vibrational and rotational motion). Now, the question is, on average, how is the total kinetic energy of the system split up amongst these various degrees of freedom? The answer is given by a theorem known as the Equipartition Theorem. If you read this small section from the Wikipedia article about it, it should give you all the information you need to answer your question.

http://en.wikipedia.org/wiki/Equipartition_theorem#Translational_energy_and_ideal_gases

As you probably already guessed just from the name of the theorem, it turns out that the total kinetic energy is split up (partitioned) equally amongst the degrees of freedom in the system.

EDIT: I know that helium exists as monatomic gas (each particle in the gas is a single helium atom). This is in constrast to other examples like hydrogren, which is a diatomic gas (each particle is a molecule consisting of two H atoms bonded together -- H₂). I believe that argon, being a noble gas just like helium, probably also exists in monatomic form. That makes this question particularly simple.