A question in statistical physics

[shakgoku]

1. A gas molecules of mass m are in thermodynamic equilibrium at a temperature T.
If $$v_{x},v_{y},v_{z}$$ are the components of velocity v, then the mean value of $$(v_{x}-{\alpha} {v_{y}}+{\beta} {v_{z}})^2$$ is:

a.$$(1+\alpha^2+\beta^2)\frac{k_{b}T}{m}$$

b.$$(1-\alpha^2+\beta^2)\frac{k_{b}T}{m}$$

c. $$(\beta^2-\alpha^2)\frac{k_{b}T}{m}$$

d.$$(\alpha^2+\beta^2)\frac{k_{b}T}{m}$$

Homework Equations

:[/B]
$$[v_{rms} \sqrt{\frac{3k_{b}T}{m}}$$

$$K.E = \frac{3k_{b}T}{2}$$

 

[Mike Pemulis]

Easiest way to do this: Multiply out $$(v_{x}-{\alpha} {v_{y}}+{\beta} {v_{z}})^2$$. Find each expectation value separately, add them up.

 

[shakgoku]

Easiest way to do this: Multiply out $$(v_{x}-{\alpha} {v_{y}}+{\beta} {v_{z}})^2$$. Find each expectation value separately, add them up.

how to find expectation values?

 

[Mike Pemulis]

Sorry, I meant mean value -- same thing.

"Okay, how do I find mean values?"

Good question, which can be answered in a couple of different ways. Can I ask what level you are? Undergrad, grad? Is this a chemistry or physics class?

 

[shakgoku]

Sorry, I meant mean value -- same thing.

"Okay, how do I find mean values?"

Good question, which can be answered in a couple of different ways. Can I ask what level you are? Undergrad, grad? Is this a chemistry or physics class?

undergrad physics

 

[Mike Pemulis]

Okay, so hopefully your book has a derivation of v_rms. Take a look at that; it should provide some clues of how to derive v_x², v_y², and v_z².

One hint from me: What is v_rms, in terms of the components of velocity? Do you we expect the mean values of the components to be different from each other? In other words, is there anything special about the x-direction that would imply that v_x² is different from v_y²?

Now, none of this helps you find the cross-terms, only the squared terms. Try to get the squared terms first, and then we can move on.