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## Homework Statement

Part (a): Explain what A and f means. Describe an experiment that shows this relation is true.

Part (b):Find the temperature of vessel 1.

Part (c): Why is temperature of vessel initially the same as vessel 1? Calculate the final temperature of vessel 2. Find the time taken for vessel 2 to reach this temperature.

## Homework Equations

## The Attempt at a Solution

__Part (b)__Letting ##T_0 = 300 K## be room temperature, and ##\alpha = \frac{m}{2kT}##,

the speed distribution of effusing molecules from 1 into 2 is:

[tex]g_{(v)} = 2 \alpha^2 v^3 \space exp\left(-\alpha v^2\right)[/tex]

The average energy of these particles is:

[tex]\langle E'\rangle = \frac{1}{2}m \int_0^{\infty} 2\alpha^2 v^5 \space exp\left(-\alpha v^2\right)[/tex]

[tex] = 2kT_0[/tex]

Now, in general the average KE of a system with temperature ##T## is ##\frac{3}{2}kT##.

[tex]\langle E' \rangle = \frac{3}{2}k\left(\frac{4}{3}T_0\right)[/tex]

[tex] = \frac{3}{2}kT_1[/tex]

This implies that temperature of box 1 is ##T_1 = \frac{4}{3}T_0 = 400 K##.

__Part (c)__In the beginning, the average KE of molecules entering box 2 should be higher than the average KE of particles in box 1, since particles effusing have higher KE? I'm not sure why it would be the same.

Now the same as above,

[tex]T_2 = \frac{4}{3}T_1 = 533 K[/tex]

The flux of particles effusing from box 1 is given by:

[tex]\phi = \frac{1}{4}n \langle v_1\rangle = \frac{1}{4} \frac{P}{kT} \sqrt{\frac{8kT_1}{\pi m}}[/tex]

Thus number of particles per unit time is ##\phi A = \pi r^2## where ##r^2## is the area of the hole.

Not sure how to find the time taken.