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ps11_soln - Physics 105 Problem Set 11 Solutions Problem...

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Physics 105 Problem Set 11 Solutions Problem 1 (3 pts) (a) When bouncing off a wall perpendicular to the x -axis, only the x -component of the atom’s velocity changes its sign. Collisions with the other walls do not affect the x -component of the atom’s velocity. Hence the atom transfers to the wall orthogonal to the x -axis the amount of momentum equal to Δ p = 2 mv x during each collision; and it happens exactly once during the time interval equal to Δ t = 2 L v x . Consequently, the time averaged force is F = Δ p Δ t = mv 2 x L . (b) Using the result from the previous part, the pressure of the gas on the wall orthogonal to the x -axis is p = N i =1 F i L 2 = 1 L 3 N X i =1 m i v 2 ix . Given that the direction of the atom’s velocity is equally likely to point in any direction, we find: 1 2 N X i =1 m i v 2 ix = 1 2 N X i =1 m i v 2 iy = 1 2 N X i =1 m i v 2 iz = N 2 k B T . This yields for the pressure pV = N N A ( k B N A ) T . Recalling that the ideal gas constant is R = k B N A , we recover the ideal gas law.
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