Consider a particle confined within a box in the shape o

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Consider a particle confined within a box in the shape of a rectangular parallelepiped of edges Lx,Ly and Lz. The possib

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Consider a particle confined within a box in the shape of a rectangular parallelepiped of edges Lx,Ly and Lz. The possible energy levels of this particle are then given by (2.1 3). (a) Suppose that the particle is in a given state specified by particular values of the three integers nx, ny, and nz. By considering how the energy of this state must change when the length Lz. of the box is changed quasistatically by a small amount dLx., show that the force exerted by the particle in this state on a wall perpendicular to the x axis is given by Fx=-dE/dLx. (b) Calculate explicitly the force per unit area (or pressure) on this wall. By averaging over all possible states, find an expression for the mean pressure on this wall. (Exploit the property that the average values nx^2 = ny^2 = nz^2 must all be equal by symmetry.) Show that this mean pressure can be very simply expressed in terms of the mean energy E of the particle and the volume V = LxLyLz of the box.

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