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Kalirren
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[SOLVED] Stat. Mech: energy-temperature relation of a perfect classical gas
Note: This is really a problem I gave myself in an attempt to make myself understand thermodynamics better. If the problem itself is flawed (which is a possibility,) then please explain to me why and how so.
Consider the system consisting of a single particle of infinitesimal volume and finite mass m confined to a cube. (Without using the equipartition theorem,) verify that the relationship between the system temperature T = (dE/dS) and the speed |v| of the particle can be written in the form
(1/2)m|v|^2 = (3/2)kT
as predicted by the equipartition theorem.
T = (dE/dS)
E = 1/2 m|v|^2
S = k ln W
The strategy is to express both energy E and entropy S as functions of the particle's speed |v|, and then differentiate and divide:
T = dE/dS = (dE/d|v|)/(dS/d|v|)
Obviously, since E = 1/2 m|v|^2, dE/d|v| = m|v|.
Also, S = k ln W, dS/d|v| = k (1/W) (dW/d|v|).
The tricky part is how to approximate W as a function of |v|, and I think that this is the part I don't really understand. W is the number of states the particle can have. Since particle motion is quantized at some level:
W = (# possible positions) * (# possible velocities)
The only information that we have on the particle is that it is confined to a cubic box (let its volume be V) and that its speed is |v|. The number of possible positions this particle could occupy is proportional to the volume, and the number of momentum-states with speed |v| varies as |v|^2, corrresponding to the numer of lattice points that exist within the spherical shell of radius |v|+d|v|; hence,
W = MV * N|v|^2 = A|v|^2, and
dS/d|v| = k (1/W) (dW/d|v|) = k * (1/A|v|^2) * (2A|v|) = k * 2/|v|.
Dividing, we have
T = dE/dS = (dE/d|v|)/(dS/d|v|) = m|v|/(2k/|v|) = m|v|^2/(2k); rearranging gives
(1/2)m|v|^2 = kT,
which is not what we wanted.
I am aware that if my approximation for W were to have the form W = Av^3 instead of W = Av^2, then everything would work, but I don't understand how to rationalize that. What is wrong with the way I am approximating the number of possible states W?
Note: This is really a problem I gave myself in an attempt to make myself understand thermodynamics better. If the problem itself is flawed (which is a possibility,) then please explain to me why and how so.
Homework Statement
Consider the system consisting of a single particle of infinitesimal volume and finite mass m confined to a cube. (Without using the equipartition theorem,) verify that the relationship between the system temperature T = (dE/dS) and the speed |v| of the particle can be written in the form
(1/2)m|v|^2 = (3/2)kT
as predicted by the equipartition theorem.
Homework Equations
T = (dE/dS)
E = 1/2 m|v|^2
S = k ln W
The Attempt at a Solution
The strategy is to express both energy E and entropy S as functions of the particle's speed |v|, and then differentiate and divide:
T = dE/dS = (dE/d|v|)/(dS/d|v|)
Obviously, since E = 1/2 m|v|^2, dE/d|v| = m|v|.
Also, S = k ln W, dS/d|v| = k (1/W) (dW/d|v|).
The tricky part is how to approximate W as a function of |v|, and I think that this is the part I don't really understand. W is the number of states the particle can have. Since particle motion is quantized at some level:
W = (# possible positions) * (# possible velocities)
The only information that we have on the particle is that it is confined to a cubic box (let its volume be V) and that its speed is |v|. The number of possible positions this particle could occupy is proportional to the volume, and the number of momentum-states with speed |v| varies as |v|^2, corrresponding to the numer of lattice points that exist within the spherical shell of radius |v|+d|v|; hence,
W = MV * N|v|^2 = A|v|^2, and
dS/d|v| = k (1/W) (dW/d|v|) = k * (1/A|v|^2) * (2A|v|) = k * 2/|v|.
Dividing, we have
T = dE/dS = (dE/d|v|)/(dS/d|v|) = m|v|/(2k/|v|) = m|v|^2/(2k); rearranging gives
(1/2)m|v|^2 = kT,
which is not what we wanted.
I am aware that if my approximation for W were to have the form W = Av^3 instead of W = Av^2, then everything would work, but I don't understand how to rationalize that. What is wrong with the way I am approximating the number of possible states W?
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