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Hello.
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So it is given to us in the question that a three -dimensional gas has n number of free electrons.
00:09
When it has kept at a temperature of 0 kelvin, we have to prove that its kinetic energy u0 equals 3 by 5 multiplied by n multiplied by ef where ef is its fermi energy.
00:23
Now the internal energy u is given by the equation u is equal to integral n of e multiplied by d of e multiplied by e de, where n of e is the fermi direct distribution function and d of e is the density of states.
00:55
Now it is given to us that the state is at, sorry, the system is at 0 kelvin and at 0 kelvin, it is in its ground state, that is, that is the state, the state, it's above the state which has the fermi energy ef are empty.
01:11
And hence, for the same reason, we will only evaluate you from zero to fermi energy.
01:18
That is, the limits of the integration are zero to ef.
01:23
And within the limit, zero to ef, the fermi direct distribution function n of e becomes the step function and is equal to 1.
01:37
That is n of e equals 1 for e less than or equal to ef.
01:47
Now rewriting the above equation by applying the limits and substituting the value for n of e, we get u equals integral 0 to ef, d of e, d of e, multiplied.
02:05
By e, de.
02:09
Now, if we evaluate this function, then we will get the value of the kinetic energy of chaos.
02:16
Now, if n of k is the number of states, number of states for a wave vector, for a wave vector k, then n of k is given by the equation, n of k equals v divided by 3 pi square multiplied by k cube.
02:58
This is our second equation and this one is our first equation.
03:05
Here v is the volume.
03:07
Now the energy e of the nth state is given by the equation e .n equals h cross square multiplied by kn square divided by 2m...