PracticeSessionSolutions - Problem I (10 points) Consider a...

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Problem I (10 points) Consider a system of N distinguishable particles. Each particle can occupy only one of the two levels ε 1 or ε 2 , where ε 1 < ε 2 . Assume N is very large. You can distribute the particles in such way that n particles reside on level ε 2 while N-n particles reside on level ε 1 when n changes from 0 to N . 1. Find the total energy of this system as a function of the occupancy n for the level ε 2 . 2. Find the number of microstates for this system as a function of the occupancy n for the level 2 . 3. Find the entropy of this system as a function of energy and draw the plot S(E) indicating the range of energies and values of the entropy in its minima and maxima. 4. Find the temperature of this system when a) occupancy n of the level 2 is very small, b) when approximately equal amounts of particles reside on both levels 1 and 2 , and c) when most of the particles reside on level 2 . Plot schematically the dependence of temperature as a function of n . 5. If you place this system into heat reservoir, and start raising the temperature from zero to infinity, draw occupancies for each level as a function of T . 6. What physical system can correspond to such model? How experimentally can you reach the situation when most particles land at the higher level ε 2 and what temperatures this situation would corresponds to? Solution. 1. The total energy is 1 212 1 [] ( ) ( ) En N n n N n εεε =− + = + 2. The number of microstates for n=0 is 1, for n=1 is N , for general n it is given by the number of possibilities how many times n arbitrary particles can be picked out of N ! () !( )! N n nN n Ω= 3. The entropy is given by 1 21 ( ) ln ( ) ln ! ln ( )! ln( ( ))! where SE k E k N k nE k N nE EN nE εε =Ω= =
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The plot S(E) start from zero at E=N ε 1 and ends at zero at E=N 2 . It is symmetrical with respect to the middle point N( 1 + 1 )/2 where it reaches its maximum equal to ( / 2) ln ! ln / 2! ln( / 2)! ln 2 / 2ln / 2 2 / 2 ln ln ln 2 ln2 S n N k N k N k N kN N kN kN N kN kN N kN N kN kN == = + =−+ = 4. The reciprocal temperature is given by the derivative of the entropy with respect to energy. Consider the limits () , 21 , , 11 ln ! ln ! ln( )!
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This note was uploaded on 05/24/2010 for the course PHYSICS statistics taught by Professor Pathria during the Spring '09 term at Jackson Community College.

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PracticeSessionSolutions - Problem I (10 points) Consider a...

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