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A quantum system has 6.022 x 10^{23} particles in the ground state. If the temperature is 300 K
and a the energy levels are evenly spaced with a separation of 1 x 10^{−20} J, how many particles would be expected to be in the second level? The third level?
eqn:nj/ni=e^(-dEij/(kb*T)) nj= Press [SOLVE]==> .nj=1.610505521378E22 ni=6.022E23 ni=6.022E23 dEij=1E-20 dEij=1E-20 kb=1.38066E-23 kb=1.38066E-23 T=200 T=200 bound={-1E99,1E99} bound={-1E99,1E99}Answer: There would be 1.611 x 10^{22} particles in the second energy level. Now we put this value in for ni and solve again. eqn:nj/ni=e^(-dEij/(kb*T)) nj= Press [SOLVE]==> .nj=4.30708740350E20 ni=1.611E22 ni=1.611E22 dEij=1E-20 dEij=2E-20 kb=1.38066E-23 kb=1.38066E-23 T=200 T=200 bound={-1E99,1E99} bound={-1E99,1E99}Answer: There would be 4.31 x 10^{20} particles in the third energy level. |
At what temperature would a quantum system having evenly spaced energy levels with a separation of 1 x 10^{−20} J show a ratio of 1/1000 for number of particles in adjacent levels? What would
the temperature have to be if there were just 1 in a million particles excited?
Even though we only know the ratio of nj/ni, we can assume there are 1000 particles in the lower level and just put 1 in for the number of particles in the upper level. eqn:nj/ni=e^(-dEij/(kb*T)) nj=1 nj=1 ni=1000 ni=1000 dEij=1E-20 dEij=1E-20 kb=1.38066E-23 kb=1.38066E-23 T= Press [SOLVE]==> .T=104.85190220697632 bound={-1E99,1E99} bound={-1E99,1E99}Answer: The temperature would have to be 105 Kelvin for nj/ni to be 1/1000. Now we change ni to 100000 and solve again. eqn:nj/ni=e^(-dEij/(kb*T)) nj=1 nj=1 ni=1000000 ni=1000000 dEij=1E-20 dEij=1E-20 kb=1.38066E-23 kb=1.38066E-23 T=104.85190220697632 Press [F5]==> .T=52.42595110348816 bound={-1E99,1E99} bound={-1E99,1E99}Answer: The temperature would now be 52 Kelvin. |