An oscillator with frequency f $=2\mathrm{.}11012$Hz $($about typical for a greenhouse gas molecule$)$ is in equilibrium with a thermal reservoir at temperature T$.$ The spacing between the energy levels of the oscillator is given by $=$ hf$,$ where h $=6\mathrm{.}62610-34$ J$*$sec$.$

$1)$

For what temperature does P$1/$P$0=1/2,$ where P$1$ is the probability that the oscillator has E$=$ $($the first excited state$)$ and P$0$ is the probability that it has E$=0($the lowest energy state$)?$

T $=$

K

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$2)$

If T is $10\%$ of the value you calculated in part $1),$ what is the ratio P$1/$P$0?$

P$1/$ P$0=$

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$3)$

At the temperature of part $2),$ what is the ratio P$2/$P$1,$ where P$2$ is the probability that the oscillator is in the second excited state $($E$=2$$)?$

P$2/$P$1=$

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$4)$

Now suppose you have a large number of these oscillators $($such as in a solid$).$ At the temperature of parts $2)$ and $3),$ what is the average thermal energy per oscillator divided by kT$?$ That is$,$ calculate $/$kT$.($Equipartition would have this ratio be $1,$ but that's not the answer here.$)$

$/$ kT $=$