Consider two$-$state systems$$at temperature $.$ All systems are identical, with one$$state at energy $0$ and the other at energy$.$Using the Boltzmann factor for the two states, compute a formula for the energy as a function of $,,$ and $.$

Using $,$ derive a formula for the heat capacity in terms of the same variables.

Use the formulas for and to answer the questions below, assuming that N$=6\mathrm{.}0221023$ and $2\mathrm{.}0710-21$ J for all questions.

$1)$

Compute the internal energy when $=200$ K$.$

J

Your submissions:

$6\mathrm{.}64$e$-22$

Computed value:

$6\mathrm{.}64$e$-22$

Submitted:

Monday, December $2$ at $9$:$57$ PM

Feedback:

$2)$

Compute the heat capacity when $=200$ K$.$

J$/$K

Your submissions:

$1\mathrm{.}09$e$23$

Computed value:

$1\mathrm{.}09$e$23$

Submitted:

Monday, December $2$ at $9$:$57$ PM

Feedback:

$3)$

Compute the heat capacity when $=250$ K$.$

J$/$K

$4)$

Compute the heat capacity when $=20$ K$.$