## Frostbite one year ago Derive an expression for the contribution to the molar heat capacity at constant volume CV,m(T) of Cl atoms from these electronic states and evaluate it for T = 500 K.

1. Frostbite

Theory: The electronic ground state of the Cl atom is fourfold degenerate, while the first electronically excited state is doubly degenerate and lies 881 cm–1 above the ground state.

2. Frostbite

Work so far; $C _{V,m}=\left( \frac{ \partial U _{m} }{ \partial T } \right)_{V}=\frac{ N _{a} \left( \langle \epsilon ^{2} \rangle - \langle \epsilon \rangle ^{2} \right) }{ kT ^{2} }$ $\langle \epsilon \rangle=\sum_{i}^{n}p _{i} \epsilon _{i}$ $\langle \epsilon \rangle=\frac{ 4 }{ q } \epsilon _{0}+\frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon=\frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon$ $\langle \epsilon ^{2} \rangle=\sum_{i=0}^{1} p _{i} \epsilon ^{2} _{i}=\frac{ 4 }{ q } \epsilon _{0} ^{2}+\frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon ^{2}=\frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon ^{2}\\$ $C _{V,m}=\frac{ N _{A}\left( \frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon ^{2}-\left( \frac{ 2e ^{- \beta \epsilon} }{ q } \epsilon \right)^{2} \right) }{ kT ^{2} }$ Now from here things are going down hill. I know I need to substitute the equation: $E=\frac{ hc }{ \lambda } \Leftrightarrow \epsilon=hc \bar{v}$ But at the same time I like to make the expression more simple.

3. Frostbite

@ataly @aaronq @blues @Preetha @amistre64

4. amistre64

if your notation is correct, i dont see way that e^2 cant be factored out of the top

5. amistre64

other than that, i got no idea what the setup is doing :)

6. Frostbite

I don't follow the exponential function to 2?

7. amistre64

its all a bit above my reading abilities :/ srry

8. Frostbite

@abb0t

9. Frostbite

p = Boltzmann distribution q = the molecular partition function. beta = the thermodynamical beta (dah?) CV,m = molar heat capacity epsilon = the energy.

10. Frostbite

The method for solving the problem I wrote up for my self was the following: 1. The heat capacity at constant volume is related to the variance of the molecular energy levels. 2. Calculate the mean value of the electronic energy of Cl atoms <epsilon> and the mean value of the square of the electronic energy of Cl atoms <epsilon^2>. 3. insert the results from step 2. into the expression from step 1.