## Frostbite Group Title 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. one year ago one year ago

1. Frostbite Group Title

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.

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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.

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@ataly @aaronq @blues @Preetha @amistre64

4. amistre64 Group Title

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

5. amistre64 Group Title

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

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I don't follow the exponential function to 2?

7. amistre64 Group Title

its all a bit above my reading abilities :/ srry

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@abb0t

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

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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.