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Frostbite

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.

  • 11 months ago
  • 11 months ago

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  1. Frostbite
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    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.

    • 11 months ago
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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.

    • 11 months ago
  3. Frostbite
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    @ataly @aaronq @blues @Preetha @amistre64

    • 11 months ago
  4. amistre64
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    if your notation is correct, i dont see way that e^2 cant be factored out of the top

    • 11 months ago
  5. amistre64
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    other than that, i got no idea what the setup is doing :)

    • 11 months ago
  6. Frostbite
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    I don't follow the exponential function to 2?

    • 11 months ago
  7. amistre64
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    its all a bit above my reading abilities :/ srry

    • 11 months ago
  8. Frostbite
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    @abb0t

    • 11 months ago
  9. Frostbite
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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.

    • 11 months ago
  10. Frostbite
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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.

    • 11 months ago
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