The better theoretical model of molecular vibrations includes the possibility of bond dissoication Eqn. \eqref{eq:morse}, where k is the proportionality constant, D is the bond energy and r-r₀ the displacement from an equilibrium position, r₀. \begin{equation} {V_{morse}} = D{\left( {1 - {e^{ - \beta \left( {r - {r_0}} \right)}}} \right)^2} \beta = \sqrt {\frac{k}{{2D}}} \label{eq:morse} \end{equation} Solving the quantum mechanical problem results in a set of evenly spaced energy levels or frequencies ν, Eqn. \eqref{eq:energies}. \begin{equation} {E_{morse}} = \nu \left( {n + {\textstyle{1 \over 2}}} \right) - \nu x{\left( {n + {\textstyle{1 \over 2}}} \right)^2} = \nu \left( {n + {\textstyle{1 \over 2}}} \right)\left( {1 - x\left( {n + {\textstyle{1 \over 2}}} \right)} \right);x = \frac{{hc\nu }}{{4D}} \label{eq:energies} \end{equation} At finite temperatures T these energy levels are populated according to a Boltzman population distribution, Eqn. \eqref{eq:pops} \begin{equation} {P_i} = \frac{{{e^{-iE/{k_B}T}}}}{Q} \label{eq:pops} \end{equation} where Q is the partition function, Eqn. \eqref{eq:part_func}. \begin{equation} Q = \sum\limits_{j = 1}^n {{e^{ - jE/{k_B}T }}} \label{eq:part_func} \end{equation}

Boltzman populations can be combined with the dependences of thermodyanmic quantities on Q to develop the temperature dependence of the average energy, Equation \eqref{eq:ave_e}, \begin{equation} \left\langle E \right\rangle = \sum\limits_s {{E_s}{P_s}} = \frac{1}{Z}\sum\limits_s {{E_s}{e^{ - {E_s}/{k_B}T}}} \label{eq:ave_e} \end{equation} the enthalpy, Eqn. \eqref{eq:enthalpy}, \begin{equation} \left\langle H \right\rangle = \left\langle E \right\rangle + nRT \label{eq:enthalpy} \end{equation} the entropy, Eqn. \eqref{eq:entropy} \begin{equation} S = - {k_B}\sum\limits_s {{P_s}\ln {P_s}} \label{eq:entropy} \end{equation} the free energy, Eqn. \eqref{eq:free} \begin{equation} \Delta G = \Delta H - T\Delta S \label{eq:free} \end{equation} and the heat capacity, Eqn. \eqref{eq:cp}. \begin{equation} {C_V} = \frac{{\partial \left\langle E \right\rangle }}{{\partial T}} = \frac{1}{{{k_B}T}}\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle ;{\rm{ }}{C_p} = {C_V} + nR \label{eq:cp} \end{equation}

Vibration frequency

Temperature

States

Vibration= 600;

Temperature= 100;

Enthalpy= 100kJ/mol;

Entropy= 100J/molK;

Free Energy= 100kJ/mol;

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