Question

# (a) For $^{18} O_{2}, \Delta \overline{G}$ values for the transitions $v=1 \leftarrow 0, 2 \leftarrow 0,$ and $3 \leftarrow 0$ are, respectively, 1556.22, 3088.28, and $4596.21\ \mathrm{cm}^{-1}.$ Calculate $\overline{v}$ and $x_{e}.$ Assume $y_{e}$ to be zero. (b) For $^{14} N_{2}, \Delta \overline{G}$ values for the transitions $v=1 \leftarrow 0, 2 \leftarrow 0,$ and $3 \leftarrow 0$ are, respectively, 2345.15, 4661.40, and $6983.73\ \mathrm{cm}^{-1}.$ Calculate $\overline{v}$ and $x_{e}.$ Assume $y_{e}$ to be zero.

Solution

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#### (a)

In this excercise we have values of $\Delta \mathrm{G}$ for the transitions $v=1 \leftarrow 0$, $v=2 \leftarrow 0$ and $v=3 \leftarrow 0$

They are:

1556.22 $\mathrm{cm}^{-1}$, 3088.28 $\mathrm{cm}^{-1}$ and 4596.18 $\mathrm{cm}^{-1}$

We have to calculate $\overline{v}$ and $x_{\mathrm{e}}$

In case of:

\begin{align*} \Delta \mathrm{G}(v=1 \leftarrow 0)&=\overline{v}-2 \overline{v} x_{e}\\ &=1556.22 \mathrm{cm}^{-1}\\ \end{align*}

In case of:

\begin{align*} \Delta G(v=2 \leftarrow 0)&=2 \overline{v}-6 \overline{v} x_{e}\\ &=3088.28 \mathrm{cm}^{-1}\\ \end{align*}

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