24.11: The Standard Gibbs Free Energy for H₂(g), I₂(g), and HI(g)
- Page ID
- 152809
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For many diatomic molecules, the data needed to calculate \(G^o_{IG}\) are readily available in various compilations. For illustration, we consider the molecules \(H_2\), \(I_2\), and \(HI\). The necessary experimental data are summarized in Table 2.
| Compound | Molar mass, g | \(D_0\), kJ mol-1 | \(\nu\), hertz | \(r_XY\), m |
|---|---|---|---|---|
| \(H_2\) | \(2.016\) | \(432.073\) | \(1.31948 \times 10^{14}\) | \(7.4144 \times 10^{−11}\) |
| \(I_2\) | \(253.82\) | \(148.81\) | \(6.43071 \times 10^{12}\) | \(2.666 \times 10^{−10}\) |
| \(HI\) | \(127.918\) | \(294.67\) | \(6.69227 \times 10^{13}\) | \(1.60916 \times 10^{−10}\) |
The terms in the simplified equation for the standard Gibbs free energy at \(298.15\) K are given in Table 3.
| Compound | \(\ln \left[ \left( \frac{2 \pi mkT}{h^2} \right)^{3/2} \frac{kT}{p_o} \right]\) | \(\ln \left( \frac{8 \pi^2 IkT}{\sigma h^2} \right)\) | \(- \ln \left( 1 - e^{-h \nu/kT} \right)\) | \(\frac{D_0}{RT}\) |
|---|---|---|---|---|
| \(H_2\) | 126.23929 | 0.6312* | 0.0000 | 174.295 |
| \(I_2\) | 133.49256 | 7.932 | 0.4388 | 60.0289 |
| \(HI\) | 132.46470 | 3.4604 | 0.00002 | 118.868 |
| *Calculated as a sum of terms (see Table 1) rather than as the integral approximation. | ||||
Finally, the standard molar Gibbs Free Energies at \(298.15\) K are summarized in Table 4.
| Compound | \(G^o_{298 ~ \text{K}}, ~ \text{kJ mole}^{-1}\) |
|---|---|
| \(H_2\) | −746.577 |
| \(I_2\) | −500.471 |
| \(HI\) | −631.622 |
These results can be used to calculate the standard Gibbs free energy change, at \(298.15\) K, for the reaction
\[H_2\left(g\right)+I_2\left(g\right)\to 2HI\left(g\right). \nonumber \]
We find
\[{\Delta }_rG^o_{298}=2G^o\left(HI,\ g,\ 298.15\ \mathrm{K}\right)-G^o\left(H_2,\ g,\ 298.15\ \mathrm{K}\right)-G^o\left(I_2,\ g,\ 298.15\ \mathrm{K}\right)=-16.20\ \mathrm{kJ} \nonumber \]