Homework 7B key
- Page ID
- 109920
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| Atom | Nuclear Charge (Z) | Electronic Configuration | Spin Multiplicity (2S+1) |
\(E_M\) (kcal/mol) |
\(E_{M^{Z+}}\) (kcal/mol) |
\(E_{M^+}\) (kcal/mol) |
\(E_{M^-}\) (kcal/mol) |
\(E_{tot}\) (kcal/mol) |
\(E_{I_1}\) (kcal/mol) |
\(E_{EA}\) (kcal/mol) |
| H | 1 | \(1s^1\) | 2 | -313 | 0 | 0 | -305.5 | 313 | 313 | 7.5 |
| He | 2 | \(1s^2\) | 1 | -1792 | 0 | -1251.5 | ___ | 1792 | 540.5 | ___ |
| Li | 3 | \(1s^2 2s^1\) | 2 | -4663.9 | 0 | -4540.7 | -4660.8 | 4663.9 | 123.2 | 3.1 |
| Be | 4 | \(1s^2 2s^2\) | 1 | -9144.3 | 0 | -8958.1 | ___ | 9144.3 | 186.2 | ___ |
| Bo | 5 | \(1s^2 2s^2 sp^1\) | 2 | -15393.2 | 0 | -15207.7 | -15358 | 15393.2 | 185.5 | 35.2 |
| C | 6 | \(1s^2 2s^2 sp^2\) | 3 | -23649.4 | 0 | -23400 | -23660.2 | 23649.4 | 294.4 | 34.2 |
| N | 7 | \(1s^2 2s^2 sp^3\) | 4 | -34132.2 | 0 | -33811 | -34089.4 | 34132.2 | 321.2 | 42.8 |
| O | 8 | \(1s^2 2s^2 sp^4\) | 3 | -46935.6 | 0 | -46657.5 | -46923.5 | 46935.6 | 278.1 | 12.1 |
| F | 9 | \(1s^2 2s^2 sp^5\) | 2 | -62362.6 | 0 | -61999 | -62392.2 | 62362.6 | 363.6 | -29.6 |
| Ne | 10 | \(1s^2 2s^2 sp^6\) | 1 | -80632.7 | 0 | -80174.9 | ___ | 80632.7 | 457.8 | ___ |
| Na | 11 | \(1s^2 2s^2 sp^6 3s^1\) | 2 | -101564.3 | 0 | -101450.1 | -101561.7 | 101564.3 | 114.2 | 2.6 |
| Mg | 12 | \(1s^2 2s^2 sp^6 3s^2\) | 1 | -125256.1 | 0 | -125103.4 | ___ | 125256.1 | 152.7 | ___ |
| Al | 13 | \(1s^2 2s^2 sp^6 3s^2 3p^1\) | 2 | -151778.4 | 0 | -151649.3 | -151778.8 | 151778.4 | 129.1 | -0.4 |
| Si | 14 | \(1s^2 2s^2 sp^6 3s^2 3p^2\) | 3 | -181256.8 | 0 | -181080.4 | -181276.7 | 181256.8 | 176.4 | -19.9 |
| P | 15 | \(1s^2 2s^2 sp^6 3s^2 3p^3\) | 4 | -213798.7 | 0 | -213569.6 | -213790.3 | 213798.7 | 229.1 | 8.4 |
| S | 16 | \(1s^2 2s^2 sp^6 3s^2 3p^4\) | 3 | -249434.4 | 0 | -249221.5 | -249455.6 | 249434.4 | 212.9 | -21.2 |
| Cl | 17 | \(1s^2 2s^2 sp^6 3s^2 3p^5\) | 2 | -288323.7 | 0 | -288051 | -288380.5 | 288323.7 | 272.7 | -56.8 |
| Ar | 18 | \(1s^2 2s^2 sp^6 3s^2 3p^6\) | 1 | -330572.6 | 0 | -330234.2 | ___ | 330572.6 | 338.4 | ___ |
Note: \(E_{M^+}\) is the electronic energy of the atom with one electron lost; \(E_{M^-}\) is the electronic energy of the atom after gaining one electron; \(E_{M^{Z+}}\) is the electronic energy of the nucleus without electron. \(E_{tot}=E_{M^{Z+}}-E_M\); \(E_{I_1}=E_{M^{+}}-E_M\); \(E_{EA}=E_{M^{-}}-E_M\).
From this plot, we can see that the first ionization energy of the elements in the same row increases as atomic number increases. As atomic number increases, the attraction of the valence electron from the nucleus also increases, thus increasing the first ionization energy. The valence electrons of the elements in the same row occupy the same sub-shell, and will experience similar attraction from the nucleus. However, the valence electrons in different sub-shells will experience different attractions. Therefore, you will observe this periodic trend.
The total binding energy increases as the atomic number increases and does not show periodic trend. This is because the total binding energy is the sum of the ionization energies for all the electrons, so the more electrons and protons there are, the larger the total energy.
The electron affinity energy has the general trend of decreasing as atomic number increases in the same row of the periodic table. The reason is that as the nuclear charge increases, the more easily it is to gain electrons. As explained above for the first ionization energies, the valence electrons for the elements in the same row experience similar attractions or repulsions from the nucleus. Therefore, elements from different rows will have a larger difference in electron affinity. These two reasons together contribute to the periodic trend that is observed.
From these three plots, we can verify that the octet rule is true. Firstly, the elements with octet electronic configurations (He, Ar, Ne...) have the highest first ionization energies within their respective rows, suggesting that they are the most stable in their respective rows. Secondly, those elements with octet configurations do not have electron affinity, which means they do not want to gain electrons, whereas all other elements can gain electrons. These observations confirm that the octet configurations are very stable and do not tend to gain or loss electrons. However, the trend of \(E_{tot}\) does not show the stability of the octet rule. The reason is the same as explained above for the lack of periodic trend of \(E_{tot}\).
The comparison between the calculated Electron Affinities to the "correct" ones suggests that the periodic trend is there, but the exact values do not match. Considering that theoretical predictions can never be 100% accurate, this is acceptable.