Question

a hydrogen - like ion is an ion containing only one electron. the energies of the electron in a hydrogen - like ion are given by

$e=-\left(2.180\times 10^{-18}\mathrm{j}\
ight)\left(\frac{z^{2}}{n^{2}}\
ight)$

where $n$ is the principal quantum number, and $z$ is the atomic number of the element. calculate the ionization energy, in $\frac{\mathrm{kj}}{\mathrm{mol}}$, of the $\mathrm{he}^{+}$ ion. be sure your answer has the correct number of significant digits.

note: reference the fundamental constants table for additional information.

$\frac{\mathrm{kj}}{\mathrm{mol}}$

Explanation:

Step1: Determine the values of \(n\) and \(Z\)

For \(He^{+}\), \(Z = 2\) (atomic number of helium). Ionization energy is the energy required to remove an electron from \(n=\infty\) (final state) to \(n = 1\) (initial state). Here, we consider the energy change \(\Delta E=E_{\infty}-E_{1}\). Since \(E_{\infty}=0\) (as \(n=\infty\), \(E =-\frac{(2.180\times 10^{-18}\text{ J})Z^{2}}{n^{2}}\) and when \(n=\infty\), \(E = 0\)), so \(\Delta E=-E_{1}\).

Step2: Calculate the energy for one ion

Substitute \(n = 1\) and \(Z = 2\) into the formula \(E=-\frac{(2.180\times 10^{-18}\text{ J})Z^{2}}{n^{2}}\).

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The ionization energy for one ion \(\Delta E=8.72\times 10^{-18}\text{ J}\)

Step3: Convert energy per ion to energy per mole

Use Avogadro's number \(N_{A}=6.022\times 10^{23}\text{ mol}^{-1}\)

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Answer:

\(\boldsymbol{5250}\)