A hydrogen-like ion is an ion containing only one electron. The energies of the electron in a hydrogenlike ion are given by $$ E_{n}=-\left(2.18 \times 10^{-18} \mathrm{~J}\right) Z^{2}\left(\frac{1}{n^{2}}\right) $$ in which \(n\) is the principal quantum number and \(Z\) is the atomic number of the element. Calculate the ionization energy (in kilojoules per mole) of the \(\mathrm{He}^{+}\) ion.

Short Answer

Expert verified
The ionization energy of the \(\mathrm{He}^{+}\) ion is 5.24824 kJ/mol.

Step by step solution

01

Identifying the variables

Identify the values for Z and n. For a \(\mathrm{He}^{+}\) ion, the value of Z (the atomic number of Helium) is 2. Since the ionization energy is the energy to remove the first electron, we will take n (the principal quantum number) as 1, corresponding to the ground state.
02

Substitute the values in Energy equation

Substitute these values into the given equation for the energy of an electron. Then we have \(E_{n}=-\left(2.18 \times 10^{-18} \cdot (2)^{2}\left(\frac{1}{(1)^{2}}\right)\right) J\).
03

Calculating the energy

Upon calculation, the energy, \(E_{n}\), is found to be approximately -8.72 x 10^-18 Joules.
04

Conversion of energy to per-molecuel basis

Convert this from energy per atom to energy per mole by multiplying by Avogadro's number \(6.022 \times 10^{23}\) . Now the energy is -8.72 x 10^-18 J/atom × \(6.022 \times 10^{23}\) atoms/mol = -5248.24 Joules/mol.
05

Converting to kJ/mol

Convert this to kilojoules per mole. This is -5248.24 J/mol ÷ 1000 J/kJ = -5.24824 kJ/mol.

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