Given that the ionization energy of the ground state of a certain hydrogen-like species is [tex]\( 106.82 \times 10^{-18} \)[/tex] J/atom, we need to determine the number of protons (denoted as [tex]\( Z \)[/tex]) in the nucleus of this species.
To solve this, we need to use the formula for the ionization energy of a hydrogen-like species:
[tex]\[ E = Z^2 \times R_H \][/tex]
where [tex]\( E \)[/tex] is the ionization energy, [tex]\( Z \)[/tex] is the number of protons in the nucleus, and [tex]\( R_H \)[/tex] is the Rydberg constant for hydrogen ([tex]\( 2.18 \times 10^{-18} \)[/tex] J).
The steps to find [tex]\( Z \)[/tex] are as follows:
1. Set up the equation: Use the given ionization energy [tex]\( E \)[/tex] and the Rydberg constant [tex]\( R_H \)[/tex]:
[tex]\[ 106.82 \times 10^{-18} = Z^2 \times 2.18 \times 10^{-18} \][/tex]
2. Solve for [tex]\( Z^2 \)[/tex]: Isolate [tex]\( Z^2 \)[/tex] by dividing both sides of the equation by [tex]\( 2.18 \times 10^{-18} \)[/tex]:
[tex]\[ Z^2 = \frac{106.82 \times 10^{-18}}{2.18 \times 10^{-18}} \][/tex]
3. Calculate the value: Perform the division:
[tex]\[ Z^2 = \frac{106.82}{2.18} \][/tex]
4. Simplify the division:
[tex]\[ Z^2 \approx 49 \][/tex]
5. Find [tex]\( Z \)[/tex]: Take the square root of [tex]\( Z^2 \)[/tex]:
[tex]\[ Z = \sqrt{49} \][/tex]
[tex]\[ Z = 7 \][/tex]
Thus, the number of protons contained in the nucleus of this hydrogen-like species is [tex]\( 7 \)[/tex].
The correct answer is:
e) 7
