Question

The energy of second Bohr orbit of the hydrogen atom is \(-328 \, \text{kJ mol}^{-1}\), hence the energy of fourth Bohr orbit would be

• A. \(-41 \, \text{kJ mol}^{-1}\)
• B. \(-1312 \, \text{kJ mol}^{-1}\)
• C. \(-164 \, \text{kJ mol}^{-1}\)
• D. \(-82 \, \text{kJ mol}^{-1}\)

Hint:

The energy of the nth orbit is inversely proportional to n\(^2\). As n increases, the energy becomes less negative (increases).

Answer 1

The Correct Option is D

To solve this problem, we need to use the Bohr model of the hydrogen atom. According to the Bohr model, the energy \((E_n)\) of an electron in the \(n^{th}\) orbit of a hydrogen atom is given by:

\(E_n = -\dfrac{1312}{n^2} \, \text{kJ mol}^{-1}\)

Where \(n\) is the principal quantum number of the orbit.

Given that the energy of the second Bohr orbit (\(n = 2\)) is \(-328 \, \text{kJ mol}^{-1}\), let's verify this using the formula:

\(E_2 = -\dfrac{1312}{2^2} = -\dfrac{1312}{4} = -328 \, \text{kJ mol}^{-1}\)

This formula gives us the correct value given in the problem, confirming its correct usage. Now, let's calculate the energy for the fourth Bohr orbit (\(n = 4\)):

\(E_4 = -\dfrac{1312}{4^2} = -\dfrac{1312}{16} \, \text{kJ mol}^{-1}\)

Simplifying the equation:

\(E_4 = -82 \, \text{kJ mol}^{-1}\)

The calculation confirms that the energy of the fourth Bohr orbit is \(-82 \, \text{kJ mol}^{-1}\), which matches the correct answer choice. Thus, option \(-82 \, \text{kJ mol}^{-1}\) is the correct answer.