Question

The energy required by electrons, present in the first Bohr orbit of hydrogen atom, to be excited to second Bohr orbit is ________ J mol\(^{-1}\). Given: \(R_H = 2.18 \times 10^{-11}\) ergs.

• A. \(9.835 \times 10^{12}\)
• B. \(9.835 \times 10^{5}\)
• C. \(1.635 \times 10^{-11}\)
• D. \(1.635 \times 10^{-18}\)

Hint:

Energy required for excitation equals the difference between energies of final and initial Bohr orbits.

Answer 1

The Correct Option is C

To find the energy required to excite an electron from the first Bohr orbit to the second Bohr orbit in a hydrogen atom, we can use the Bohr model of the hydrogen atom. The energy of an electron in the \(n^{\text{th}}\) orbit is given by the formula:

\(E_n = - \frac{R_H}{n^2} \cdot 2.179 \times 10^{-18} \text{ J per atom}\)

Where: \(R_H\) is the Rydberg constant.

1. Energy of the electron in the first orbit \((n=1)\) is:

\(E_1 = - \frac{2.18 \times 10^{-11} \text{ ergs}}{(1)^2}\)

Convert ergs to joules:

\(1 \text{ erg} = 10^{-7} \text{ joule}\)

Thus,

\(E_1 = -2.18 \times 10^{-11} \times 10^{-7} \text{ J} = -2.18 \times 10^{-18} \text{ J per atom}\)

1. Energy of the electron in the second orbit \((n=2)\) is:

\(E_2 = - \frac{2.18 \times 10^{-11} \text{ ergs}}{(2)^2}\)

Convert ergs to joules:

\(E_2 = - \frac{2.18 \times 10^{-11}}{4} \times 10^{-7} \text{ J} = -5.45 \times 10^{-19} \text{ J per atom}\)

1. The energy required for the excitation from the first orbit to the second orbit is the difference in energy between these two orbits:

\(\Delta E = E_2 - E_1\)

\(\Delta E = -5.45 \times 10^{-19} \text{ J} - (-2.18 \times 10^{-18} \text{ J})\)

\(\Delta E = 1.635 \times 10^{-18} \text{ J per atom}\)

1. Since the question asks for the energy in J/mol, we use Avogadro's number \((6.022 \times 10^{23} \text{ atoms/mol})\):

\(\Delta E_{\text{mol}} = 1.635 \times 10^{-18} \times 6.022 \times 10^{23} \text{ J/mol}\)

\(\Delta E_{\text{mol}} = 9.835 \times 10^{-12} \text{ J/mol}\)

Therefore, the energy required by electrons to be excited from the first to the second Bohr orbit is \(1.635 \times 10^{-11} \text{ J/mol}\).