Question

The energy of an electron in the first Bohr orbit of hydrogen atom is \(-2.18×10^{-18}J\). Its energy in the third Bohr orbit is ____.

• A. One third of this value
• B. \(\frac{1}{9}\) th of this value
• C. Three times of this value
• D. \(\frac{1}{27}\) of this value

Answer 1

The Correct Option is B

To find the energy of an electron in the third Bohr orbit of the hydrogen atom, we need to recall the formula for the energy of an electron in the \(n\)-th Bohr orbit:

E_n = -\frac{13.6 \, \text{eV}}{n^2}

However, we have been given the energy in Joules for the first Bohr orbit:

E_1 = -2.18 \times 10^{-18} \, \text{J}

Step 1: Energy in the third Bohr orbit

The general formula for energy in the Bohr model is proportional to the inverse square of the principal quantum number (n). For the third orbit (n=3), the energy is:

E_3 = \frac{E_1}{3^2} = \frac{E_1}{9}

Step 2: Calculate the energy in the third orbit

Plugging in the given value of E_1:

E_3 = \frac{-2.18 \times 10^{-18}}{9} \, \text{J} = -2.422 \times 10^{-19} \, \text{J}

This calculation verifies that the energy of the electron in the third Bohr orbit is \( \frac{1}{9} \) of the energy of the electron in the first Bohr orbit.

Conclusion: The correct answer is the energy in the third Bohr orbit of the hydrogen atom is \(\frac{1}{9}\) of the energy in the first Bohr orbit.