Question

The energy required to excite an electron from the first Bohr orbit of a hydrogen atom to the second Bohr orbit is:

• A. \(1.634\times 10^{-18}\,\text{J}\)
• B. \(1.2\times 10^{-19}\,\text{J}\)
• C. \(0.2\times 10^{-18}\,\text{J}\)
• D. \(1.2\times 10^{-20}\,\text{J}\)

Hint:

For hydrogen atom problems:
Use \(E_n=-\dfrac{13.6}{n^2}\,\text{eV}\)
Excitation energy is always positive (energy absorbed)
Ionization energy corresponds to transition from \(n=1\) to \(n=\infty\)

Answer 1

The Correct Option is A

Concept:  
The energy required to excite an electron from one orbit to another in the Bohr model is given by the energy difference between the two orbits. According to Bohr’s model of the hydrogen atom, the energy levels are quantized and given by the formula: \[ E_n = -\frac{k e^2}{2r_n} = - \frac{13.6}{n^2} \, \text{eV} \] where \( E_n \) is the energy of the nth orbit, \( n \) is the principal quantum number, and 13.6 eV is the energy of the first orbit (n = 1). 
Step 1: Energy of the first orbit (n = 1): \[ E_1 = - \frac{13.6}{1^2} = -13.6 \, \text{eV} \] 
Step 2: Energy of the second orbit (n = 2): \[ E_2 = - \frac{13.6}{2^2} = - \frac{13.6}{4} = -3.4 \, \text{eV} \] 
Step 3: Energy required for excitation: The energy required to excite the electron from the first orbit to the second orbit is the difference between the energies of the second and first orbits: \[ \Delta E = E_2 - E_1 = (-3.4 \, \text{eV}) - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \] 
Step 4: Convert the energy to joules: To convert from eV to joules, we use the conversion factor: \[ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \] Therefore, \[ \Delta E = 10.2 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = 1.634 \times 10^{-18} \, \text{J} \] 
Final Answer: \[ \boxed{1.634 \times 10^{-18} \, \text{J}} \]