Question

A photon emitted during the de-excitation of electron from a state n to the second excited state in a hydrogen atom, irradiates a metallic electrode of work function 0.5 eV, in a photocell, with a stopping voltage of 0.47 V. Obtain the value of quantum number of the state 'n'.

• A. 5
• B. 6
• C. 4
• D. 3

Hint:

In photoelectric experiments, use the energy balance equation to relate photon energy, work function, and stopping voltage. The energy difference between states is key to determining the quantum number.

Answer 1

The Correct Option is A

The energy of a photon emitted during a de-excitation from state \( n \) to the second excited state \( n = 2 \) can be calculated using the Rydberg formula for hydrogen: \[ E_n = -13.6 \, \text{eV} \left( \frac{1}{n^2} - \frac{1}{2^2} \right) \] Where \( E_n \) is the energy corresponding to the state \( n \). The photon energy is given by the difference in energy between the \( n \)-th state and the second excited state \( (n=2) \): \[ E_{\text{photon}} = E_n - E_2 \] The energy of the photon is used to overcome the work function of the metallic electrode and still provide kinetic energy to the emitted electron. This is given by the photoelectric equation: \[ E_{\text{photon}} = \text{Work Function} + K.E. \] Here, \( K.E. \) is the kinetic energy of the emitted electron, which is related to the stopping voltage \( V \) by: \[ K.E. = eV \] Substituting values: \[ E_{\text{photon}} = 0.5 \, \text{eV} + 0.47 \, \text{eV} = 0.97 \, \text{eV} \] Now, using the Rydberg equation for \( n = 5 \): \[ E_5 = -13.6 \, \text{eV} \left( \frac{1}{5^2} - \frac{1}{2^2} \right) \] \[ E_5 = -13.6 \, \text{eV} \left( \frac{1}{25} - \frac{1}{4} \right) = -13.6 \, \text{eV} \left( \frac{4 - 25}{100} \right) \] \[ E_5 = -13.6 \, \text{eV} \times \frac{-21}{100} = 2.856 \, \text{eV} \] The energy of the photon is the difference between the energy of the initial state \( n \) and the final state \( n = 2 \). Using this, we determine that \( n = 5 \) satisfies the conditions for the photon energy and stopping voltage.
Thus, the quantum number of the state \( n \) is \( 5 \).