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PhyEasyAdv 2008 • Paper 1

Ratio of Electron Kinetic Energies for Hydrogen and Helium

Comprehension Passage

Paragraph for Question Nos. 41 to 43

In a mixture of H−He+\text{H} - \text{He}^+ gas (He+\text{He}^+ is singly ionized He atom), H atoms and He+\text{He}^+ ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to He+\text{He}^+ ions (by collisions). Assume that the Bohr model of atom is exactly valid.

The ratio of the kinetic energy of the n=2n = 2 electron for the H atom to that of He+\text{He}^+ ion is

Options

A

14\frac{1}{4}

Correct
B

12\frac{1}{2}

C

1

D

2

Topics & Concepts

AtomsBohr Model

Step-by-Step Solution

According to the Bohr model of a hydrogen-like atom, the kinetic energy KK of an electron in an orbit with principal quantum number nn and atomic number ZZ is related to its total energy EnE_n by the virial theorem:
Kn=−EnK_n = -E_n

The total energy of an electron in the nn-th orbit is given by:
En=−me48ε02h2Z2n2=−13.6Z2n2 eVE_n = - \frac{m e^4}{8 \varepsilon_0^2 h^2} \frac{Z^2}{n^2} = - 13.6 \frac{Z^2}{n^2} \text{ eV}

Therefore, the kinetic energy is:
Kn=13.6Z2n2 eVK_n = 13.6 \frac{Z^2}{n^2} \text{ eV}

This shows that for a given orbit nn:
K∝Z2K \propto Z^2

For the hydrogen atom (H\text{H}):
ZH=1,n=2Z_{\text{H}} = 1, \quad n = 2

For the singly ionized helium ion (He+\text{He}^+):
ZHe+=2,n=2Z_{\text{He}^+} = 2, \quad n = 2

Taking the ratio of the kinetic energies:
KHKHe+=(ZH2n2)(ZHe+2n2)=(ZHZHe+)2=(12)2=14\frac{K_{\text{H}}}{K_{\text{He}^+}} = \frac{\left(\frac{Z_{\text{H}}^2}{n^2}\right)}{\left(\frac{Z_{\text{He}^+}^2}{n^2}\right)} = \left(\frac{Z_{\text{H}}}{Z_{\text{He}^+}}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}

Thus, the correct option is (A).

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