Question 648: Atoms & Nuclei

Question

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

Accepted Answer

To determine the ratio of the kinetic energy of the $n=2$ electron for the H atom to that of the He$^+$ ion, we need to consider the Bohr model of the atom. According to the Bohr model, the kinetic energy (KE) of an electron in a hydrogen-like ion in the nth orbit can be given by:
$ KE_n = \frac{1}{2} m v_n^2 = \frac{Z^2 e^4 m}{8 \epsilon_0^2 h^2 n^2} $
where:
$ m $ is the mass of the electron
$ Z $ is the atomic number of the nucleus
$ e $ is the charge of the electron
$ \epsilon_0 $ is the permittivity of free space
$ h $ is Planck's constant
$ n $ is the principle quantum number, which is 2 in this case
For a hydrogen atom (H) ($ Z = 1 $) in the $ n = 2 $ state:
$ KE_{H (n=2)} = \frac{1^2 \cdot e^4 m}{8 \epsilon_0^2 h^2 \cdot 2^2} = \frac{e^4 m}{32 \epsilon_0^2 h^2} $
For a singly ionized helium ion (He$^+$) ($ Z = 2 $) in the $ n = 2 $ state:
$ KE_{He^+ (n=2)} = \frac{2^2 \cdot e^4 m}{8 \epsilon_0^2 h^2 \cdot 2^2} = \frac{4 e^4 m}{32 \epsilon_0^2 h^2} = \frac{e^4 m}{8 \epsilon_0^2 h^2} $
To find the ratio of the kinetic energy of the $ n=2 $ electron for the H atom to that of the He$^+$ ion, we divide the kinetic energies:
$ 	ext{Ratio} = \frac{KE_{H (n=2)}}{KE_{He^+ (n=2)}} = \frac{\frac{e^4 m}{32 \epsilon_0^2 h^2}}{\frac{e^4 m}{8 \epsilon_0^2 h^2}} = \frac{1}{4} $
Therefore, the ratio of the kinetic energy of the $ n=2 $ electron for the H atom to that of the He$^+$ ion is:
Option A: $\frac{1}{4}$

Answer

$\frac{1}{4}$

Answer

$\frac{1}{2}$

Answer

1

Answer

2

Atoms & Nuclei

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

Choose the correct answer: