Question

When a photon is emitted by a hydrogen atom, the photon carries a momentum with it. (a) Calculate the momentum carries by the photon when a hydrogen atom emits light of wavelength 656.3 nm. (b) With what speed does the atom recoil during this transition? Take the mass of the hydrogen atom = 1.67 × 10⁻²⁷ kg. (c) Find the kinetic energy of recoil of the atom.

Answer 1

Given:
Wavelength of light emitted by hydrogen, λ = 656.3 nm
Mass of hydrogen atom, m = 1.67 × 10⁻²⁷ kg

(a) Momentum $\left(p\right)$ is given by
$P=\frac{h}{\lambda }$
Here,
h = Planck's constant
λ = Wavelength of light

$$\begin{gathered} \therefore p=\frac{6.63\times {10}^{-34}}{656.3\times {10}^{-9}} \\ p=0.01\times {10}^{-25} \\ p=1\times {10}^{-27}\mathrm{kgm}/\mathrm{s} \end{gathered}$$

(b) Momentum, p = mv
Here,
m = Mass of hydrogen atom
v = Speed of atom
$\therefore$ 1 × 10⁻²⁷ = (1.67 × 10⁻²⁷)× υ
$$\begin{gathered} \Rightarrow \upsilon =\frac{1}{1.67} \\ =0.598=0.6\mathrm{m}/\mathrm{s} \end{gathered}$$

(c) Kinetic energy $\left(K\right)$ of the recoil of the atom is given by
$K=\frac{1}{2}m{v}^2$
Here,
m = Mass of the atom
v = Velocity of the atom
$\therefore K=\frac{1}{2}\times \left(1.67\times {10}^{-27}\right)\times {\left(0.6\right)}^2\mathrm{J}$
$$\begin{gathered} K=\frac{0.3006\times {10}^{-27}}{1.6\times {10}^{-19}}\mathrm{eV} \\ K=1.9\times {10}^{-9}\mathrm{eV} \end{gathered}$$