Question

If a hydrogen atom emits a Photon of wave length \(\lambda\). the recoil speed of the atom of mass \(\mathrm{m}\) is given by (A) \((\mathrm{h} / \mathrm{m} \lambda)\) (B) \((\mathrm{mh} / \lambda)\) (C) \(\operatorname{mh} \lambda\) (D) \((\mathrm{m} \lambda / \mathrm{h})\)

Short answer

The short answer is: The recoil speed of the hydrogen atom is given by v = \(\frac{h}{m \lambda}\), which matches option (A).

Step-by-step solution

Understand the given parameters and variables

We are given the following variables and parameters: - Wavelength of the emitted photon: \(\lambda\) - Mass of the hydrogen atom: m - Planck's constant: h Our task is to find the correct expression for the recoil speed of the hydrogen atom, given these parameters.

Find the momentum of the emitted photon

According to the de Broglie hypothesis, the momentum of a photon can be expressed as: Momentum of the photon (p_photon) = \(\frac{h}{\lambda}\) Where h is the Planck's constant and \(\lambda\) is the wavelength of the emitted photon.

Apply the conservation of momentum to find the momentum of the hydrogen atom

According to conservation of momentum, the momentum of the hydrogen atom after the emission of the photon is equal and opposite to the momentum of the emitted photon. Thus, the momentum of the hydrogen atom can be expressed as: Momentum of the hydrogen atom (p_hydrogen) = -p_photon

Find the recoil speed of the hydrogen atom

The momentum of the hydrogen atom can be related to its mass, m, and its recoil speed, v, by the following expression: p_hydrogen = m * v We can now substitute the expression for p_hydrogen from Step 3: m * v = -\(\frac{h}{\lambda}\) Now, we need to find the expression for the recoil speed, v. We can do this by isolating v: v = -\(\frac{h}{m \lambda}\) Considering that only the magnitudes are shown in the options, we can disregard the negative sign: v = \(\frac{h}{m \lambda}\) Comparing this expression to the provided options, we can see that it matches option (A). Thus, the recoil speed of the hydrogen atom is given by: v = \(\frac{h}{m \lambda}\)