Question

If the de Broglie wavelengths associated with a proton and an $\alpha$-particle are equal, then the ratio of the velocities of the proton and the $\alpha$-particle will be:
$A)\text{ }1:4$
$B)\text{ }1:2$
$C)\text{ 4}:1$
$D)\text{ 2}:1$

Answer 1

This problem can be solved by using the direct formula for the de Broglie wavelength of a particle in terms of its mass and velocity. By using the direct formula, we can get the wavelength of the proton and the $\alpha$-particle and get the required ratio.
Formula used:
$\lambda =\dfrac{h}{mv}$

Complete step-by-step answer:
We will use the direct formula for the de Broglie wavelength of a particle in terms of its mass and velocity.
The de Broglie wavelength $\lambda$ of a particle of mass $m$ moving with velocity $v$ is given by
$\lambda =\dfrac{h}{mv}$ --(1)
Where $h=6.636\times {{10}^{-34}}J.s$ is the Planck’s constant.
Now, let us analyze the question.
Let the velocities of the proton and the $\alpha$-particle be ${{v}_{p}}$ and ${{v}_{\alpha }}$ respectively.
Let the masses of the proton and the $\alpha$-particle be ${{m}_{p}}$ and ${{m}_{\alpha }}$ respectively.
The de Broglie wavelengths of the proton and the $\alpha$-particle are given to be equal. Let the de Broglie wavelengths be $\lambda$.
Using (1), we get,
$\dfrac{h}{{{m}_{p}}{{v}_{p}}}=\dfrac{h}{{{m}_{\alpha }}{{v}_{\alpha }}}$
$\therefore \dfrac{{{v}_{p}}}{{{v}_{\alpha }}}=\dfrac{{{m}_{\alpha }}}{{{m}_{p}}}$ --(2)
Now, an alpha particle is nothing but a helium nuclei with two protons and two neutrons.
The mass ${{m}_{\alpha }}$ of an alpha particle is therefore, ${{m}_{\alpha }}=4u$
The mass of a proton is ${{m}_{p}}=u$.
Using this information in (2), we get
$\therefore \dfrac{{{v}_{p}}}{{{v}_{\alpha }}}=\dfrac{4u}{u}=4=4:1$
Therefore, the required ratio is $4:1$.
Therefore, the correct option is $C)\text{ 4}:1$.

Note: We could also have solved this problem by writing the de Broglie wavelength to be inversely proportional to the mass and the velocity of a particle and thereby, solving the problem. In this way we would have got the result in the calculation directly and also avoid the use of the unnecessary variable of the Planck’s constant. However, it is better to write the full formula as students often make mistakes while writing these proportionality equations.