Question

A proton and an electron have the same kinetic energy. Which one has a smaller de-Broglie wavelength and why?

Answer 1

Hint- To compare the de-Broglie wavelength of proton and neutron we should have some relation which can compare them with each other. The de-Broglie wavelength of a particle is inversely proportional to its momentum.

Step By Step Answer:
We know the relation,

$$\lambda ={\displaystyle \frac{h}{p}}$$

Where,

$\lambda =$de-Broglie wavelength
$p=$momentum of the particle
 $\lambda ={\displaystyle \frac{h}{\sqrt{2m(K.E.)}}}$--------equation (1)

Where,

$(K.E.)=$kinetic energy of the particle
$m=$mass of the particle
$h=$Planck's constant.

Here it is a given condition that the kinetic energy of both the particle (proton and electron) are equal.

Hence kinetic energy of proton = kinetic energy of the electron.

$(K.E.{)}_p=(K.E.{)}_e$
Where, $(K.E.{)}_p=$kinetic energy of the proton
$(K.E.{)}_e=$kinetic energy of the electron

So now we can see that in the equation (1) we conclude that the planck's constant is a constant which will not give any relation between de-Broglie wavelength of the proton and electron, and also kinetic energy of both particles are equal so only one quantity is remaining for getting the relation as follows,

$\lambda \propto {\displaystyle \frac{1}{m}}$,

This implies that the de-Broglie wavelength of the particle is inversely proportional to the mass of that particular particle.

As we know that the mass of the proton is more than the mass of the electron.

$m_e<m_p$

Where $m_e=$mass of the electron

$m_p=$mass of the proton

So finally, we will get the relation as follows,

${\lambda }_e>{\lambda }_p$

Hence the de-Broglie wavelength of the proton will be smaller than that of the electron.

Note- The mass of the proton is about 1800 times more than the mass of the electron, in other words the proton is approximately 1800 times more massive than the electron. Its momentum (at the same speed) is 1800 times that of an electron and therefore the de-Broglie wavelength is 1800 times smaller than that of the electron.