Question

The speed of a proton is one-hundredth of the speed of light in vacuum. What is its de-Broglie wavelength?
Assume that one mole of protons has a mass equal to one gram. $ \left[ {h = 6.626 \times {{10}^{ - 34}}Js} \right] $
A. $ 1.3 \times {10^{ - 3}}{A^o} $
B. $ 13.33 \times {10^{ - 3}}{A^o} $
C. $ 13.13 \times {10^{ - 13}}{A^o} $
D. $ 1.33 \times {10^{ - 13}}{A^o} $

Answer 1

Hint: Calculation of the de Broglie wavelength is done for a specific proton with the help of velocity. Taking the value of the one mole of a proton to have the mass of equal to one gram, the mass of each of the protons can be determined. The wavelength equation for de Broglie wavelength is used as there is only one variable.

Complete step by step solution
The speed of the proton is given as $ \dfrac{1}{{100}} $ of the speed of light in vacuum. This means that the speed of the proton is $ v = \dfrac{{3 \times {{10}^8}}}{{100}} = 3 \times {10^6}m/s $ .
Based on the velocity of the protons involved the de Broglie wavelength can be calculated in the specific formula for the wavelength. The formula for determining the de Broglie wavelength is:
 $ \lambda = \dfrac{h}{{mv}} $
Here in the given formula the $ \lambda $ is the de Broglie wavelength, the Planck's constant is denoted by $ h $ and the mass of the proton involved in the process is denoted by $ m $ . The mass of the proton has a constant value, which is $ m = 1.672 \times {10^{ - 27}}kg $ and the value for the Plank’s constant is $ h = 6.626 \times {10^{ - 34}}Js $ . But the velocity is the variable in the equation and the velocity is previously determined, which is why the given value for de Broglie wavelength will be:
 $ \lambda = \dfrac{{6.626 \times {{10}^{ - 34}}}}{{(1.672 \times {{10}^{ - 27}}) \times (3 \times {{10}^6})}} $
From here we get,
 $ \Rightarrow \lambda = \dfrac{{6.626 \times {{10}^{ - 13}}}}{{5.016}} $
After dividing we get,
 $ \Rightarrow \lambda = 1.321 \times {10^{ - 13}} $
The value of de Broglie wavelength that is calculated from every given proton is almost close to that of the value $ 1.33 \times {10^{ - 13}}m $ which when converted in units the value will be $ 1.33 \times {10^{ - 3}}{A^o} $ , which proves that the correct choice for this is A. $ 1.3 \times {10^{ - 3}}{A^o} $ which is the approximate value of de Broglie wavelength.

Hence, option A is correct

Note
The importance of de Broglie wavelength can be determined based on the velocity at which the fundamental particles are moving inside the atom. The changes in the wavelength are determined according to the changes in the velocity of the fundamental particles that are present in the atom.