Question

Calculate the wavelength (in nanometre) associated with a proton moving at $1.0\times {{10}^{3}}m{{s}^{-1}}$.
($mass$ $of$ $proton =$ $1.67\times {{10}^{-27}}Kg$ and $h =$ $6.63\times {{10}^{-34}}Js$)
(A) 0.032 nm
(B) 0.40 nm
(C) 2.5 nm
(D) 14.0 nm

Answer 1

Hint: The wavelength associated with the moving proton can be calculated by de Broglie’s principle. The pre-knowledge of waves is necessary while solving a given problem.

Complete step by step solution:
Let us see some basics of structure of atom as stated by quantum mechanical model,
The developments for the new model of structure of an atom after Bohr’s model were;
1. The dual nature of matter
2. Heisenberg’s Uncertainty Principle
Here, we will see about dual nature of matter,
de-Broglie’s principle-
This states that light exhibits dual nature i.e. particle and wave; every microscopic matter such as electron, proton, etc. also has dual nature.
Thus, all material particles in motion possess wave characteristics.
According to de-Broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by,
\[\lambda =\dfrac{h}{p}=\dfrac{h}{mv}\]
where, p = momentum of the particle in motion.
Although the dual nature of matter is applicable for all material objects, it is significant for microscopic bodies only. For larger bodies, the wavelength of the associated waves is very small and cannot be measured by the above method.
Illustration-
Given that,
mass of proton = $1.67\times {{10}^{-27}}Kg$
$h =$ $6.63\times {{10}^{-34}}Js$
velocity of moving proton = $1.0\times {{10}^{3}}m{{s}^{-1}}$
Thus, using above equation
\[\lambda =\dfrac{h}{p}=\dfrac{h}{mv}\]
$\lambda =\dfrac{6.63\times {{10}^{-34}}}{1.67\times {{10}^{-27}}\times 1\times {{10}^{3}}}=0.40\times {{10}^{-9}}m$
Hence, the wavelength of the moving proton is 0.40 nm.

Therefore, option (B) is correct.

Note: Check the units while calculating for any given specific unit (here in nm).
Use the wavelength equation according to the given data as we have two types of equations for wavelength calculations i.e. one for given planck's constant (h) and other for velocity of light (c).