Chapter 27, Problem 61P

To determine

Justify the use of non-relativistic formulas in the Bohr atom.

Answer to Problem 61P

Solution:

Yes, the use of non-relativistic formulas in the Bohr atom are justified.

Explanation of Solution

To the check the use of non-relativistic formulas in the Bohr atom, first we will calculate the electron’s velocity by the quantum condition as

$\begin{array}{l}mv{r}_1=\frac{nh}{2\pi }\\ \text{or,}v=\frac{nh}{2\pi m{r}_1}=\frac{(1)(6.63\times {10}^{-34}\text{J}\text{.s})}{2\pi (9.11\times {10}^{-31}\text{kg})(0.529\times {10}^{-10}\text{m})}=2.18\times {10}^6\text{m/s}\\ \text{or}\text{7}\text{.3}\times {\text{10}}^{-3}c\end{array}$

Now, the relativistic factor is given as

${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}=1-\frac{1}{2}{\left(\frac{v}{c}\right)}^2=1-2.7\times {10}^{-5}=0.99997$

This is approximately equals to 1 so we can say that, the use of non-relativistic formulas in the Bohr atom are justified.