Given below are two statements:
Statement I : According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus.
Statement II : According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in principle quantum number.
In the light of the above statements, choose the most appropriate answer from the options given below:

First we recall the basic relations of Bohr’s model for a hydrogen-like (one-electron) species of nuclear charge $$+Ze$$.

Bohr postulated that in the stationary orbit of principal quantum number $$n$$, the necessary centripetal force for circular motion of the electron is provided by the Coulombic attraction between the nucleus and the electron. Hence we write

$$\frac{m\,v_n^{\,2}}{r_n}= \frac{1}{4\pi\varepsilon_0}\;\frac{Z\,e^{2}}{r_n^{\,2}},$$

where $$m$$ is the electron mass, $$v_n$$ its speed in the $$n^{\text{th}}$$ orbit and $$r_n$$ the radius of that orbit.

Rearranging the above expression, we get

$$m\,v_n^{\,2}= \frac{1}{4\pi\varepsilon_0}\;\frac{Z\,e^{2}}{r_n}.$$

Bohr also gave the radius of the $$n^{\text{th}}$$ orbit as

$$r_n = \frac{\varepsilon_0\,h^{2}}{\pi\,m\,e^{2}}\; \frac{n^{2}}{Z},$$

where $$h$$ is Planck’s constant.

Substituting this $$r_n$$ into the previous force balance, we obtain

$$m\,v_n^{\,2}= \frac{1}{4\pi\varepsilon_0}\; \frac{Z\,e^{2}}{ \dfrac{\varepsilon_0\,h^{2}}{\pi\,m\,e^{2}}\; \dfrac{n^{2}}{Z}}.$$ Cancelling and simplifying step by step, we have

$$m\,v_n^{\,2}= \frac{1}{4\pi\varepsilon_0}\; \frac{Z^{2}e^{4}\pi\,m}{\varepsilon_0\,h^{2}\,n^{2}}.$$

Dividing both sides by $$m$$,

$$v_n^{\,2}= \frac{1}{4\pi\varepsilon_0}\; \frac{Z^{2}e^{4}\pi}{\varepsilon_0\,h^{2}\,n^{2}}.$$

Taking square root,

$$v_n = \frac{Z\,e^{2}}{2\varepsilon_0\,h}\;\frac{1}{n}.$$

Thus we have derived the well-known proportionality

$$v_n \propto \frac{Z}{n}.$$ Now we analyse the two statements in the light of the above relation.

• For Statement I we examine the dependence on $$Z$$. The formula shows that when $$Z$$ decreases (we move to a nucleus with fewer positive charges), the numerator $$Z$$ becomes smaller, so $$v_n$$ becomes smaller. Therefore the magnitude of velocity actually decreases with a decrease in positive nuclear charge. Hence Statement I is false.

• For Statement II we examine the dependence on $$n$$. Because velocity is inversely proportional to $$n$$, a decrease in the principal quantum number (i.e., moving to an inner orbit) makes $$n$$ smaller and hence $$v_n$$ larger. Therefore the magnitude of velocity indeed increases with decrease in $$n$$. Hence Statement II is true.

We have found that Statement I is false while Statement II is true. Among the given choices, this corresponds to Option A.

Hence, the correct answer is Option A.