Question 479: Atoms & Nuclei

Question

The force acting on the electrons in hydrogen atom (Bohr's theory) is related to the principle quantum number $$n$$ as

Accepted Answer

In Bohr's theory of the hydrogen atom, the force acting on the electrons is the electrical force between the positively charged nucleus (proton) and the negatively charged electron. This force is given by Coulomb's law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
The relevant formula from Coulomb's law is
$$F = k \frac{e^2}{r^2}$$
where $F$ is the force between the electron and the nucleus, $k$ is Coulomb's constant, $e$ is the charge of the electron (the same magnitude for the proton), and $r$ is the distance between the nucleus and the electron.
In Bohr's model, the radius of the $n$th orbit (where $n$ is the principal quantum number) is directly proportional to $n^2$. This relation is given by the formula for the radius of the $n$th orbit as,
$$r_n = n^2 \cdot a_0$$
where $a_0$ is the Bohr radius, the radius of the electron's orbit in its ground state (when $n=1$).
Substituting the expression for $r_n$ into the formula for $F$, we get
$$F = k \frac{e^2}{(n^2 a_0)^2} = k \frac{e^2}{n^4 a_0^2}$$
This shows that the force $F$ is inversely proportional to $n^4$, indicating that as the principal quantum number $n$ increases, the force acting on the electrons decreases. Thus, the force acting on the electrons in a hydrogen atom, according to Bohr's theory, is related to the principal quantum number $n$ as $n^{-4}$.
Therefore, the correct option is:
Option A: $$n^{-4}$$

Answer

$$n^{-4}$$

Answer

$$n^4$$

Answer

$$n^{-2}$$

Answer

$$n^2$$

Atoms & Nuclei

The force acting on the electrons in hydrogen atom (Bohr's theory) is related to the principle quantum number $n$ as

Choose the correct answer: