Question 178: Atoms & Nuclei

Question

The radius of third stationary orbit of electron for Bohr's atom is R. The radius of fourth stationary orbit will be:

Accepted Answer

The radius of the nth stationary orbit in the Bohr model of an atom is directly proportional to the square of its principal quantum number n and inversely proportional to the atomic number Z. This relationship is represented by the formula $$r_n = \frac{n^2h^2}{4\pi^2kme^2Z},$$ where :
$n$ is the principal quantum number,
$h$ is Planck's constant,
$k$ is the Coulomb constant,
$m$ is the mass of the electron,
$e$ is the charge of the electron, and
$Z$ is the atomic number of the nucleus (for hydrogen, Z=1).
To find the radius of the fourth orbit ($r_4$), in comparison to the third orbit ($r_3$), we apply the formula with $n=4$ for the fourth orbit and $n=3$ for the third orbit, and simplify as follows:
$$\begin{aligned}

& \frac{r_4}{r_3} = \frac{(4^2)h^2}{4\pi^2kme^2Z} \div \frac{(3^2)h^2}{4\pi^2kme^2Z} = \frac{(4^2)}{(3^2)} \\

& = \frac{16}{9}

\end{aligned}$$
Thus, the radius of the fourth stationary orbit is $\frac{16}{9}$ times the radius of the third stationary orbit, $R$. Therefore, $$r_4 = \frac{16}{9} R$$.

Answer

$$\frac{4}{3} \mathrm{R}$$

Answer

$$\frac{16}{9} R$$

Answer

$$\frac{3}{4} R$$

Answer

$$\frac{9}{16} \mathrm{R}$$

Atoms & Nuclei

The radius of third stationary orbit of electron for Bohr's atom is R. The radius of fourth stationary orbit will be:

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