Question 444: Atoms & Nuclei

Question

In a hydrogen atom in its ground state, the first Bohr orbit has radius $r_1$. The electron's orbital speed becomes one-third when the atom is raised to one of its excited states. The radius of the orbit in that excited state is

Accepted Answer

The radius of the Bohr orbit in a hydrogen atom is given by the equation:
$$ r_n = n^2 \cdot r_1 $$
where $ r_n $ is the radius of the orbit in the $ n $-th excited state, and $ r_1 $ is the radius of the first Bohr orbit in the ground state, which is approximately $ 5.29 	imes 10^{-11} $ meters.
The orbital speed of an electron in the Bohr model is inversely proportional to the principal quantum number $ n $:
$$ v_n = \frac{v_1}{n} $$
If the electron's orbital speed becomes one-third of its speed in the ground state, we have:
$$ \frac{v_1}{n} = \frac{v_1}{3} $$
This simplifies to:
$$ n = 3 $$
Substituting $ n = 3 $ into the equation for the radius:
$$ r_n = 3^2 \cdot r_1 = 9 \cdot r_1 $$
Thus, the radius of the orbit in the excited state is:
Option C
$$ 9 \cdot r_1 $$

Answer

$3 r_1$

Answer

$4 r_1$

Answer

$9 \pi_1$

Answer

$16 \mathrm{r}_1$

Atoms & Nuclei

In a hydrogen atom in its ground state, the first Bohr orbit has radius r_1. The electron's orbital speed becomes one-third when the atom is raised to one of its excited states. The radius of the orbit in that excited state is

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