Question

Using Bohr atomic model drive expression for calculating orbits in He+ using this expression calculate the radius of fourth orbital of he+ion

Answer 1

Answer on Question #75754, Chemistry / Inorganic Chemistry

Using Bohr atomic model drive expression for calculating orbits in He+ using this expression calculate the radius of fourth orbital of he+ion

Solution

1. Expression for calculating orbits in He+.

Coulomb force is centripetal when electron moves in orbit. Then

$$\frac{\mathrm{Ze} \cdot \mathrm{e}}{4\pi\varepsilon_0 r^2} = \frac{m v^2}{r};$$ $$r = \frac{Z e^2}{4\pi\varepsilon_0 m v^2}.$$

According to Bohr postulates the angular momentum of stationary electron is quantized:

$$m \nu r = n h / 2\pi; \quad \Rightarrow \quad \nu = n h / 2\pi m r.$$

Then

$$r = \frac{Z e^2}{4\pi\varepsilon_0 m v^2} = \frac{Z e^2 4\pi^2 m^2 r^2}{4\pi\varepsilon_0 m n^2 h^2} = \frac{Z e^2 \pi m r^2}{\varepsilon_0 n^2 h^2} \Rightarrow r = \frac{\varepsilon_0 h^2}{Z e^2 \pi m} \cdot n^2, \text{ where } n = 1, 2, 3, \dots$$

$Z(\mathrm{He}^{+}) = 2$, then

$$r = \frac{\varepsilon_0 h^2}{2 e^2 \pi m} \cdot n^2, \text{ where } n = 1, 2, 3, \dots$$

2. The radius of fourth orbital of $\mathrm{He}^{+}$ ion

$$r_4 = \frac{\varepsilon_0 h^2}{Z e^2 \pi m} \cdot n^2 = \frac{8.85 \cdot 10^{-12} \cdot (6.626 \cdot 10^{-34})^2}{2 \cdot (1.6 \cdot 10^{-19})^2 \cdot 3.14 \cdot 9.1 \cdot 10^{-31}} \cdot 4^2 = 4.25 \cdot 10^{-10} \, (\mathrm{m}).$$

Answer: 1. $r = \frac{\varepsilon_0 h^2}{2 e^2 \pi m} \cdot n^2$, where $n = 1, 2, 3, \ldots$

2. $r_4 = 4.25 \cdot 10^{-10} \, \mathrm{m}$

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