Question

According to Bohr's theory the radius of electron in an orbit described by Principal quantum number \(\mathrm{n}\) and atomic number \(Z\), is Proportional to. (A) \(Z^{2} n^{2}\) (B) \(\left(\mathrm{n}^{2} / \mathrm{Z}\right)\) (C) \(\left(\mathrm{Z}^{2} / \mathrm{n}\right)\) (D) \(\left(\mathrm{n} / \mathrm{Z}^{2}\right)\)

Short answer

The correct answer is (B) \(\frac{n^2}{Z}\), as the radius of the electron's orbit in a hydrogen-like atom is proportional to this expression according to Bohr's theory.

Step-by-step solution

Recall Bohr's model for electron radius

According to Bohr's theory for hydrogen-like atoms, the radius of the electron's orbit (r) is determined by the following formula: \[r = \frac{n^2 h^2 \epsilon_0}{\pi Z e^2}\] Where: - n is the principal quantum number - h is Planck's constant - \(\epsilon_0\) is the vacuum permittivity - Z is the atomic number - e is the elementary charge - \(\pi\) is the mathematical constant

Simplify the formula to find the proportionality relationship

In order to find the proportionality relationship between r, n, and Z, we can simplify the existing formula by focusing on the terms containing n and Z: \[r \propto \frac{n^2}{Z}\] This means that the radius of the electron's orbit is proportional to the expression \(\frac{n^2}{Z}\).

Compare the simplified expression to the given options

Now we check which of the given options matches the proportionality relationship that we derived: (A) \(Z^2 n^2\) (B) \(\frac{n^2}{Z}\) (C) \(\frac{Z^2}{n}\) (D) \(\frac{n}{Z^2}\) From our analysis, we can see that option (B) \(\frac{n^2}{Z}\) matches the proportionality relationship we derived in Step 2. Therefore, the correct answer is (B) \(\frac{n^2}{Z}\).