If in hydrogen atom, radius of n^th Bohr orbit is $r_n$ . Time period and frequency of revolution of electron in nth orbit are ${\mathrm{T}}_{\mathrm{n}}$ and ${\mathrm{f}}_{\mathrm{n}}\text{,}$ choose the correct option.

In the Bohr model of the hydrogen atom, the radius of nth orbit is given by:

r_n ∝ n²

This relationship indicates that the radius of the nth orbit increases proportionally to the square of the principal quantum number (n). The logarithmic form of this relationship can be expressed as:

ln(r_n/r₁) = 2ln(n)

This equation can be compared to a linear equation in the form y = mx, where:

• y → ln(r_n/r₁)
• x → ln(n)
• m → 2

The slope of this graph is positive, confirming the proportional relationship between the logarithm of the radius of nth orbit and the logarithm of the quantum number.

Time Period and Frequency in the nth Orbit

The time period of revolution (T_n) is directly proportional to n³, as:

T_n ∝ n³

This leads to the selection of option (a). The frequency of revolution (f_n), being the inverse of the time period, follows:

f_n ∝ 1/n³

When plotting ln(f_n/f₁) against ln(n), the graph has a negative slope, as the frequency decreases with increasing quantum number. This confirms that the graph between ln(f_n/f₁) and ln(n) does not have a positive slope.

Key Observations:

• The radius of nth orbit grows with the square of the quantum number.
• The time period of revolution increases with the cube of the quantum number.
• The frequency of revolution decreases with the cube of the quantum number.

Thus, the correct options are:

• Option (a) for the time period of revolution.
• Option (b) for the logarithmic relationship of the radius of nth orbit.

This detailed explanation highlights the significance of the radius of nth orbit in understanding the behavior of the hydrogen atom's electron.