The user wants me to solve a JEE Chemistry question based on Bohr's model of the atom. The question is a matching type, involving two lists. List-I has physical quantities related to an electron in the nth orbit, and List-II describes their dependence on the principal quantum number 'n'.

List-I: (I) Radius of the nth orbit (II) Angular momentum of the electron in the nth orbit (III) Kinetic energy of the electron in the nth orbit (IV) Potential energy of the electron in the nth orbit

List-II: (P) $\propto n^{-2}$ (Q) $\propto n^{-1}$ (R) $\propto n^{0}$ (S) $\propto n^{1}$ (T) $\propto n^{2}$ (U) $\propto n^{1/2}$

I need to find the correct combination from the given options.

Let's analyze each quantity in List-I and determine its dependence on 'n' based on Bohr's model for a one-electron atom with atomic number Z.

Step 1: Analyze the Radius of the nth orbit (I)

According to Bohr's model, the radius of the nth orbit ($r_n$) is given by: $r_n = \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2}$ For a given atom, Z is constant. All other terms ($h, \epsilon_0, \pi, m, e$) are constants. Therefore, the radius is directly proportional to the square of the principal quantum number. $r_n \propto n^2$ So, the combination (I) matches with (T).

Step 2: Analyze the Angular momentum of the electron in the nth orbit (II)

Bohr's second postulate states that the angular momentum ($L_n$) of an electron in a stationary orbit is an integral multiple of $h/(2\pi)$. $L_n = \frac{nh}{2\pi}$ Since h and $2\pi$ are constants, the angular momentum is directly proportional to the principal quantum number. $L_n \propto n$ So, the combination (II) matches with (S).

Step 3: Analyze the Kinetic energy of the electron in the nth orbit (III)

The total energy of an electron in the nth orbit ($E_n$) is given by: $E_n = -\frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2}$ $E_n \propto \frac{1}{n^2} \propto n^{-2}$ In the Bohr model, there's a relationship between total energy ($E_n$), kinetic energy ($KE_n$), and potential energy ($PE_n$): $KE_n = -E_n$ $PE_n = 2E_n$ Therefore, the kinetic energy is: $KE_n = -E_n = -\left(-\frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2}\right) = \frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2}$ For a given atom, the kinetic energy is inversely proportional to the square of the principal quantum number. $KE_n \propto n^{-2}$ So, the combination (III) matches with (P).

Step 4: Analyze the Potential energy of the electron in the nth orbit (IV)

Using the relationship $PE_n = 2E_n$: $PE_n = 2 \left(-\frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2}\right) = -\frac{m Z^2 e^4}{4 \epsilon_0^2 h^2 n^2}$ For a given atom, the potential energy is also inversely proportional to the square of the principal quantum number. $PE_n \propto n^{-2}$ So, the combination (IV) also matches with (P).

Step 5: Evaluate the given options

Let's check the provided options based on our analysis:

• (I) Radius $\propto n^2$ (Matches T)
• (II) Angular momentum $\propto n$ (Matches S)
• (III) Kinetic energy $\propto n^{-2}$ (Matches P)
• (IV) Potential energy $\propto n^{-2}$ (Matches P)

Now, let's evaluate the options:

• A: (III), (P) - This states that Kinetic energy is proportional to $n^{-2}$. Our analysis confirms this. This is a correct combination.
• B: (III), (S) - This states that Kinetic energy is proportional to $n^1$. This is incorrect. Angular momentum is proportional to $n^1$.
• C: (IV), (U) - This states that Potential energy is proportional to $n^{1/2}$. This is incorrect. Potential energy is proportional to $n^{-2}$.
• D: (IV), (Q) - This states that Potential energy is proportional to $n^{-1}$. This is incorrect. The velocity of the electron is proportional to $n^{-1}$.

Based on the evaluation, the only correct combination among the choices is (III), (P).

Final Answer is A.