Frequancy `=f_(1)`, Time period = T, Energy of `n^(th)` orbit `= E_(n)`, radius of `n^(th)` orbit `=r^(n)`, Atomic number = Z, Orbit number = n :
`{:(,"Column-I",,"Column-II",),((A),f,(p),n^(3),),((B),T,(q),Z^(2),),((E),E_(n),(r ),(1)/(n^(2)),),((D),(1)/(r_(n)),(s),Z,):}`

Answer

To solve the matching question, we will analyze the relationships between the terms in Column I and Column II based on the principles of atomic structure.

### Step 1: Analyze Frequency (f)
The frequency \( f \) is related to the time period \( T \) and can be expressed as:
\[ f \propto \frac{1}{T} \]
From the relationship of time period, we know:
\[ T \propto \frac{n^3}{Z^2} \]
Thus, substituting this into the frequency equation:
\[ f \propto \frac{Z^2}{n^3} \]
This means that frequency \( f \) is proportional to \( Z^2 \) and inversely proportional to \( n^3 \). Therefore, we can match:
- \( f \) with \( Z^2 \) (option Q).

### Step 2: Analyze Time Period (T)
From the previous analysis, we have:
\[ T \propto \frac{n^3}{Z^2} \]
This indicates that time period \( T \) is directly proportional to \( n^3 \). Therefore, we can match:
- \( T \) with \( n^3 \) (option P).

### Step 3: Analyze Energy of nth Orbit (E_n)
The energy \( E_n \) of the nth orbit is given by:
\[ E_n \propto \frac{Z^2}{n^2} \]
This means that energy \( E_n \) is proportional to \( Z^2 \) and inversely proportional to \( n^2 \). Therefore, we can match:
- \( E_n \) with \( Z^2 \) (option Q).

### Step 4: Analyze Radius of nth Orbit (r_n)
The radius \( r_n \) of the nth orbit is given by:
\[ r_n \propto \frac{n^2}{Z} \]
Taking the reciprocal gives:
\[ \frac{1}{r_n} \propto \frac{Z}{n^2} \]
This means that \( \frac{1}{r_n} \) is proportional to \( Z \) and inversely proportional to \( n^2 \). Therefore, we can match:
- \( \frac{1}{r_n} \) with \( Z \) (option S).

### Step 5: Final Matching
Now, let's summarize the matches:
- \( f \) matches with \( Z^2 \) (Q)
- \( T \) matches with \( n^3 \) (P)
- \( E_n \) matches with \( Z^2 \) (Q)
- \( \frac{1}{r_n} \) matches with \( Z \) (S)

### Final Answers:
- A -> Q
- B -> P
- E -> Q
- D -> S