Match List I with List II and select the correct answer using the codes given below the lists: | List I | List II | | --- | --- | | $P$. Boltzmann constant | 1. $[ML^{2}T^{-1}]$ | | $Q$. Coefficient of viscosity | 2. $[ML^{-1}T^{-1}]$ | | $R$. Planck constant | 3. $[MLT^{-3}K^{-1}]$ | | $S$. Thermal conductivity | 4. $[ML^{2}T^{-2}K^{-1}]$ | $PQRS$ is

Step 1: Find the dimensional formula for Boltzmann constant (P).

The Boltzmann constant $k_B$ relates energy and temperature.

$$k_B = \frac{\text{Energy}}{\text{Temperature}}$$

Substitute the dimensions of energy $[ML^{2}T^{-2}]$ and temperature $[K]$.

$$[k_B] = \frac{[ML^{2}T^{-2}]}{[K]} = [ML^{2}T^{-2}K^{-1}]$$

This matches with entry 4 in List II.

$$P \to 4$$

Step 2: Find the dimensional formula for the coefficient of viscosity (Q).

From Newton's law of viscosity, the viscous force is given by:

$$F = \eta A \frac{dv}{dx}$$ $$\eta = \frac{F}{A \left(\frac{dv}{dx}\right)}$$

Substitute the dimensions of force $[MLT^{-2}]$, area $[L^{2}]$, and velocity gradient $[T^{-1}]$.

$$[\eta] = \frac{[MLT^{-2}]}{[L^{2}][T^{-1}]} = [ML^{-1}T^{-1}]$$

This matches with entry 2 in List II.

$$Q \to 2$$

Step 3: Find the dimensional formula for Planck constant (R).

Using the energy of a photon:

$$E = h\nu$$ $$h = \frac{E}{\nu}$$

Substitute the dimensions of energy $[ML^{2}T^{-2}]$ and frequency $[T^{-1}]$.

$$[h] = \frac{[ML^{2}T^{-2}]}{[T^{-1}]} = [ML^{2}T^{-1}]$$

This matches with entry 1 in List II.

$$R \to 1$$

Step 4: Find the dimensional formula for thermal conductivity (S).

From Fourier's law of heat conduction, the rate of heat flow is:

$$\frac{dQ}{dt} = \kappa A \frac{dT}{dx}$$ $$\kappa = \frac{\frac{dQ}{dt}}{A \left(\frac{dT}{dx}\right)}$$

Substitute the dimensions of heat flow rate (power) $[ML^{2}T^{-3}]$, area $[L^{2}]$, and temperature gradient $[K L^{-1}]$.

$$[\kappa] = \frac{[ML^{2}T^{-3}]}{[L^{2}][K L^{-1}]} = [MLT^{-3}K^{-1}]$$

This matches with entry 3 in List II.

$$S \to 3$$

Step 5: Combine the matches to find the code for PQRS.

Combining the individual results:

$$P \to 4, \quad Q \to 2, \quad R \to 1, \quad S \to 3$$

The correct sequence for $PQRS$ is $4, 2, 1, 3$.

Answer: $4, 2, 1, 3$ (Option 3)