Match the following columns, `{:("Column1","Coloumn2"),("a Specific heat","p [MLT^(-3)K^(-1)]"),("b Coefficient of thermal conductivity","q [MT^(-3)K^(-4)]"),("c Boltzmann constant","r [L^(2)T^(-2)K^(-1)]"),("d Stefan's constat","s [ML^(2)T^(-2)K^(-1)]"):}`

Answer

To solve the problem of matching the columns based on the physical quantities and their dimensions, we will analyze each item in Column 1 and find its corresponding dimension from Column 2.

### Step-by-Step Solution:

1. **Specific Heat (a)**:
- Specific heat (C) is defined as the amount of heat (Q) required to raise the temperature of a unit mass (m) by one degree (T).
- The formula is:
\[ C = \frac{Q}{m \Delta T} \]
- The dimension of heat (Q) is \( [M L^2 T^{-2}] \).
- The dimension of mass (m) is \( [M] \).
- The dimension of temperature (T) is \( [K] \).
- Therefore, the dimension of specific heat becomes:
\[ [C] = \frac{[M L^2 T^{-2}]}{[M][K]} = [M^0 L^2 T^{-2} K^{-1}] \]
- This matches with option **(p)**.

2. **Coefficient of Thermal Conductivity (b)**:
- The coefficient of thermal conductivity (K) is defined as:
\[ K = \frac{Q \cdot L}{A \cdot \Delta T} \]
- Here, \( A \) is the area, and \( L \) is the length.
- The dimension of area (A) is \( [L^2] \).
- Therefore, substituting the dimensions:
\[ [K] = \frac{[M L^2 T^{-2}] \cdot [L]}{[L^2] \cdot [K]} = \frac{[M L^3 T^{-2}]}{[L^2][K]} = [M L T^{-3} K^{-1}] \]
- This matches with option **(q)**.

3. **Boltzmann Constant (c)**:
- The Boltzmann constant (k) is defined as the energy per temperature:
\[ k_B = \frac{E}{T} \]
- The dimension of energy (E) is \( [M L^2 T^{-2}] \).
- Therefore, the dimension of the Boltzmann constant becomes:
\[ [k_B] = \frac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}] \]
- This matches with option **(s)**.

4. **Stefan's Constant (d)**:
- Stefan's constant (σ) is defined in relation to the energy emitted per unit area per unit time per unit temperature raised to the fourth power:
\[ \sigma = \frac{E}{A \cdot t \cdot T^4} \]
- Therefore, substituting the dimensions:
\[ [\sigma] = \frac{[M L^2 T^{-2}]}{[L^2] \cdot [T] \cdot [K^4]} = \frac{[M L^2 T^{-2}]}{[L^2 T K^4]} = [M L^0 T^{-3} K^{-4}] \]
- This matches with option **(r)**.

### Final Matching:
- a → p
- b → q
- c → s
- d → r