The related equations are : `Q=mc(T_(2)-T_(1)), l_(1)=l_(0)[1+alpha(T_(2)-T_(1))]` and `PV-nRT`,
where the symbols have their usual meanings. Find the dimension of
(A) specific heat capacity (C) (B) coefficient of linear expansion `(alpha)` and (C) the gas constant (R).

Explanation

To find the dimensions of specific heat capacity (C), coefficient of linear expansion (α), and the gas constant (R), we will analyze each equation step by step.

### Step 1: Specific Heat Capacity (C)

1. **Start with the equation**:
\[ Q = mc(T_2 - T_1) \]
Here, \(Q\) is the heat energy, \(m\) is the mass, \(C\) is the specific heat capacity, and \(T_2 - T_1\) is the change in temperature.

2. **Rearranging the equation for C**:
\[ C = \frac{Q}{m(T_2 - T_1)} \]

3. **Identify the dimensions**:
- The dimension of heat energy \(Q\) is the same as energy, which is:
\[ [Q] = [E] = M^1 L^2 T^{-2} \]
- The dimension of mass \(m\) is:
\[ [m] = M^1 \]
- The dimension of temperature change \((T_2 - T_1)\) is:
\[ [T] = K^1 \]

4. **Substituting dimensions into the equation for C**:
\[ [C] = \frac{[Q]}{[m][T]} = \frac{M^1 L^2 T^{-2}}{M^1 K^1} \]

5. **Simplifying the dimensions**:
\[ [C] = L^2 T^{-2} K^{-1} \]

### Step 2: Coefficient of Linear Expansion (α)

1. **Start with the equation**:
\[ L_1 = L_0(1 + \alpha(T_2 - T_1)) \]

2. **Rearranging for α**:
\[ \alpha = \frac{L_1 - L_0}{L_0(T_2 - T_1)} \]

3. **Identify the dimensions**:
- The dimension of length \((L_1 - L_0)\) is:
\[ [L] = L^1 \]
- The dimension of temperature change \((T_2 - T_1)\) is:
\[ [T] = K^1 \]

4. **Substituting dimensions into the equation for α**:
\[ [\alpha] = \frac{[L]}{[L][T]} = \frac{L^1}{L^1 K^1} \]

5. **Simplifying the dimensions**:
\[ [\alpha] = K^{-1} \]

### Step 3: Gas Constant (R)

1. **Start with the equation**:
\[ PV = nRT \]

2. **Rearranging for R**:
\[ R = \frac{PV}{nT} \]

3. **Identify the dimensions**:
- The dimension of pressure \(P\) is:
\[ [P] = M^1 L^{-1} T^{-2} \]
- The dimension of volume \(V\) is:
\[ [V] = L^3 \]
- The dimension of number of moles \(n\) is:
\[ [n] = \text{mol} \]
- The dimension of temperature \(T\) is:
\[ [T] = K^1 \]

4. **Substituting dimensions into the equation for R**:
\[ [R] = \frac{[P][V]}{[n][T]} = \frac{(M^1 L^{-1} T^{-2})(L^3)}{(\text{mol})(K^1)} \]

5. **Simplifying the dimensions**:
\[ [R] = \frac{M^1 L^{2} T^{-2}}{\text{mol} \cdot K^1} \]

### Final Results
- **Dimension of Specific Heat Capacity (C)**:
\[ [C] = L^2 T^{-2} K^{-1} \]
- **Dimension of Coefficient of Linear Expansion (α)**:
\[ [\alpha] = K^{-1} \]
- **Dimension of Gas Constant (R)**:
\[ [R] = M^1 L^{2} T^{-2} \text{mol}^{-1} K^{-1} \]