Gas constant 'R' has dimensions :

Options

A. (a)`M^(1)L^(2)T^(2)K^(-1)`
B. (b)`M^(1)L^(2)T^(-2)K^(-1)`
C. (c)`M^(-1)L^(-2)T^(-2)K^(-1)`
D. (d)`M^(1)L^(1)T^(2)K^(1)`

Answer

To find the dimensions of the gas constant 'R', we start with the ideal gas law equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Gas constant
- \( T \) = Temperature

### Step 1: Rearranging the Ideal Gas Law
We can rearrange the equation to solve for \( R \):

\[ R = \frac{PV}{nT} \]

### Step 2: Finding the Dimensions of Each Quantity
Now, we need to find the dimensions of each quantity involved in the equation.

1. **Pressure (P)**:
- Pressure is defined as force per unit area.
- The dimension of force is \( M L T^{-2} \) (mass × acceleration).
- Area has dimensions \( L^2 \).
- Therefore, the dimension of pressure is:
\[ [P] = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]

2. **Volume (V)**:
- Volume is defined as length cubed.
- Therefore, the dimension of volume is:
\[ [V] = L^3 \]

3. **Number of Moles (n)**:
- The number of moles is a dimensionless quantity.
- Therefore, the dimension is:
\[ [n] = 1 \]

4. **Temperature (T)**:
- The dimension of temperature is represented as:
\[ [T] = K \quad (\text{where K is the dimension for temperature}) \]

### Step 3: Substituting Dimensions into the Equation for R
Now we can substitute the dimensions we found into the equation for \( R \):

\[ [R] = \frac{[P][V]}{[n][T]} = \frac{(M L^{-1} T^{-2})(L^3)}{1 \cdot K} \]

### Step 4: Simplifying the Expression
Now, we simplify the expression:

\[ [R] = \frac{M L^{-1} T^{-2} \cdot L^3}{K} = \frac{M L^{2} T^{-2}}{K} \]

### Final Result
Thus, the dimensions of the gas constant \( R \) are:

\[ [R] = M L^{2} T^{-2} K^{-1} \]

### Conclusion
The correct answer is option B: \( M L^{2} T^{-2} K^{-1} \).

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