In the ideal gas equation, the gas constant `R` has the dimension of `-`

Options

A. Mole`-atm//K`
B. Litre`//` mole
C. Litre`-atm//K//`mole
D. `erg//K`

Explanation

To find the dimensions of the gas constant \( R \) in the ideal gas equation \( PV = nRT \), we can follow these steps:

### Step 1: Write down the ideal gas equation
The ideal gas equation is given by:
\[ PV = nRT \]

### Step 2: Rearrange the equation to isolate \( R \)
To find the gas constant \( R \), we can rearrange the equation:
\[ R = \frac{PV}{nT} \]

### Step 3: Identify the dimensions of each variable
- **Pressure (P)**: The unit of pressure is typically expressed in atmospheres (atm) or pascals (Pa). In terms of base units, pressure can be expressed as:
\[ [P] = \text{Force/Area} = \frac{\text{Mass} \cdot \text{Acceleration}}{\text{Length}^2} = \frac{M \cdot L \cdot T^{-2}}{L^2} = M \cdot L^{-1} \cdot T^{-2} \]

- **Volume (V)**: The unit of volume is typically liters (L) or cubic meters (m³). In terms of base units:
\[ [V] = L^3 \]

- **Number of moles (n)**: The unit of the amount of substance is moles (mol):
\[ [n] = \text{mol} \]

- **Temperature (T)**: The unit of temperature is Kelvin (K):
\[ [T] = K \]

### Step 4: Substitute the dimensions into the equation for \( R \)
Now substituting the dimensions into the equation for \( R \):
\[ [R] = \frac{[P][V]}{[n][T]} = \frac{(M \cdot L^{-1} \cdot T^{-2})(L^3)}{(\text{mol})(K)} \]

### Step 5: Simplify the dimensions
Now simplifying the expression:
\[ [R] = \frac{M \cdot L^{-1} \cdot T^{-2} \cdot L^3}{\text{mol} \cdot K} = \frac{M \cdot L^{2} \cdot T^{-2}}{\text{mol} \cdot K} \]

### Conclusion
Thus, the dimensions of the gas constant \( R \) are:
\[ [R] = \frac{M \cdot L^{2}}{\text{mol} \cdot T^{2}} \]

### Final Answer
The correct option for the dimensions of the gas constant \( R \) is:
\[ \text{Option C: } \frac{M \cdot L^{2}}{\text{mol} \cdot K} \]

---