What are the dimensions of gas constant ?

Options

A. `[M L T^(-2) K^(-1)]`
B. `[M^(0) L T^(-2) K^(-1)]`
C. `[M L^(2) T^(-2) K^(-1) mol^(-1)]`
D. `[M^(0) L^(2) T^(-2) K^(-1)]`

Explanation

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 in Kelvin

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

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

### Step 2: Identifying the Dimensions of Each Variable
Now, we need to identify the dimensions of each variable in the equation.

- **Pressure (P)**: The dimension of pressure is given by force per unit area.
- Force (F) has dimensions \( [M L T^{-2}] \) (mass × acceleration).
- Area (A) has dimensions \( [L^2] \).
- Therefore, the dimension of pressure is:

\[ [P] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

- **Volume (V)**: The dimension of volume is:

\[ [V] = [L^3] \]

- **Number of moles (n)**: The dimension of the number of moles is:

\[ [n] = [mol] \]

- **Temperature (T)**: The dimension of temperature is:

\[ [T] = [K] \]

### Step 3: Substituting the Dimensions into the Expression for R
Now, we substitute the dimensions of \( P \), \( V \), \( n \), and \( T \) into the expression for \( R \):

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

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

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

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

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