A quantity X is given by `X = ((R )/( N_(A)K))^(2)` where R is the universal gas constant, `N_(A)` is Avagadro number & K is the Boltzmann's constant Dimension of X is that of

Options

A. Energy
B. Length
C. Moles
D. None of these

Explanation

To find the dimension of the quantity \( X \) given by the formula:

\[ X = \left( \frac{R}{N_A K} \right)^2 \]

where:
- \( R \) is the universal gas constant,
- \( N_A \) is Avogadro's number,
- \( K \) is the Boltzmann's constant.

### Step 1: Identify the dimensions of each component

1. **Universal Gas Constant \( R \)**:
- The dimension of \( R \) is given in Joules per Kelvin per mole, which can be expressed as:
\[ [R] = [E][T]^{-1}[N]^{-1} = [M L^2 T^{-2}][\Theta]^{-1}[N]^{-1} \]
where:
- \( [E] \) is energy (Joules),
- \( [T] \) is temperature (Kelvin),
- \( [N] \) is amount of substance (moles).
Thus, the dimension of \( R \) is:
\[ [R] = [M L^2 T^{-2} \Theta^{-1} N^{-1}] \]

2. **Avogadro's Number \( N_A \)**:
- \( N_A \) is a count of particles per mole, hence it is dimensionless:
\[ [N_A] = [N]^{-1} \]

3. **Boltzmann's Constant \( K \)**:
- The dimension of \( K \) is Joules per Kelvin, which can be expressed as:
\[ [K] = [E][T]^{-1} = [M L^2 T^{-2}][\Theta]^{-1} \]
Thus, the dimension of \( K \) is:
\[ [K] = [M L^2 T^{-2} \Theta^{-1}] \]

### Step 2: Substitute the dimensions into the formula for \( X \)

Now, substituting the dimensions into the expression for \( X \):

\[ X = \left( \frac{[R]}{[N_A][K]} \right)^2 \]

Substituting the dimensions we have:

\[ X = \left( \frac{[M L^2 T^{-2} \Theta^{-1} N^{-1}]}{[N]^{-1} [M L^2 T^{-2} \Theta^{-1}]} \right)^2 \]

### Step 3: Simplify the expression

The dimensions simplify as follows:

\[ X = \left( \frac{[M L^2 T^{-2} \Theta^{-1} N^{-1}]}{[N]^{-1} [M L^2 T^{-2} \Theta^{-1}]} \right)^2 = \left( \frac{1}{[N]^{-1}} \right)^2 = [1]^2 = [1] \]

### Conclusion

Thus, the dimension of \( X \) is dimensionless.

### Final Answer

The dimension of \( X \) is dimensionless.

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