Dimcnsional analysis of the relation (Energy) = `("Pressure difference")^(3//2) ("Volume")^(3//2)` gives the value of n as

Options

A. 3
B. 2
C. `3//2`
D. `1//2`

Answer

To solve the problem using dimensional analysis, we will follow these steps:

### Step 1: Write down the dimensions of each quantity involved.

1. **Energy (E)**: The dimension of energy is given by:
\[ [E] = [M][L^2][T^{-2}] \]

2. **Pressure difference (ΔP)**: The dimension of pressure is:
\[ [P] = [M][L^{-1}][T^{-2}] \]
Therefore, the dimension of pressure difference is the same:
\[ [ΔP] = [M][L^{-1}][T^{-2}] \]

3. **Volume (V)**: The dimension of volume is:
\[ [V] = [L^3] \]

### Step 2: Set up the equation based on the given relation.

The relation given in the question is:
\[ E = (ΔP)^{\frac{3}{2}} (V)^{\frac{3}{2}} \]

Substituting the dimensions we found:
\[ [M][L^2][T^{-2}]^n = \left([M][L^{-1}][T^{-2}]\right)^{\frac{3}{2}} \cdot \left([L^3]\right)^{\frac{3}{2}} \]

### Step 3: Simplify the right-hand side.

Calculating the right-hand side:
1. For pressure difference:
\[ (ΔP)^{\frac{3}{2}} = \left([M][L^{-1}][T^{-2}]\right)^{\frac{3}{2}} = [M^{\frac{3}{2}}][L^{-\frac{3}{2}}][T^{-3}] \]

2. For volume:
\[ (V)^{\frac{3}{2}} = \left([L^3]\right)^{\frac{3}{2}} = [L^{\frac{9}{2}}] \]

Now, combining these:
\[ [M^{\frac{3}{2}}][L^{-\frac{3}{2}}][T^{-3}] \cdot [L^{\frac{9}{2}}] = [M^{\frac{3}{2}}][L^{\left(-\frac{3}{2} + \frac{9}{2}\right)}][T^{-3}] = [M^{\frac{3}{2}}][L^{3}][T^{-3}] \]

### Step 4: Equate the dimensions from both sides.

Now we have:
\[ [M][L^2][T^{-2}]^n = [M^{\frac{3}{2}}][L^{3}][T^{-3}] \]

This gives us:
\[ [M^n][L^{2n}][T^{-2n}] = [M^{\frac{3}{2}}][L^{3}][T^{-3}] \]

### Step 5: Compare the powers of M, L, and T.

From the comparison, we can set up the following equations:
1. For mass (M):
\[ n = \frac{3}{2} \]

2. For length (L):
\[ 2n = 3 \implies n = \frac{3}{2} \]

3. For time (T):
\[ -2n = -3 \implies n = \frac{3}{2} \]

### Conclusion

From all three comparisons, we consistently find that:
\[ n = \frac{3}{2} \]

Thus, the value of \( n \) is \( \frac{3}{2} \).