According to Laplace's formula, the velocity (V) of sound in a gas is given by `v=sqrt((gammaP)/(rho))`, where P is the pressure and `rho` is the density of the gas. What is the dimensional formula for `gamma` ?

Options

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

Explanation

To find the dimensional formula for \( \gamma \) in Laplace's formula for the velocity of sound in a gas, we start with the equation:

\[ V = \sqrt{\frac{\gamma P}{\rho}} \]

Where:
- \( V \) is the velocity of sound,
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

### Step 1: Square both sides of the equation

To eliminate the square root, we square both sides:

\[ V^2 = \frac{\gamma P}{\rho} \]

### Step 2: Rearrange the equation to solve for \( \gamma \)

Rearranging the equation gives us:

\[ \gamma = \frac{V^2 \rho}{P} \]

### Step 3: Determine the dimensional formulas for \( V^2 \), \( \rho \), and \( P \)

1. **Velocity \( V \)**:
- The unit of velocity is meters per second (m/s).
- The dimensional formula for velocity is:
\[ [V] = [L][T^{-1}] \]
- Therefore, for \( V^2 \):
\[ [V^2] = [L^2][T^{-2}] \]

2. **Density \( \rho \)**:
- Density is mass per unit volume. The unit of density is kilograms per cubic meter (kg/m³).
- The dimensional formula for density is:
\[ [\rho] = [M][L^{-3}] \]

3. **Pressure \( P \)**:
- Pressure is force per unit area. The unit of pressure is Pascals (N/m²), where 1 N = 1 kg·m/s².
- The dimensional formula for pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

### Step 4: Substitute the dimensional formulas into the equation for \( \gamma \)

Now substituting the dimensional formulas into the equation for \( \gamma \):

\[ \gamma = \frac{[V^2] [\rho]}{[P]} = \frac{[L^2][T^{-2}] \cdot [M][L^{-3}]}{[M][L^{-1}][T^{-2}]} \]

### Step 5: Simplify the expression

Now, we simplify the expression:

\[ \gamma = \frac{[L^2][T^{-2}][M][L^{-3}]}{[M][L^{-1}][T^{-2}]} \]

Cancelling out the common terms:

- The \( [M] \) cancels out.
- The \( [T^{-2}] \) cancels out.
- The \( [L^2] \) and \( [L^{-3}] \) gives \( [L^{-1}] \) in the numerator and \( [L^{-1}] \) in the denominator cancels out.

Thus, we are left with:

\[ \gamma = [L^0][M^0][T^0] \]

### Conclusion: Dimensional formula for \( \gamma \)

The dimensional formula for \( \gamma \) is:

\[ \gamma = [M^0][L^0][T^0] \]

This indicates that \( \gamma \) is a dimensionless quantity.