The velocity of sound in a gas depends on its pressure and density . Obtain the relation between velocity , pressure and density.

Answer

To find the relationship between the velocity of sound (v), pressure (P), and density (ρ) in a gas, we can start by expressing the dependencies in terms of dimensional analysis.

### Step 1: Identify the variables
We know that:
- Velocity of sound (v)
- Pressure (P)
- Density (ρ)

### Step 2: Write the dimensional formulas
1. **Velocity (v)** has dimensions of length per time:
\[ [v] = L^1 T^{-1} \]

2. **Pressure (P)** is defined as force per unit area. The dimensional formula for force (F) is mass times acceleration:
\[ [F] = M^1 L^1 T^{-2} \]
Therefore, pressure can be expressed as:
\[ [P] = \frac{[F]}{[A]} = \frac{M^1 L^1 T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]

3. **Density (ρ)** is mass per unit volume:
\[ [ρ] = \frac{M^1}{L^3} = M^1 L^{-3} T^0 \]

### Step 3: Establish the relationship
Assuming the velocity of sound (v) is proportional to the pressure (P) and inversely proportional to the square root of density (ρ), we can write:
\[ v \propto \frac{P^x}{ρ^y} \]
where x and y are constants to be determined.

### Step 4: Write the dimensional equation
Substituting the dimensions into the equation gives:
\[ [L^1 T^{-1}] = \frac{[M^1 L^{-1} T^{-2}]^x}{[M^1 L^{-3}]^y} \]

### Step 5: Simplify the right-hand side
This expands to:
\[ [L^1 T^{-1}] = \frac{M^{x} L^{-x} T^{-2x}}{M^{y} L^{-3y}} \]
This can be simplified to:
\[ [L^1 T^{-1}] = M^{x-y} L^{-x+3y} T^{-2x} \]

### Step 6: Equate dimensions
Now we equate the dimensions on both sides:
1. For mass (M):
\[ x - y = 0 \quad \text{(1)} \]
2. For length (L):
\[ -x + 3y = 1 \quad \text{(2)} \]
3. For time (T):
\[ -2x = -1 \quad \Rightarrow \quad x = \frac{1}{2} \quad \text{(3)} \]

### Step 7: Solve the equations
From equation (3), substituting \(x = \frac{1}{2}\) into equation (1):
\[ \frac{1}{2} - y = 0 \quad \Rightarrow \quad y = \frac{1}{2} \]

Now substituting \(x = \frac{1}{2}\) and \(y = \frac{1}{2}\) into equation (2):
\[ -\frac{1}{2} + 3 \left(\frac{1}{2}\right) = 1 \quad \Rightarrow \quad 1 = 1 \quad \text{(True)} \]

### Conclusion
Thus, the final relationship between the velocity of sound (v), pressure (P), and density (ρ) is:
\[ v = k \sqrt{\frac{P}{ρ}} \]
where \(k\) is a proportionality constant.