The pressure (p) and the dencity `rho` of given mass of a gas expressed by Boyle's law, `p = K rho` holds true.

Options

A. for any gas under any condition
B. for same gas under any condition
C. Only if the tamperature is kept constant.
D. None of these

Explanation

To solve the question regarding the relationship between pressure (p) and density (ρ) of a gas as expressed by Boyle's law, we will follow these steps:

### Step 1: Understand Boyle's Law
Boyle's law states that for a given mass of gas at constant temperature, the product of pressure (P) and volume (V) is constant. This can be mathematically expressed as:
\[ PV = \text{constant} \]

### Step 2: Express Density in Terms of Mass and Volume
Density (ρ) is defined as mass (m) per unit volume (V):
\[ \rho = \frac{m}{V} \]
From this, we can express volume in terms of mass and density:
\[ V = \frac{m}{\rho} \]

### Step 3: Substitute Volume in Boyle's Law
Substituting the expression for volume (V) into Boyle's law gives:
\[ P \left(\frac{m}{\rho}\right) = \text{constant} \]
This simplifies to:
\[ P = \frac{\text{constant} \cdot \rho}{m} \]

### Step 4: Rearranging the Equation
Let’s denote the constant as K (where K = constant/m). We can rewrite the equation as:
\[ P = K \cdot \rho \]
This shows that pressure (P) is directly proportional to density (ρ) when the mass (m) is constant.

### Step 5: Conclusion on Conditions
It is important to note that Boyle's law and the derived relationship hold true only under constant temperature conditions. Therefore, the relationship \( P = K \rho \) is valid only if the temperature remains constant.

### Final Answer
The correct answer to the question is that the relationship \( P = K \rho \) holds true **only if the temperature is constant**.

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