If P is the pressure and d is the density of gas, then P and d are related as :

Options

A. `P alpha 1"/"d`
B. `P alpha d`
C. `P alpha d^2`
D. `P alpha 1"/"d^2`

Answer

To solve the problem of how pressure (P) and density (d) of a gas are related, we can follow these steps:

### Step 1: Understand Boyle's Law
Boyle's Law states that for a given mass of gas at constant temperature, the volume (V) of the gas is inversely proportional to the pressure (P). This can be expressed mathematically as:
\[ V \propto \frac{1}{P} \]
or
\[ PV = k \]
where \( k \) is a constant.

**Hint:** Remember that Boyle's Law applies when temperature is constant and relates pressure and volume.

### Step 2: Relate Volume to Density
The density (d) of a gas is defined as its mass (m) divided by its volume (V):
\[ d = \frac{m}{V} \]
From this equation, we can express volume in terms of density:
\[ V = \frac{m}{d} \]

**Hint:** Density is mass divided by volume; rearranging this gives you volume in terms of mass and density.

### Step 3: Substitute Volume in Boyle's Law
Since we know from Boyle's Law that \( V \) is inversely proportional to \( P \), we can substitute the expression for volume from the density equation:
\[ P \propto \frac{1}{V} \]
Substituting \( V = \frac{m}{d} \) into this gives:
\[ P \propto \frac{d}{m} \]
This indicates that if the mass (m) is constant, pressure (P) is directly proportional to density (d).

**Hint:** When you substitute the expression for volume, pay attention to how the relationships change.

### Step 4: Conclusion
From the relationship derived, we conclude that pressure (P) is directly proportional to density (d) when mass is constant:
\[ P \propto d \]
Thus, if density increases, pressure also increases, assuming the mass of the gas does not change.

**Hint:** Think about how changes in one variable affect the other when they are directly proportional.

### Final Answer
Therefore, the correct relationship between pressure (P) and density (d) of a gas is:
\[ P \propto d \]