If P is the pressure and `rho` is the density of a gas, then P and `rho` are related as :

Options

A. ` P prop rho `
B. `P prop ( 1//rho)`
C. `P prop rho^(2)`
D. `P prop (1 // rho^(2))`

Explanation

To find the relationship between pressure (P) and density (ρ) of a gas, we can use the Ideal Gas Law, which states:

\[ PV = nRT \]

Where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature of the gas in Kelvin

### Step 1: Express the number of moles (n) in terms of mass (m) and molar mass (M)
The number of moles \( n \) can be expressed as:

\[ n = \frac{m}{M} \]

Where:
- \( m \) = mass of the gas
- \( M \) = molar mass of the gas

### Step 2: Substitute n in the Ideal Gas Law
Substituting \( n \) into the Ideal Gas Law gives:

\[ PV = \left( \frac{m}{M} \right) RT \]

### Step 3: Rearrange the equation
Rearranging the equation to isolate \( P \):

\[ P = \frac{mRT}{MV} \]

### Step 4: Express density (ρ)
Density \( \rho \) is defined as mass per unit volume:

\[ \rho = \frac{m}{V} \]

### Step 5: Substitute density into the equation
From the definition of density, we can express mass \( m \) as:

\[ m = \rho V \]

Substituting this into the equation for pressure:

\[ P = \frac{\rho V RT}{MV} \]

### Step 6: Simplify the equation
The volume \( V \) cancels out:

\[ P = \frac{\rho RT}{M} \]

### Step 7: Analyze the relationship
From the final equation, we can see that pressure \( P \) is directly proportional to density \( \rho \):

\[ P \propto \rho \]

### Conclusion
Thus, the correct relationship between pressure and density of a gas is that pressure is directly proportional to density.