Which of the following is a graph of the relation of the density `(rho)` of a gas sample to the absolute tempeature keeping constant pressure.

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the relationship between the density (ρ) of a gas sample and the absolute temperature (T) while keeping the pressure constant, we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Absolute temperature

### Step 2: Relate Density to Volume
Density (ρ) is defined as:
\[ \rho = \frac{m}{V} \]
Where:
- \( m \) = Mass of the gas
- \( V \) = Volume of the gas

We can express the volume in terms of density:
\[ V = \frac{m}{\rho} \]

### Step 3: Substitute Volume in the Ideal Gas Law
Substituting the expression for volume into the ideal gas law gives:
\[ P \left(\frac{m}{\rho}\right) = nRT \]

### Step 4: Express Moles in Terms of Density
The number of moles \( n \) can be expressed as:
\[ n = \frac{m}{M} \]
Where \( M \) is the molar mass of the gas. Substituting this into the equation gives:
\[ P \left(\frac{m}{\rho}\right) = \left(\frac{m}{M}\right)RT \]

### Step 5: Simplify the Equation
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{P}{\rho} = \frac{RT}{M} \]

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ \rho = \frac{PM}{RT} \]

### Step 7: Analyze the Relationship
From the equation \( \rho = \frac{PM}{RT} \), we can see that:
- When pressure \( P \) and molar mass \( M \) are constant, density \( \rho \) is inversely proportional to temperature \( T \).
- This means that as temperature increases, density decreases, and vice versa.

### Step 8: Determine the Graph
The relationship \( \rho \propto \frac{1}{T} \) indicates that the graph of density versus temperature will be a hyperbola that decreases as temperature increases. Therefore, the correct graph will show a decrease in density with an increase in temperature.

### Conclusion
The correct answer to the question is option A, which reflects the inverse relationship between density and temperature at constant pressure.

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