At a constant pressure, of the following graphs that one which represents the variation of the density of an ideal gas with the absolute temperature T, is

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the density of an ideal gas varies with absolute temperature at constant pressure, 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 the Ideal Gas Law
We can express the number of moles \( n \) in terms of density \( \rho \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Where \( m \) is the mass of the gas. The volume \( V \) can be expressed in terms of density:
\[ V = \frac{m}{\rho} \]
Substituting these into the ideal gas law gives:
\[ P \left(\frac{m}{\rho}\right) = \frac{m}{M} RT \]

### Step 3: Simplify the Equation
Rearranging the equation, we can eliminate \( m \) (assuming it is not zero):
\[ P = \frac{RT \rho}{M} \]
From this, we can express density \( \rho \):
\[ \rho = \frac{PM}{RT} \]

### Step 4: Analyze the Relationship Between Density and Temperature
At constant pressure \( P \):
\[ \rho \propto \frac{1}{T} \]
This indicates that density \( \rho \) is inversely proportional to the absolute temperature \( T \).

### Step 5: Determine the Graphical Representation
Since density is inversely proportional to temperature, the graph of density \( \rho \) versus temperature \( T \) will be a hyperbolic curve that decreases as temperature increases. This is characteristic of an inverse relationship.

### Conclusion
The correct graph representing the variation of density with absolute temperature at constant pressure will show a downward curve, indicating that as temperature increases, density decreases.

---