An ideal gas is initially at temperature T and volume V. Its volume is increased by `DeltaV` due to an increase in temperature `DeltaT,` pressure remaining constant. The quantity `delta=(DeltaV)/(VDeltaT)` varies with temperature as

Options

A.
B.
C.
D.

Explanation

To solve the problem step by step, we will use the ideal gas law and the relationship between volume, temperature, and pressure.

### 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 of gas
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 2: Analyze the Situation
We are given that:
- The initial volume of the gas is \( V \).
- The volume increases by \( \Delta V \) due to an increase in temperature \( \Delta T \).
- The pressure remains constant.

### Step 3: Apply the Ideal Gas Law
Since the pressure is constant, we can express the change in volume as:
\[ P \Delta V = nR \Delta T \]

### Step 4: Rearranging the Equation
Dividing both sides by \( PV \) gives:
\[ \frac{\Delta V}{V} = \frac{nR \Delta T}{PV} \]

### Step 5: Substitute for Pressure
From the ideal gas law, we can express pressure as:
\[ P = \frac{nRT}{V} \]
Substituting this into the equation gives:
\[ \frac{\Delta V}{V} = \frac{nR \Delta T}{\left(\frac{nRT}{V}\right)V} \]

### Step 6: Simplifying the Equation
This simplifies to:
\[ \frac{\Delta V}{V} = \frac{\Delta T}{T} \]
Thus, we can express the change in volume per unit volume and temperature as:
\[ \frac{\Delta V}{V \Delta T} = \frac{1}{T} \]

### Step 7: Define the Quantity \( \delta \)
Let:
\[ \delta = \frac{\Delta V}{V \Delta T} \]
From our previous step, we have:
\[ \delta = \frac{1}{T} \]

### Step 8: Analyze the Variation of \( \delta \) with Temperature
As \( T \) increases, \( \delta \) decreases, indicating an inverse relationship. Therefore, \( \delta \) varies with temperature as:
\[ \delta \propto \frac{1}{T} \]

### Step 9: Graphical Representation
The relationship \( \delta T = \text{constant} \) indicates that if we plot \( \delta \) against \( T \), we will get a rectangular hyperbola.

### Conclusion
Thus, the quantity \( \delta = \frac{\Delta V}{V \Delta T} \) varies with temperature \( T \) as \( \delta \propto \frac{1}{T} \), and the correct option is that the graph of \( \delta \) versus \( T \) is a rectangular hyperbola.