An ideal gas is initially at a thermodynamic temperature T and has a volume V. Its volume is increased by ΔV due to an increase in temperature ΔT, pressure remaining constant. Which of the following graphs shows how the quantity $\delta =\frac{∆V}{(V∆T)}$ varies with temperature?

The ideal gas equation is

$PV=nRT$

In isobaric process pressure will remain constant throughout the process.

With the help of these equation we can determine the required relationship

Given,

The ideal gas have a initial temperature $T$. The initial volume is $V$

The volume is increased by $\Delta V$. Pressure is constant.

For isobaric process; $P=$constant

The ideal gas equation is,

$PV=nRT$

Where

$P,n$are constant

$$\begin{gathered} V\propto T \\ V=KT \end{gathered}$$---(1)

Differentiate the equation w.r.t $V$

$dV=KdT$ ----(2)

Now divide the equation (2) with (1) we get,

$\frac{dV}{V}=\frac{KdT}{KT}$

$\frac{dV}{V}=\frac{dT}{TW}$

When the change is more it is represented as,

$\frac{\Delta V}{V}=\frac{\Delta T}{T}$

$\frac{\Delta V}{V\Delta T}=\frac{1}{T}$

We know that,

$\delta =\frac{1}{T}$

It means that $\delta$ is inversely proportional to temperature.

The option (3) satisfy the given condition.

The option (1),( 2) & (4) do not satisfy the given condition.

Hence the correct option is (3).