Which of the following graphs represent the behaviour of an ideal gas ? (Symbols have their usual me

Q: Why is the slope positive?

Because the number of moles $n$ and the universal gas constant $R$ are always positive values. Their product, which represents the slope, must therefore be positive.

Q: Why does the graph pass through the origin?

At absolute zero temperature ($T = 0$ K), the product $pV$ for an ideal gas approaches zero, meaning the line must intersect the origin $(0,0)$.

Q: What if the gas is not ideal?

Real gases deviate from this straight-line behavior, especially at high pressures and low temperatures, due to intermolecular forces and finite molecular volume.

Q: Does the slope depend on the type of gas?

No, for an ideal gas, the slope depends only on the number of moles $n$. It is independent of the chemical nature of the gas.

Q: What does the slope of this graph represent physically?

The slope represents $nR$, which is the product of the number of moles and the universal gas constant. It is directly proportional to the amount of gas present.

Question

Which of the following graphs represent the behaviour of an ideal gas ? (Symbols have their usual meanings.)

Sigma Percentile · Accepted Answer

### The Graphical Challenge

Graphs are the visual language of physics. They allow us to see relationships between macroscopic variables at a single glance. In this problem, we are presented with four different graphs and asked to identify which one correctly represents the behavior of an ideal gas when we plot the product of pressure and volume ($pV$) against the absolute temperature ($T$).

To solve this, we cannot just guess; we need to rely on the fundamental laws that govern the behavior of gases.

### The Master Equation

The cornerstone of kinetic theory and thermodynamics is the Ideal Gas Equation. It beautifully ties together all the state variables of a gas:

$$pV = nRT$$

Here, $p$ is the absolute pressure, $V$ is the volume, $n$ is the number of moles of the gas, $R$ is the universal gas constant, and $T$ is the absolute temperature in Kelvin. This single equation is all we need to decode the correct graph.

### Mapping Math to Geometry

Let's look at our axes. The $y$-axis represents the entire term $pV$, and the $x$-axis represents the temperature $T$.

For a given, closed sample of an ideal gas, the number of moles $n$ is a constant. The universal gas constant $R$ is, of course, always constant. Therefore, the product $nR$ is simply a constant value. Let's call this constant $m$. We can rewrite our ideal gas equation as:

$$pV = (nR)T$$

Now, let's map this to the standard equation of a straight line in coordinate geometry, which is $y = mx + c$.

By substituting our variables, we get $y = pV$ and $x = T$. The equation becomes $y = mx$, where the slope $m = nR$ and the $y$-intercept $c = 0$.

### The Final Verdict

What does the equation $y = mx$ tell us visually? It represents a **straight line that passes exactly through the origin** $(0,0)$. Furthermore, since the number of moles $n$ and the gas constant $R$ are both positive quantities, the slope $m$ must be positive. This means the line will go upwards as we move to the right.

Looking at our options, only graph (c) shows a straight line with a positive slope originating from the origin. Therefore, it perfectly captures the direct proportionality between $pV$ and $T$ for an ideal gas.

Animated Solution for Physics - Kinetic Theory: Which of the following graphs represent the behaviour of an ideal gas ? (Symbols have their usual meanings.)

Select Answer:

Visualized Solution

Visual Anchor

Ideal Gas Equation

Mapping Variables to Axes

Equation of Straight Line

Deducing the Graph

Conclusion

The Sigma Insight: Kinetic Theory of Gases and Gas Laws

The Graphical Challenge

The Master Equation

Mapping Math to Geometry

The Final Verdict

Similar Questions

For an ideal gas the instantaneous change in pressure $p$ with volume $V$ is given by the equation $\frac{d p}{d V} = - a p$ . If $p = p_{0}$ at $V = 0$ is the given boundary condition, then the maximum temperature one mole of gas can attain is (Here $R$ is the gas constant)

One mole of an ideal gas passes through a process, where pressure and volume obey the relation $p = p_{0} [1 - \frac{1}{2} (\frac{V _{0}}{V})^{2}]$ . Here, $p_{0}$ and $V_{0}$ are constants. Calculate the change in the temperature of the gas, if its volume changes from $V_{0}$ to $2 V_{0}$ .