Animated Solution for Chemistry - States of Matter: Which one of the following graphs is not correct for ideal gas? $d=\text{density}$, $p=\text{pressure}$, $T=\text{temperature}$

To determine which graph incorrectly represents the behavior of an ideal gas, we must first establish a mathematical relationship between density ($d$), pressure ($p$), and temperature ($T$). We begin our journey with the fundamental equation of state for an ideal gas:

Here, $p$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

Since the question asks us to analyze density, we need to introduce mass into our equation. We know that the number of moles $n$ is equal to the given mass $w$ divided by the molar mass $M$ ($n=\frac{w}{M}$). Substituting this into the ideal gas equation gives:

Density ($d$) is defined as mass per unit volume ($d=\frac{w}{V}$). To bring density into our equation, we can rearrange the terms to group $w$ and $V$ together:

Replacing $\frac{w}{V}$ with $d$, we get:

Now, let's isolate density ($d$) to make it the subject of our formula. By cross-multiplying, we arrive at our master equation:

This elegant equation is the key to unlocking the behavior of the gas under various conditions. Let's use it to analyze each graph one by one.

If we keep the pressure ($p$) constant for a specific gas (constant $M$), the term $\frac{pM}{R}$ becomes a constant. Our master equation simplifies to:

This shows that density is inversely proportional to temperature. Mathematically, an equation of the form $y=\frac{k}{x}$ represents a rectangular hyperbola. Graph (I) correctly depicts this hyperbolic curve. However, Graph (II) shows a straight line passing through the origin, implying $d\propto T$. This would mean heating a gas makes it denser at constant pressure, which physically violates Charles's Law. Therefore, Graph (II) is incorrect.

What happens if we plot density against $\frac{1}{T}$? Let's rewrite our master equation slightly:

If we treat $\frac{1}{T}$ as our independent variable (like $x$ on the x-axis), the equation takes the form $y=mx$, where the slope $m=\frac{pM}{R}$. This is the equation of a straight line passing through the origin. Graph (III) perfectly illustrates this linear relationship.

Finally, let's examine the relationship between density and pressure at a constant temperature ($T$). If $T$ is constant, the entire term $\frac{M}{RT}$ becomes a constant. Our equation becomes:

This indicates that density is directly proportional to pressure ($d\propto p$). As you compress a gas, its density increases linearly. This relationship is represented by a straight line passing through the origin, which perfectly matches Graph (IV).

By systematically deriving the density formula from the ideal gas law, we have verified that Graphs (I), (III), and (IV) are physically and mathematically sound. Graph (II), which incorrectly suggests a direct proportionality between density and temperature, is the flawed representation. Thus, Graph (II) is our final answer.