Animated Solution for Physics - Thermodynamics: A real gas behaves like an ideal gas if its

The world of thermodynamics is built upon the elegant and mathematically pristine concept of an ideal gas. It is a theoretical construct that simplifies the chaotic, microscopic dance of billions of molecules into a neat, predictable equation: $pV=nRT$.

However, nature is rarely as perfect as our equations. Real gases—like the oxygen we breathe or the carbon dioxide we exhale—do not always play by the rules. They have their own quirks and physical realities.

But under certain specific conditions, these rebellious real gases can be coaxed into behaving almost exactly like their perfect, ideal counterparts. To understand when and why this happens, we must take a microscopic journey into the heart of the gas itself.

To understand the deviation of real gases, we first need to look at the foundation of the ideal gas model: the Kinetic Theory of Gases.

This theory makes two massive, simplifying assumptions. First, it assumes that gas molecules are infinitesimally small point masses. In other words, the actual volume occupied by the molecules themselves is considered to be absolutely zero compared to the vast empty space of the container.

Second, it assumes that these molecules are completely indifferent to one another. There are absolutely no intermolecular forces of attraction or repulsion. They only interact during perfectly elastic collisions.

In reality, neither of these assumptions is strictly true. Molecules do have a physical size, and they do exert weak attractive forces (like Van der Waals forces) on each other.

So, the question becomes: under what conditions do these physical realities become so insignificant that we can safely ignore them?

Expanding the Void

Let us tackle the first assumption regarding the volume of the molecules. Imagine a crowded room. If you pack a hundred people into a tiny elevator, the physical size of each person matters a lot. There is very little empty space, and people are constantly bumping into each other.

This is analogous to a gas at high pressure. The gas is compressed, the total volume $V$ is small, and the actual volume of the molecules becomes a significant fraction of the total space. The ideal gas assumption breaks down completely.

Now, imagine taking those same hundred people and placing them in a massive, empty football stadium. Suddenly, the physical size of each individual is completely negligible compared to the vast emptiness of the stadium.

This is exactly what happens to a gas at low pressure. When the pressure is low, the gas is highly expanded. The total volume $V$ of the container is huge. In this vast microscopic void, the tiny volume occupied by the molecules themselves is mathematically insignificant.

Furthermore, because the molecules are spread so far apart, they rarely come close enough to feel each other's attractive forces. Therefore, low pressure is the first critical condition that forces a real gas to mimic an ideal gas.

Overpowering the Pull

Now, let us address the second assumption: the absence of intermolecular forces. Even if molecules are far apart, they still have a slight tendency to attract one another.

Imagine two magnets sliding past each other on a table. If they are moving very slowly, their attractive force will catch them, and they will snap together.

This is what happens in a gas at low temperature. The molecules have low kinetic energy. They move sluggishly. When they pass near each other, the weak intermolecular forces have enough time and influence to pull them off their straight-line paths. The gas deviates significantly from ideal behavior and, if the temperature is low enough, it will condense into a liquid!

But what if we shoot those magnets past each other at the speed of a bullet? They are moving so incredibly fast that their kinetic energy completely overpowers the brief attractive pull. They fly past each other almost as if the magnetic force wasn't even there.

This is the magic of high temperature. Temperature is a direct measure of the average kinetic energy of the molecules. At high temperatures, the molecules are zipping around with massive thermal agitation.

Their kinetic energy is so overwhelmingly large that the tiny potential energy from intermolecular attraction becomes completely negligible. They smash and bounce off the walls and each other without being significantly deflected by attractive forces.

Therefore, high temperature is the second critical condition for ideal behavior.

Visualizing the Truth

Physicists and chemists use a mathematical tool called the compressibility factor, denoted by $Z$, to quantify how much a real gas deviates from ideal behavior.

It is defined as $Z=\frac{pV}{nRT}$.

For a perfectly ideal gas, $Z$ is exactly equal to $1$ under all conditions of temperature and pressure. If we plot a graph of $Z$ versus pressure $P$, an ideal gas would just be a flat, horizontal line at $Z=1$.

For real gases, the graph tells a fascinating story. At very low pressures (approaching zero), the curves for all real gases converge exactly at $Z=1$. This mathematically proves our earlier intuition: at low pressure, real gases behave ideally.

As we increase the pressure, the curves deviate. However, if we look at the curve for a gas at a very high temperature, we notice something remarkable. The high-temperature curve stays much closer to the $Z=1$ line over a much wider range of pressures compared to a gas at a low temperature.

The brilliant physicist Johannes Diderik van der Waals formulated an equation that mathematically corrects the ideal gas law by accounting for these two physical realities.

His famous equation is:

Here, the term $\frac{a{n}^2}{V^2}$ corrects for the intermolecular forces of attraction, effectively adding to the measured pressure. The term $nb$ corrects for the actual volume occupied by the gas molecules, subtracting it from the total volume.

Let us apply our two conditions to this equation. At low pressure, the volume $V$ is extremely large. Because $V$ is in the denominator of the pressure correction term, $\frac{a{n}^2}{V^2}$ becomes vanishingly small. Similarly, because $V$ is so large, subtracting the tiny molecular volume $nb$ makes almost no difference ($V-nb\approx V$).

At high temperature, the right side of the equation ($nRT$) becomes very large, making the small correction terms on the left side mathematically insignificant in comparison.

In both cases, the Van der Waals equation elegantly collapses back into the simple, beautiful ideal gas law: $pV=nRT$.

The journey from a real gas to an ideal gas is a story of minimizing physical constraints. By expanding the volume through low pressure, we render the physical size of the molecules insignificant. By injecting massive kinetic energy through high temperature, we shatter the influence of intermolecular forces.

Together, these two conditions—low pressure and high temperature—create the perfect microscopic environment for a chaotic real gas to achieve the mathematical perfection of an ideal gas.