Working Note · Thermodynamics · Part XI
Ideal Gas Equation: Molecules as a Drumroll on the Walls
A single four-letter formula — pV=nRT — governs both the inflation of a balloon in the sun, the operation of a car engine, and weather forecasting.
In 1834, Émile Clapeyron combined three independent experimental laws (Boyle, Charles, Gay-Lussac) into a single equation of state for an ideal gas. Behind the simple formula pV = nRT lies an honest microscopic picture: pressure is the statistics of billions of molecules colliding with the container walls. In this note — all three partial laws, a derivation from kinetic theory, and a running numerical example with a balloon.
The equation in a nutshell
| Quantity | Meaning | SI unit |
|---|---|---|
| p | pressure | Pa |
| V | volume | m³ |
| n | amount of substance | mol |
| T | absolute temperature | K (not °C!) |
An "ideal" gas is a model where molecules are considered as points with no intrinsic volume, not attracting each other (interacting only through elastic collisions). Real gases (air, nitrogen, oxygen) under ordinary room conditions follow this equation with accuracy sufficient for engineering calculations — deviations become noticeable only at very high pressure or very low temperature (§6).
Three Partial Laws Within One
pV = nRT "contains" three earlier experimental laws as special cases — each is obtained by fixing one variable:
| Law | Condition | Relation |
|---|---|---|
| Boyle’s (1662) | T = const | pV = const |
| Charles’s (1787) | p = const | V/T = const |
| Gay-Lussac’s (1802) | V = const | p/T = const |
Three scientists, working at different times and independently, were actually measuring the same formula from different angles — like three blind men feeling an elephant by its different parts. Clapeyron in 1834 was the first to see the whole.
Microscopic Picture
Click on the modes above: molecules fly faster when heated (more collisions with the wall per second AND each collision is stronger — both effects raise the pressure) or collide with the walls more frequently in a smaller volume during compression. From the kinetic theory of gases, one can rigorously derive:
where nV is the number of molecules per unit volume, ⟨Ek⟩ is the average kinetic energy of a single molecule. Pressure is not some separate substance, but a direct statistical consequence of the fact that trillions of molecules bombard the container walls every second.
The Combined Law
For a fixed amount of gas (n does not change), from pV = nRT follows a convenient practical form — comparison of two states of the same gas without needing to know R or n at all:
Numerical example (Boyle’s law, T = const): a gas at pressure p₁ = 100 kPa occupies volume V₁ = 2 L. It is compressed to V₂ = 0.5 L at the same temperature. Find the new pressure.
Compressed 4 times in volume — the pressure increased exactly 4 times. Simple inverse proportionality of Boyle’s law.
Running Example: Balloon
A balloon on the ground (p₁ = 100 kPa, V₁ = 3 L, T₁ = 300 K) is brought into a warm room by a radiator, where it is simultaneously slightly compressed by neighbors in a crowd to V₂ = 2 L and heated to T₂ = 330 K. Find the new pressure inside the balloon.
All three effects — initial pressure, compression, and heating — are accounted for by one formula at once, without needing to compute n or R. This is exactly how engineers and meteorologists work with gases in practice: not through absolute values, but through the ratio of two states.
Where It Leads: When Gas Is Not Ideal
The ideal gas model is an approximation that works worse the denser the gas and the closer it is to the condensation point into a liquid. Two assumptions that break down in real conditions: molecules HAVE their own volume (not points), and there IS weak attraction between them (van der Waals forces) — at high pressure, molecules pack more tightly, and the attraction begins to noticeably pull the gas together, reducing the effective pressure on the walls compared to the ideal model.
In 1873, Johannes van der Waals corrected the equation of state with two additional terms accounting for the intrinsic volume of molecules and their mutual attraction: (p + a/V²)(V − b) = nRT. At sufficiently low temperature, this equation predicts not only the gas phase but also the liquid phase of the same substance — a model from which the modern understanding of phase transitions and the critical point was born (for which van der Waals received the 1910 Nobel Prize).
On the site, this is a separate law if you want to go deeper:
Home Experiment
Inflate a balloon to medium size at room temperature, mark the approximate diameter with a marker, then place it in the fridge for 15-20 minutes (not in the freezer — the rubber will stiffen there).
What to notice the balloon will noticeably shrink in size (with the same amount of air inside and almost the same atmospheric pressure outside) — a direct demonstration of Charles’s/Gay-Lussac’s law: lowering T at almost constant p requires a decrease in V. Take it back to warmth — the balloon will expand back to its original size.
Take a medical syringe without a needle, draw in some air, tightly close the opening with your finger (so air cannot escape), and try to push the plunger.
What to notice the more you compress (decrease V), the more the syringe resists your finger — the increasing pressure exactly follows Boyle’s law pV = const (§4): half the volume — twice the pressure, you physically feel it with your hand.
Practice problems
First think for yourself, then open the solution. The solution everywhere follows the scheme: Given → Law → Solution → Answer.
1. A gas at pressure 200 kPa and volume 4 L (constant T) expands to 8 L. Find the new pressure.
Given p₁ = 200 kPa, V₁ = 4 L, V₂ = 8 L, T = const.
Law Boyle’s law (§2): p₁V₁ = p₂V₂.
Solution p₂ = p₁V₁/V₂ = 200·4/8 = 100 kPa.
Answer 100 kPa — volume twice as large, pressure half as much.
2. A gas at constant pressure occupies 2 L at a temperature of 300 K. To what volume will it expand when heated to 450 K?
Given V₁ = 2 L, T₁ = 300 K, T₂ = 450 K, p = const.
Law Charles’s law (§2): V₁/T₁ = V₂/T₂.
Solution V₂ = V₁·T₂/T₁ = 2·450/300 = 3 L.
Answer 3 L.
3. A sealed cylinder with gas at 300 K has a pressure of 150 kPa. To what temperature must it be heated so that the pressure rises to 200 kPa (cylinder volume does not change)?
Given p₁ = 150 kPa, T₁ = 300 K, p₂ = 200 kPa, V = const.
Law Gay-Lussac’s law (§2): p₁/T₁ = p₂/T₂.
Solution T₂ = T₁·p₂/p₁ = 300·200/150 = 400 K.
Answer 400 K (127 °C).
4. Why, according to kinetic theory (§3), does gas pressure increase when heated even if the volume and number of molecules do not change?
Law formula (1), §3: p = (2/3)nV⟨Ek⟩.
Solution temperature is essentially a measure of the average kinetic energy of molecules ⟨Ek⟩. Upon heating (nV does not change, volume and number of molecules the same) the molecules on average move faster — each collision with the wall is stronger AND collisions occur more frequently (the particle makes it to the wall and back faster) — both effects raise p.
Answer increase in average kinetic energy of molecules — stronger and more frequent collisions with the walls.
5. Under what conditions (high/low pressure, high/low temperature) is a real gas worst described by the ideal gas equation? Why?
Law limitations of the model (§6).
Solution worst — at HIGH pressure and LOW temperature: molecules come closer, their intrinsic volume ceases to be negligibly small compared to the free space, and attractive forces (van der Waals) become noticeable — both assumptions of the ideal model (point-like, non-interacting particles) are violated. Under these same conditions, the gas usually approaches the point of condensation into a liquid.
Answer high pressure + low temperature — conditions close to condensation.
Typo, incorrect formula, numbers don’t add up, poorly explained — write to us: feedback@bridge42worlds.org. Every message is actually read, analyzed, and corrections are made based on it — this is not "into a black hole".