Gas molecules fly around without rest and strike the walls of their container; the push of those collisions on the walls is pressure.
Hold temperature and the number of molecules fixed and pressure and volume are inversely proportional (Boyle's law); hold pressure and the number of molecules fixed and volume is proportional to absolute temperature (Charles's law).
Adding the number of molecules and folding these relationships together gives the ideal gas equation PV = nRT, and in a gas mixture the partial pressures of the gases add up to the total pressure.
Changing the pressure and temperature in the diagram moves the volume of 1 mol of gas and its state on the isotherm together.
How Do Gas Molecules Move?
The state of a gas is set by four quantities, pressure, volume, temperature and number of molecules, and if all four change at once you cannot tell which change caused which. So Boyle held temperature and the number of molecules fixed and compared only pressure and volume, finding them inversely proportional, while Charles held pressure and the number of molecules fixed and found volume proportional to absolute temperature. Add Avogadro's law, that at the same temperature and pressure volume is proportional to the number of molecules, and the three relationships fold into one equation, PV = nRT. The T in it is absolute temperature in kelvins, and Dalton's law of partial pressures, in which each gas in a mixture carries its mole fraction's share of the total pressure, follows straight from it.
💡 A Room Full of Ping-Pong Balls
①Gas molecules are like ping-pong balls flying in every direction around a room
②Pressure is the force of the balls striking the walls, divided by the wall area
③Make the room smaller and the balls hit the walls more often in the same time, so the pressure rises
④Raise the temperature and the balls fly faster, hitting harder and more often
⑤Add more balls (moles) and the hits also become more frequent, raising the pressure
Following the Ideal Gas Equation
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Container and PV graph for 1 mol of gas: volume as pressure and temperature change
On the left is a container closed by a piston, and its height is the volume V of 1 mol of gas. The arrows on top are the pressure P on the piston, and the 15 dots inside mark that the number of molecules does not change. The length of each dot's tail stands for the average speed of the molecules, so raising the temperature lengthens the tails while the dots stay the same size. In the graph on the right, the gold curve is the isotherm P = nRT / V for the current temperature and the red dot is the current state. Raising the pressure from 1 atm to 5 atm slides the dot up and to the left along the same isotherm and cuts the volume to one fifth. Raising the temperature from 0 °C to 200 °C pushes the isotherm itself outward from the dashed 0 °C curve, so the volume grows at the same pressure.
1.0 atm
27 °C
Ideal Gas Equation
PV = nRT
P: pressure (atm), V: volume (L), n: moles, R = 0.0821 L·atm/(mol·K), T: absolute temperature (K)
🔍 What to Check with the Sliders
①Hold the temperature and double the pressure, and the volume halves: Boyle's law
②Hold the pressure and raise the temperature from 0 °C (273 K) to 200 °C (473 K), and the volume grows by 473 / 273 ≈ 1.73 times: Charles's law
③An isotherm on the PV graph is an inverse curve along which P × V stays constant
④The starting state (1 atm, 27 °C) has a volume of 24.6 L, the same as the answer to Example 2 below
Three Laws in One
The most common slip is putting Celsius temperatures straight into the equation. Going from 10 °C to 20 °C doubles the number, but the absolute temperature rises only from 283 K to 293 K, about 3.5 %, so the volume grows by a little over 3.5 %, not twofold. The T in the gas laws is always converted with K = °C + 273. Within a problem, first write down which quantities are held constant; that decides whether Boyle's or Charles's law applies. The 22.4 L figure pairs with 1 mol only at 0 °C and 1 atm; at 27 °C the same mole takes up about 24.6 L. At very high pressure or very low temperature, attractions between molecules and the volume of the molecules themselves make real gases depart from this equation.
Boyle's Law
P₁V₁ = P₂V₂ (T, n constant)
At constant temperature and moles, pressure and volume are inversely proportional
Charles's Law
V1T1 = V2T2 (P, n constant)
At constant pressure and moles, volume is proportional to absolute temperature
Avogadro's Law
V1n1 = V2n2 (P, T constant)
At the same temperature and pressure, equal volumes of gas hold equal numbers of molecules
💡 Why the Three Laws Become One Equation
①Boyle: V ∝ 1/P; squeeze a balloon and it shrinks
②Charles: V ∝ T; heat the air in a hot-air balloon and it swells
③Avogadro: V ∝ n; blow more air into a balloon and it grows
④Combining the three gives V ∝ nT / P, that is PV = nRT, with the gas constant R as the constant of proportionality
⑤Always use absolute temperature for T (K = °C + 273)
Gas Mixtures and Partial Pressure
Dalton's Law of Partial Pressures
Ptotal = P₁ + P₂ + P₃ + ⋯
The total pressure is the sum of the pressures each gas would exert if it filled the container alone
Mole Fraction and Partial Pressure
Pi = xi × Ptotal = nintotal × Ptotal
The partial pressure of a gas is its mole fraction times the total pressure
💡 Why Real Gases Differ
①An ideal gas is a model in which molecules exert no forces on each other and take up no volume themselves
②At high pressure molecules are crowded close, so their own volume can no longer be ignored
③At low temperature molecules slow down and the attractions between them start to matter
④So at high pressure and low temperature, real gases depart from PV = nRT
⑤At 0 °C and 1 atm, 1 mol of ideal gas occupies 22.4 L
Worked Examples
Example 1
At constant temperature, a gas occupies 3 L at 2 atm. What is its volume when compressed to 6 atm?
1
Temperature and moles are constant, so use Boyle's law, P₁V₁ = P₂V₂.
P₁V₁ = P₂V₂
2
Substitute into V₂ = P₁V₁ / P₂.
V₂ = 2 × 36 = 1 L
▸ 1 L
The pressure tripled, so the volume drops to one third. It is the same change as the dot moving along the isotherm in the diagram when the pressure is raised at fixed temperature.
Example 2
What is the volume of 1 mol of ideal gas at 27 °C and 1 atm? (R = 0.0821 L·atm/(mol·K))
1
Rearrange PV = nRT for V and convert the temperature to kelvins.
V = nRTP, T = 27 + 273 = 300 K
2
Substitute n = 1, R = 0.0821, T = 300 and P = 1.
V = 1 × 0.0821 × 3001 ≈ 24.6 L
▸ about 24.6 L
It is larger than the 22.4 L at 0 °C because the temperature rose from 273 K to 300 K; 22.4 × 300 / 273 ≈ 24.6 L gives the same answer.
Summary
Ideal Gas Equation
PV = nRT
R = 0.0821 L·atm/(mol·K) = 8.314 J/(mol·K)
CSAT-style problem, adapted
A steel container holds 2 mol of N₂ and 3 mol of O₂ at a total pressure of 5 atm. What is the partial pressure of O₂?
①1 atm
②2 atm
③3 atm
④4 atm
⑤5 atm
▸ ③ 3 atm
1
By Dalton's law, partial pressure = mole fraction × total pressure.
PO_2 = nO_2ntotal × Ptotal
2
The mole fraction of O₂ is 3 / (2 + 3) = 3/5, and the total pressure is 5 atm.
PO_2 = 35 × 5 = 3 atm
🎯 Exam Points
①PV = nRT and the two values of R (0.0821 L·atm/(mol·K) and 8.314 J/(mol·K))
②Boyle: PV constant; Charles: V / T constant
③T is always absolute temperature (K = °C + 273)
④Dalton's law of partial pressures: P_total = P₁ + P₂ + ⋯, and partial pressure = mole fraction × total pressure