Gas molecules fly in all directions and hit the walls; the sum of those impacts is pressure.
Boyle’s law keeps PV constant at fixed T and n, and Charles’s law makes V proportional to absolute temperature at fixed P and n.
Together they form the ideal gas equation PV = nRT, and mixed gases follow Dalton’s law of partial pressures.
Change pressure and temperature and watch volume respond, then build intuition for PV = nRT.
How Do Gas Molecules Behave?
If you nudge pressure, volume, absolute temperature, and moles all at once, the pair that is actually causing the change is hidden. Boyle's law leaves P and V inverse only after temperature and moles are held; Charles's law leaves V and T direct only after pressure and moles are held. Stack Avogadro's relation on the same conditions and the four quantities collapse to the single line PV = nRT, where T is kelvin, not Celsius. For a mixture, Dalton's place sits beside that equation: each gas adds to the total pressure by its own mole fraction. Write the held conditions first, or the same letters will point at different laws.
④Smaller room (volume) → hit more often → pressure ↑
⑤More balls (moles) → more frequent hits → pressure ↑
Visualizing the Ideal Gas Law
seegongsik.com
Gas container and PV diagram — volume change with pressure and temperature
On the left is a vessel with a piston; fifteen dots inside sit on fixed seeds, and only their radius and brightness grow when temperature is higher. The height of the vessel writes volume V in L, and three arrows above write pressure P in atm. On the right a gold isotherm is drawn on the PV axes and a red dot marks the present state. Raising the pressure slider from 0.5 atm to 5.0 atm at the same temperature seats the piston lower so V shrinks and the red dot is restamped along the isotherm. Changing the temperature slider from -50°C to 200°C converts T = °C + 273 into kelvin and redraws the isotherm itself. In this figure n is fixed at 1 mol and R is 0.0821.
1 atm
27 °C
Ideal Gas Equation
PV = nRT
P (atm), V (L), n (mol), R = 0.0821, T: absolute temperature (K)
🔍 Try the Two Sliders!
①P↑ → V↓ (Boyle's law: PV constant at fixed T)
②T↑ → V↑ (Charles's law: V/T constant at fixed P)
③Isotherms on PV graph are hyperbolas
④The red dot shows the current state
⑤This sketch keeps n = 1 mol fixed
Combining Three Laws → Ideal Gas Equation
Multiplying with the Celsius mark as written collapses the relation near 0°C. T in the gas laws is absolute, so V/T and PV = nRT hold only after you shift with K = °C + 273. Change pressure and volume freely at the same time and you break Boyle's premise (fixed T, n) and Charles's premise (fixed P, n) together, so write which cell is held beside the equation first. At high pressure and low temperature, molecular volume and attractions start to pull a real gas off the ideal line, and the 22.4 L mark of STP is the same number only at 0°C and 1 atm. Align units and the kelvin shift before you compute a ratio.
Boyle's Law
P₁V₁ = P₂V₂ (T, n constant)
At fixed T → P and V are inversely proportional
Charles's Law
V1T1 = V2T2 (P, n constant)
At fixed P → V proportional to absolute T
Avogadro's Law
V1n1 = V2n2 (P, T constant)
Equal volumes of gas at same conditions contain equal molecules
💡 Why Do the Three Laws Combine into One?
①Boyle: P, V inverse (push a balloon, it shrinks)
②Charles: V, T proportional (heat a balloon, it expands)
③Avogadro: V, n proportional (more air → larger balloon)
④Combine three → PV = nRT (ideal gas)
⑤T must be in K: K = °C + 273
Gas Mixtures and Partial Pressure
Dalton's Law of Partial Pressures
Ptotal = P₁ + P₂ + P₃ + ⋯
Sum of independent pressures from each gas
Mole Fraction and Partial Pressure
Pi = xi × Ptotal = nintotal × Ptotal
mole fraction × total pressure = partial pressure
💡 Why Do Real Gases Differ?
①'Ideal gas' = no intermolecular forces + negligible molecular volume
②At high P or low T, real gases deviate from ideal
③Reason: intermolecular attraction and finite molecular volume
④At STP (0°C, 1 atm), 1 mol ideal gas = 22.4 L
Worked Examples
Example 1
At constant temperature, a gas occupying 3 L at 2 atm is compressed to 6 atm. What is its volume?
1
Constant T and moles → Boyle’s law P₁V₁ = P₂V₂.
P₁V₁ = P₂V₂
2
Substitute into V₂ = P₁V₁/P₂.
V₂ = 2 × 36 = 1 L
▸ 1 L
Boyle’s law: at constant T, pressure and volume are inversely proportional. 3× pressure → 1/3 volume.
Example 2
What volume does 1 mol of ideal gas occupy at 27°C and 1 atm? (R = 0.0821)
1
From PV = nRT, V = nRT/P. Use absolute temperature in K.
V = nRTP, T = 27 + 273 = 300 K
2
Substitute n = 1, R = 0.0821, T = 300, P = 1.
V = 1 × 0.0821 × 3001 ≈ 24.6 L
▸ about 24.6 L
When using PV = nRT, T must be in kelvin. 27°C = 300 K.
Summary
Ideal Gas Equation
PV = nRT
R = 0.0821 L·atm/(mol·K) = 8.314 J/(mol·K)
exam-style
A steel vessel holds 2 mol N₂ and 3 mol O₂ with total pressure 5 atm. What is the partial pressure of O₂?
①1 atm
②2 atm
③3 atm
④4 atm
⑤5 atm
▸ ③ 3 atm
1
Dalton’s law of partial pressures: partial pressure = mole fraction × total pressure.