Playground
Interactive piston-cylinder with bouncing gas molecules: adjust volume, moles, and temperature to see pressure change in real time.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Pressureoutput Absolute pressure of the gas | Pa | M·L⁻¹·T⁻² | 1000 – 1000000 | |
| Volume Volume occupied by the gas | m³ | L³ | 0.001 – 1 | |
| Amount of substance Number of moles of gas | mol | N | 0.01 – 10 | |
| Temperature Absolute temperature of the gas | K | Θ | 100 – 1000 |
Deep dive
Derivation
Combines three empirical laws: Boyle's law (PV = const at fixed T), Charles's law (V/T = const at fixed P), and Avogadro's law (V/n = const at fixed P,T). Merging: PV/nT = const ≡ R. The constant R = 8.314 J/(mol·K) is determined experimentally.
Experimental verification
Verified by Regnault (1847) using precision gas thermometry. Modern confirmation via pVT measurements on dilute gases (He, Ar) at low pressure, where intermolecular forces vanish. Agreement to <0.1% for pressures below 1 atm.
Common misconceptions
- The ideal gas law is not a single discovery but a synthesis of Boyle's, Charles's, and Avogadro's laws
- T must be absolute temperature (Kelvin), not Celsius — using °C gives wrong results
- Real gases deviate significantly at high pressure or low temperature; the law is an idealization, not universal
Real-world applications
- Weather balloon altitude prediction — volume increases as atmospheric pressure drops
- Scuba diving tank pressure calculations
- Internal combustion engine cylinder analysis
- Designing HVAC systems and pneumatic machinery
Worked examples
Hot-car balloon pressure
Given:
- n:
- 0.05
- T_i:
- 293
- T_f:
- 333
- P_i:
- 101300
Find: P_f
Solution
P_f = P_i × (T_f / T_i) = 101300 × (333 / 293) = 115100 Pa ≈ 115.1 kPa
Volume of 1 mol at STP
Given:
- n:
- 1
- T:
- 273.15
- P:
- 101325
Find: V
Solution
V = nRT/P = 1 × 8.314 × 273.15 / 101325 = 0.02241 m³ = 22.41 L
Scenarios
What if…
- scenario:
- What if you double the temperature while keeping volume fixed?
- answer:
- Pressure doubles. At constant V and n, P ∝ T. Doubling T from 300 K to 600 K doubles P — this is why sealed containers can burst when heated.
- scenario:
- What if you halve the volume at constant temperature?
- answer:
- Pressure doubles (Boyle's law). P ∝ 1/V at constant T and n. This is the principle behind syringes and hydraulic presses.
- scenario:
- What if you use the ideal gas law for water vapor near 100°C?
- answer:
- It works reasonably well at low pressures (< 0.5 atm), but fails near the boiling point at 1 atm because intermolecular hydrogen bonds cause large deviations. Use the van der Waals equation instead.
Limiting cases
- condition:
- V → ∞
- result:
- P → 0
- explanation:
- Gas expands into infinite volume, pressure vanishes.
- condition:
- T → 0
- result:
- P → 0
- explanation:
- At absolute zero, ideal gas pressure drops to zero (real gases liquefy first).
- condition:
- n → 0
- result:
- PV → 0
- explanation:
- No gas means no pressure-volume product.
Context
Émile Clapeyron · 1834
Clapeyron unified Boyle's, Charles's, and Avogadro's laws into a single equation. The constant R was later determined precisely by experiment.
Hook
Why does a balloon pop when you leave it in a hot car?
A sealed balloon contains 0.05 mol of air at 20°C and 101.3 kPa. Find the new pressure if the temperature rises to 60°C.
Dimensions: [P]·[V] = [n]·[R]·[T] → (M·L⁻¹·T⁻²)(L³) = (mol)(J·mol⁻¹·K⁻¹)(K) → M·L²·T⁻² = M·L²·T⁻² ✓
Validity: Valid for dilute gases far from condensation. Breaks down at high pressures or low temperatures where intermolecular forces and molecular volume matter (use van der Waals equation instead).