Gas Laws — the Hidden Relationship Between Pressure, Volume, and Temperature 🎈

A few puzzles: why does a balloon shrink in the fridge and puff up on the radiator? Why does an aerosol can explode if you leave it in the sun? Why does water boil at 86°C at 4000 m altitude instead of 100°C? Why does a pressure cooker cook food twice as fast? They all share one root cause: the three-way relationship between pressure, volume, and temperature in a gas — captured by one elegant equation.

The core idea in one paragraph 📌

Three quantities — pressure (\( P \)), volume (\( V \)), absolute temperature (\( T \)) — of a fixed amount of gas are coupled. Boyle: at constant \( T \), \( PV=\text{const} \). Charles: at constant \( P \), \( V/T=\text{const} \). Gay-Lussac: at constant \( V \), \( P/T=\text{const} \). Combine the three ⇒ the combined gas law: \( PV/T=\text{const} \). Add Avogadro (\( V \propto n \) at fixed \( P, T \)) and you get the ideal gas equation \( \boxed{PV = nRT} \), with \( R = 8.314\,\text{J/(mol·K)} \). Temperature must always be in kelvin; using Celsius is catastrophically wrong. Kinetic theory derives all this from bouncing particles: average kinetic energy per molecule \( = \tfrac{3}{2}k_B T \).

Four key laws 📐

\[ \boxed{\text{Boyle (const T):} \quad P_1 V_1 = P_2 V_2} \]

Inverse P–V — compress ⇒ higher pressure. A sealed syringe you push on.

\[ \boxed{\text{Charles (const P):} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2}} \]

Volume proportional to absolute T — heat ⇒ expand. Balloon on a warm radiator.

\[ \boxed{\text{Gay-Lussac (const V):} \quad \frac{P_1}{T_1} = \frac{P_2}{T_2}} \]

Pressure proportional to absolute T — heating a rigid can ⇒ higher pressure. Sun-baked aerosol.

\[ \boxed{\text{Combined:} \quad \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \qquad\Rightarrow\qquad PV = nRT} \]

Ideal gas law — \( n \): moles, \( R = 8.314\,\text{J/(mol·K)} = 0.0821\,\text{L·atm/(mol·K)} \).

How the laws relate 📊

Law	Fixed	Relation	Everyday example
Boyle	\( T \)	\( PV = \text{const} \)	Squeezing a syringe
Charles	\( P \)	\( V/T = \text{const} \)	Balloon in the fridge
Gay-Lussac	\( V \)	\( P/T = \text{const} \)	Aerosol in the sun
Avogadro	\( P, T \)	\( V/n = \text{const} \)	Doubling gas in a balloon
Combined	—	\( PV/T = \text{const} \)	Everything changes at once
Ideal gas	—	\( PV = nRT \)	The universal form

Why temperature must be in kelvin ⚠️

The single biggest student mistake: plugging in Celsius. Example: a balloon at \( 27° \)C warmed to \( 54° \)C. How much does its volume grow?

• ❌ Wrong: \( V_2/V_1 = 54/27 = 2 \) → double
• ✅ Right: \( V_2/V_1 = (54+273)/(27+273) = 327/300 = 1.09 \) → only 9% bigger

Huge difference! Because Celsius \( T=0 \) is an arbitrary point, not a physical zero. Charles’s law is anchored to absolute zero.

Example 1: Squeezing a syringe (\( PV=\text{const} \)) 💉

You have a syringe holding \( V_1 = 60\,\text{mL} \) of air at \( P_1 = 1\,\text{atm} \). Seal the tip and push the plunger to \( V_2 = 20\,\text{mL} \). New pressure?

\[ P_2 = \frac{P_1 V_1}{V_2} = \frac{1 \times 60}{20} = 3\ \text{atm} \]

Three times higher — the plunger pushes back strongly.

Example 2: Balloon from warm to cold (\( V/T=\text{const} \)) 🎈

A balloon \( V_1 = 2\,\text{L} \) at \( T_1 = 27°\text{C} = 300\,\text{K} \) goes into a \( T_2 = 3°\text{C} = 276\,\text{K} \) fridge. New volume?

\[ V_2 = V_1 \cdot \frac{T_2}{T_1} = 2 \times \frac{276}{300} = 1.84\,\text{L} \]

About 8% shrinkage — clearly visible.

Example 3: Aerosol can in the sun (\( P/T=\text{const} \)) 💥

A can at \( T_1 = 20° \)C = \( 293 \) K has \( P_1 = 3\,\text{atm} \). Left on a car dashboard it reaches \( T_2 = 70° \)C = \( 343 \) K. New pressure?

\[ P_2 = P_1 \cdot \frac{T_2}{T_1} = 3 \times \frac{343}{293} = 3.51\ \text{atm} \]

Still below the ~10 atm burst limit, but if \( T \) hits \( 200° \)C (a fire), \( P \to 4.8 \) atm and pyrolysis of the contents can ignite. The “keep away from heat” label is not decorative.

Example 4: How much O₂ in a hospital cylinder? (\( PV=nRT \)) 🩺

A hospital oxygen cylinder, \( V = 40\,\text{L} \), \( P = 150\,\text{atm} \) (~15 MPa), \( T = 20° \)C = \( 293 \) K.

\[ n = \frac{PV}{RT} = \frac{150 \times 40}{0.0821 \times 293} \approx 249\ \text{mol} \]

With \( M(\text{O}_2)=32 \) g/mol: \( m = 249 \times 32 \approx 8.0\,\text{kg} \) of oxygen. At 2 L/min (~240 L of gas at 1 atm) for a patient, that’s ~2500 minutes = 42 hours 🕐.

Kinetic theory — why these laws are correct 🧪

Boyle, Charles, and the rest were empirical. Then Clausius, Maxwell, and Boltzmann (1860s) derived them from a simple model: point particles bouncing elastically. The famous result:

\[ \boxed{\bar{E}_k = \tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}k_B T} \]

\( k_B = 1.38\times 10^{-23}\,\text{J/K} \) is Boltzmann’s constant. So temperature is literally the average kinetic energy per molecule. And \( PV=nRT \) is just its macroscopic form.

The root-mean-square speed of a molecule:
\[ v_\text{rms} = \sqrt{\frac{3RT}{M}} \]

For room air (M ≈ 29 g/mol, T = 300 K): \( v_\text{rms} \approx 507\,\text{m/s} \) — faster than sound! Molecules move that fast, but the mean free path is only ~70 nm, so they diffuse in every direction.

Python analysis 🐍

1) Boyle’s law — PV diagram at three temperatures

import numpy as np, matplotlib.pyplot as plt

R = 0.0821   # L·atm/(mol·K)
n = 1        # 1 mol
V = np.linspace(0.5, 25, 200)   # L

for T in [200, 300, 500]:
    P = n * R * T / V
    plt.plot(V, P, label=f"T = {T} K")

plt.xlabel("Volume V (L)"); plt.ylabel("Pressure P (atm)")
plt.title("Boyle's law: PV = nRT (isotherms)")
plt.grid(alpha=0.3); plt.legend(); plt.show()

2) Balloon at altitude — how much does it grow?

# Standard atmosphere: P and T fall with altitude
def atm_conditions(h_m):
    """P (Pa) and T (K) at altitude h (m) — simple model."""
    T = 288.15 - 0.0065 * h_m      # linear lapse (troposphere)
    P = 101325 * (T/288.15)**5.256 # barometric formula
    return P, T

def balloon_volume(V0, P0, T0, P, T):
    """Combined law: PV/T = const"""
    return V0 * (P0/P) * (T/T0)

V0, P0, T0 = 1.0, 101325, 288.15   # 1 L at sea level

for h in [0, 1000, 3000, 5000, 10000, 20000]:
    P, T = atm_conditions(h)
    V = balloon_volume(V0, P0, T0, P, T)
    print(f"h={h:>5d}m  →  P={P/101325:.3f} atm, T={T-273.15:.1f}°C, V={V:.2f} L")

# At h = 20 km a balloon grows ~15× — that's why weather balloons burst
# at high altitude.

3) Maxwell-Boltzmann distribution — molecular speeds

import numpy as np, matplotlib.pyplot as plt

def maxwell_boltzmann(v, M_kg_per_mol, T):
    """Molecular speed distribution (m/s)"""
    R = 8.314
    return (M_kg_per_mol/(2*np.pi*R*T))**1.5 * 4*np.pi*v**2 \
           * np.exp(-M_kg_per_mol*v**2/(2*R*T))

v = np.linspace(0, 2000, 500)
for T in [200, 300, 500, 1000]:
    plt.plot(v, maxwell_boltzmann(v, 0.029, T), label=f"T = {T} K")

plt.xlabel("Speed (m/s)"); plt.ylabel("Probability density")
plt.title("Maxwell-Boltzmann distribution — air (M = 29 g/mol)")
plt.legend(); plt.grid(alpha=0.3); plt.show()

# rms speed
R, M = 8.314, 0.029
for T in [200, 300, 500, 1000]:
    v_rms = np.sqrt(3*R*T/M)
    print(f"T = {T} K → v_rms = {v_rms:.0f} m/s")

Take-home summary 🎁

The three quantities \( P, V, T \) of a fixed gas are fully coupled. Boyle (\( PV=k \), isothermal), Charles (\( V/T=k \), isobaric), Gay-Lussac (\( P/T=k \), isochoric), and Avogadro (\( V/n=k \)) combine into \( PV=nRT \). Temperature is always in kelvin — Celsius gives errors of 5–10×. Kinetic theory says \( \bar E_k = \tfrac{3}{2}k_B T \) ⇒ temperature is literally molecular motion. Room-air molecules move at ~500 m/s — faster than sound! These equations drive not just balloons but weather, high-altitude cooking, engine design, and industrial chemistry 🎯.

Congratulations — Chapter 4 complete 🎉

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“Nice to know” box: Why does a pressure cooker cook food 3× faster? 🍲

Water boils at 100°C only at atmospheric pressure (~1 atm). At 4000 m altitude the pressure is ~0.6 atm, so water boils at 86°C — and food cooks 2–3× slower (chemical reactions speed up exponentially with temperature: the Arrhenius law says roughly every 10°C = 2× faster). A pressure cooker does the opposite: sealed steam pushes the internal pressure to ~2 atm ⇒ water boils at ~120°C ⇒ food cooks ~4× faster. The same gas physics that makes altitude cooking slow makes pressure cooking fast ✨. Fun fact: Denis Papin who built the first “steam digester” in 1679 reported that his prototype exploded with “a bang louder than 20 cannons”! Modern designs have a proper safety valve.

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Test yourself 📝

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References and further exploration 📚

Articles and reference

• Wikipedia: Gas laws, Ideal gas law, Boyle’s law, Charles’s law, Gay-Lussac’s law, Kinetic theory of gases, Maxwell–Boltzmann distribution
• HyperPhysics — Kinetic theory
• Feynman Lectures — Vol. I, Ch. 39: Kinetic theory of gases

Videos (YouTube)

• Veritasium: What is temperature really?
• MinutePhysics: Boyle, Charles, and the ideal gas
• Steve Mould: Marshmallow in a vacuum
• Real Engineering: How internal combustion engines use gas laws
• Crash Course Chemistry — The ideal gas law

External simulators

• PhET — Gas Properties — interactive P, V, T, n
• PhET — Diffusion
• oPhysics — Boyle’s Law

On this site 🔗

• نسخه‌ی فارسی — قوانین گازها
• Previous section — Heat transfer
• Related — Temperature and thermometry

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🎉 Congratulations — Chapter 4 (Temperature and Heat) is complete. You now have temperature, expansion, heat, phase changes, the three transfer modes, and gas laws. Ready for Chapter 5 — Thermodynamics 🚀, which weaves all of this together with work and energy?