Properties of Gases & Gas Laws

Introduction & Everyday Importance of Gases

• Gases play critical roles in daily life:
  • Air mixture inhaled in respiration.
  • Natural gas used as fuel in combustion (cooking, heating).
  • Medical uses (e.g., anesthetic gases) due to their ability to diffuse through tissues & the blood–brain barrier.

Kinetic Molecular Theory (KMT)

• Describes molecular-level behavior of gases.
• Core postulates:
  • Gas particles (atoms or molecules) are in constant, random, straight-line motion.
  • Average speed (and thus kinetic energy, $KE$) scales with absolute temperature $T$.
  • Higher $T \rightarrow$ higher speed $\rightarrow$ greater $KE$.
  • Collisions are perfectly elastic (no net energy loss to surroundings).
  • Size of individual particles is negligible compared with total container volume ⇒ large intermolecular distances; attractive forces essentially zero.
  • Result: gas particles fill entire available volume.

Unique Macroscopic Properties of Gases

• Highly compressible: volume can be significantly reduced with applied pressure (movable piston demonstration).
• Expand to occupy full container volume because of random motion & negligible attraction.
• Variable density (much lower than liquids/solids).

Pressure: Definition, Units & Measurement

• Molecular view: pressure arises from billions of particle collisions with container walls.
• Macroscopic definition: $P = \frac{F}{A}$ (force per unit area).
• Common units & conversions (exact by definition):
  $1\;\text{atm} = 760\;\text{mmHg} = 760\;\text{Torr} = 14.7\;\text{psi} = 101\,325\;\text{Pa}$
• Measurement device – barometer (mercury column):
  • Atmospheric pressure pushes Hg up a sealed tube; height at sea level ≈ $760\;\text{mmHg}$ (defines $1\,\text{atm}$).
  • Lower external $P$ (mountains) ⇒ lower Hg height.

Atmospheric Pressure & Altitude Effects

• Composition of dry air (by volume): $78\%\,N<em>2$, $21\%\,O</em>2$, $\sim1\%$ others (Ar, CO$2$, H$2$O, \dots).
• $P_{\text{atm}}$ decreases with elevation:
  • Denver (≈$3000\,\text{m}$): $P \approx 0.83\,\text{atm}$.
  • Mt. Everest summit: $P \approx 0.33\,\text{atm}$ (≈ one-third of sea-level air & O$_2$).
• Physiological consequences:
  • Less $O<em>2$ per breath ⇒ rapid breathing (hyperventilation), need for supplemental O$2$ tanks in extreme altitudes.
  • In aircraft (~$10\,000\,\text{m}$ cruising): cabin is pressurised to $0.75\,\text{atm}$ while outside $P \approx 0.27\,\text{atm}$; ear-popping results from delayed equalisation across eardrum.

Fundamental Gas Variables

• Four inter-related macroscopic variables:
  • Pressure $P$ (atm, Pa, etc.)
  • Volume $V$ (L, m$^3$)
  • Temperature $T$ (K)
  • Amount (moles $n$)
• Holding some constant while varying others leads to simple gas laws discovered in the 17th–19th centuries.

Boyle’s Law (Pressure–Volume, $P\text{–}V$)

• Conditions: $n$ & $T$ constant.
• Statement: $P$ and $V$ are inversely proportional.
  $P \uparrow \Rightarrow V \downarrow$ and vice-versa.
• Mathematical form: $P<em>1 V</em>1 = P<em>2 V</em>2$.
• Graphs:
  • $P$ vs $V$ → hyperbola.
  • $P$ vs $\frac{1}{V}$ → straight line.
• Demonstration: increasing piston mass compresses gas ⇒ $P<em>{\text{final}} > P</em>{\text{initial}}$, $V<em>{\text{final}} < V</em>{\text{initial}}$.
• Sample problems addressed:
  1. Moving 1-L gas at $1\,\text{atm}$ into larger container ⇒ $V$ ↑ so $P$ ↓ (qualitative).
  2. Oxygen cylinder: given $P<em>i = 8.5\,\text{atm}$, $V</em>i = 10.7\,\text{L}$, $P<em>f = 0.92\,\text{atm}$ ⇒ $V</em>f = 98.6\,\text{L}$ using Boyle’s equation.

Charles’s Law (Volume–Temperature, $V\text{–}T$)

• Conditions: $n$ & $P$ constant.
• Statement: $V$ directly proportional to absolute $T$.
  $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$ (temperatures in Kelvin only).
• Graph: straight line through origin; extrapolation meets $V = 0$ at $T = 0\,\text{K}$ (absolute zero = $-273\,^{\circ}!\text{C}$).
• Examples:
  1. Doubling $T$ ((25\,^{\circ}\text{C} \rightarrow 50\,^{\circ}\text{C})) doubles $V$ for fixed $P,n$.
  2. Neon sample: $V<em>i = 19.5\,L$, $T</em>i = 76^{\circ}!\text{C}$, cool to $38^{\circ}!\text{C}$ ⇒ $V_f = 17.4\,L$.

Gay-Lussac’s Law (Pressure–Temperature, $P\text{–}T$)

• Conditions: $n$ & $V$ constant.
• Statement: $P$ directly proportional to $T$.
  $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$.
• Illustration: Barrel fixed volume heated from $200\,K$ to $400\,K$ ⇒ $P$ doubles.

Combined Gas Law

• Consolidates Boyle, Charles & Gay-Lussac for fixed $n$:
  $\frac{P<em>1 V</em>1}{T<em>1} = \frac{P</em>2 V<em>2}{T</em>2}$.
• Example (diver’s nitrogen bubble):
  • $P<em>i = 4.2\,\text{atm}$, $T</em>i = 11^{\circ}!\text{C}$, $V_i = 0.01\,\text{mL}$.
  • Upon ascent to sea level $P<em>f = 1\,\text{atm}$, $T</em>f = 29^{\circ}!\text{C}$ ⇒ $V_f = 0.045\,\text{mL}$.
  • Medical relevance: rapid ascent may produce large gas bubbles ⇢ decompression sickness.

Avogadro’s Law (Volume–Moles, $V\text{–}n$)

• Conditions: $P$ & $T$ constant.
• Statement: $V$ directly proportional to number of moles $n$.
  $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$.
• Everyday example: inflating a balloon – adding gas molecules increases volume proportionally.
• Calculation: Start with $n<em>i = 2\,\text{mol}$, $V</em>i = 1.0\,L$, add $1\,\text{mol}$ ⇒ $n<em>f = 3\,\text{mol}$; $V</em>f = 1.5\,L$.

Standard Temperature & Pressure (STP) & Molar Volume

• Defined conditions:
  • $T = 0^{\circ}!\text{C} = 273.15\,K$ (often $273\,K$).
  • $P = 1\,\text{atm}$.
• Molar volume: At STP, $1\,\text{mol}$ of any ideal gas occupies $22.4\,L$.
  • Equality statement (conversion factor): $1\,\text{mol}\;\xleftrightarrow{\text{STP}}\;22.4\,L$.
• Example: Find volume for $2.0\,\text{mol}$ O$_2$ at STP ⇒ $2.0\;\text{mol} \times 22.4\;\text{L/mol} = 44.8\,L$.

Density of Gases at STP

• General formula: $\rho = \frac{m}{V}$.
• Substituting STP molar quantities:
  $\rho_{\text{STP}} = \frac{M}{22.4\,L}$ where $M$ = molar mass.
• Examples:
  • Helium: $M = 4.00\,g\,mol^{-1} \Rightarrow \rho = 0.179\,g\,L^{-1}$.
  • CO$_2$: $M = 44.01\,g\,mol^{-1} \Rightarrow \rho = 1.965\,g\,L^{-1}$.
  • Since \rho{\text{CO}2} > \rho{\text{air}} (1.29\,g\,L^{-1}) ⇒ CO$2$ sinks in air.

Dalton’s Law of Partial Pressures

• In a gas mixture, each component exerts a partial pressure as if it alone occupied the volume.
• Total pressure:
  $P<em>{\text{tot}} = P</em>1 + P<em>2 + P</em>3 + \dots$
• If mole (or volume) percentages known: partial pressure of gas $i$ is
  $P<em>i = x</em>i P<em>{\text{tot}}$ where $x</em>i$ = mole or volume fraction (e.g., $\frac{\%}{100}$).
• Applications & examples:
  1. Air at sea level: $P<em>{\text{O}</em>2} = 0.21 \times 1\,\text{atm} = 0.21\,\text{atm}$.
  2. Cylinder with $P<em>{\text{tot}} = 3.00\,atm$, $P</em>{\text{O}<em>2} = 0.63\,atm$ ⇒ $P</em>{\text{N}_2} = 2.37\,atm$ (by subtraction).
  3. Heliox mixture (60% He, 40% O$2$) at $700\,\text{mmHg}$ ⇒ $P</em>{\text{O}_2} = 0.40 \times 700 = 280\,\text{mmHg}$.

Worked‐Problem Themes & Strategies

• Always translate word problems to variable lists
  (e.g., $P<em>i,V</em>i,T<em>i,n</em>i$ ➜ identify unknown).
• Convert all temperatures to Kelvin before substitution.
• Match units: use consistent pressure units or convert via equality factors.
• For Boyle/Charles/Gay-Lussac: hold two variables constant explicitly.
• For STP or molar volume conversions: verify the gas is indeed at STP.
• Use dimensional analysis to ensure units cancel correctly.

Real-World & Physiological Implications

• Mountain climbing & aviation: decreasing $P<em>{\text{O}</em>2}$ ⇒ hypoxia; supplemental O$_2$ or cabin pressurisation required.
• Scuba diving: combined gas law predicts nitrogen bubble expansion ⇒ controlled ascent prevents decompression sickness (“the bends”).
• Medical gas mixtures (heliox) reduce airway resistance; Dalton’s law permits calculation of therapeutic $P<em>{\text{O}</em>2}$.
• Gas compressibility underpins piston engines & industrial gas storage (Boyle’s law in action).

Key Takeaways

• Molecular motion underlies all gas properties; temperature is the average kinetic-energy gauge.
• Four variables $P,V,T,n$ interact via predictable gas laws; holding two constant yields simple two-variable relations.
• STP provides a universal yardstick (molar volume $22.4\,L$); handy for density & stoichiometric calculations.
• Mixtures obey Dalton’s law: total pressure is additive; fractions determine partial pressures crucial for respiration & industry.
• Correct unit handling (Kelvin, atm, L, mol) is essential for accurate problem solving.