Ideal Gas Law & Kinetic Theory

Atomic-Level Mass Units

  • Definition of the atomic mass unit (u)

    • 1 u=112 of the mass of one 12C atom=1.6605×10−27 kg1\,\text{u}=\frac{1}{12} \text{ of the mass of one }{}^{12}\text{C atom}=1.6605\times10^{-27}\,\text{kg}

    • Rationale for choosing 12C^{12}\text{C}

    • Most abundant carbon isotope

    • Historically convenient reference when mass spectrometry emerged

  • Practical meaning

    • Enables direct, ratio-based comparison of individual atoms without always reverting to SI kilograms

    • Serves as the bridge between microscopic (single atom) and macroscopic (gram-mole) measurements

  • Illustrative comparison

    • Average atomic masses (experimental, weighted by natural isotope abundance)

    • Magnesium (Mg) – 24.305 u24.305\,\text{u}

    • Lithium (Li) – 6.941 u6.941\,\text{u}

    • Relative heaviness

    • 24.3056.941=3.502\frac{24.305}{6.941}=3.502

    • → One Mg atom is 3.5023.502 times as massive as one Li atom

Molecular (Formula) Mass

  • Definition: sum of the atomic masses in a molecule

  • Example – water

    • Composition: H2O\text{H}_2\text{O} (2 H, 1 O)

    • Atomic masses

    • H:1.00794 u\text{H}: 1.00794\,\text{u}

    • O:15.9994 u\text{O}: 15.9994\,\text{u}

    • Calculation

    • 2×1.00794 u+15.9994 u=18.0153 u2\times1.00794\,\text{u}+15.9994\,\text{u}=18.0153\,\text{u}

    • Significance: molecular-mass values feed directly into molar-mass (g mol−1^{-1}) tables used in stoichiometry and gas-law work.

The Mole Concept

  • Motivation: macroscopic samples contain astronomical atom counts ⇒ need a collective counting unit

  • Formal definition

    • One mole = number of atoms in exactly 12 g12\,\text{g} of 12C^{12}\text{C}.

    • This number is Avogadro’s number NAN_A.

  • Avogadro’s number

    • NA=6.022×1023  mol−1N_A = 6.022\times10^{23}\;\text{mol}^{-1} (current CODATA value)

    • Universally links micro (atom) and macro (gram) scales across all substances

  • Mole/particle relation

    • Number of moles nn in any sample: n=NNAn = \frac{N}{N_A} where NN = total particles.

Mass per Mole & Converting Between n, m, & M

  • Manipulating n=NN<em>An=\tfrac{N}{N<em>A} by multiplying top & bottom by individual particle mass </em>:</p><ul><li><p></em>:</p><ul><li><p>n=\frac{mp N}{mp N_A}=\frac{\text{total mass }(m)}{\text{mass per mole }(M)}</p></li></ul></li><li><p>Therefore</p><ul><li><p></p></li></ul></li><li><p>Therefore</p><ul><li><p>n=\frac{m}{M}</p></li><li><p>Molarmass</p></li><li><p>Molar massM(gmol(g mol^{-1})=numericalvalueofatomicormolecularmass(inu).</p></li></ul></li><li><p>Concreteexamples</p><ul><li><p>) = numerical value of atomic or molecular mass (in u).</p></li></ul></li><li><p>Concrete examples</p><ul><li><p>^{12}\text{C}: M=12\,\text{g mol}^{-1} \;(by\,definition)</p></li><li><p></p></li><li><p>\text{Na}: M = 22.9898\,\text{g mol}^{-1}</p></li><li><p></p></li><li><p>\text{Ratio of single-atom masses} = \frac{22.9898\,\text{u}}{12.0000\,\text{u}} = 1.91582</p></li><li><p>Identicalratioappearsformolarmasses:Nais</p></li><li><p>Identical ratio appears for molar masses: Na is1.91582×heavierpermolethan× heavier per mole than^{12}\text{C}.</p></li></ul></li></ul><h3collapsed="false"seolevelmigrated="true">Concept−CheckProblems(Pages9–11)</h3><ul><li><p>Q1–“Whichhasmoremolecules,1molN.</p></li></ul></li></ul><h3 collapsed="false" seolevelmigrated="true">Concept-Check Problems (Pages 9–11)</h3><ul><li><p>Q1 – “Which has more molecules, 1 mol N2or1molOor 1 mol O2?”</p><ul><li><p>Bothcontainexactly?”</p><ul><li><p>Both contain exactlyN_Amolecules⇒Samecount.</p></li></ul></li><li><p>Q2–“1molHmolecules ⇒ Same count.</p></li></ul></li><li><p>Q2 – “1 mol H2vs1molOvs 1 mol O2: molecule count & mass?”

    • Same molecule count (NAeach)butOeach) but O2masslargerbecausemass larger becauseM{O2}=32\,\text{g mol}^{-1}>M{H2}=2\,\text{g mol}^{-1} ⇒ Answer A.

  • Q3 – Hope diamond atom count

    • Mass: 44.5 carats $\times$ 0.200 g carat^{-1}==8.90\,\text{g}</p></li><li><p>Moles:</p></li><li><p>Moles:n=\frac{8.90\,\text{g}}{12.011\,\text{g mol}^{-1}}\approx0.7406\,\text{mol}</p></li><li><p>Atomcount:</p></li><li><p>Atom count:N= nN_A \approx 0.7406\times6.022\times10^{23}\approx4.46\times10^{23}atoms.</p></li></ul></li></ul><h3collapsed="false"seolevelmigrated="true">IdealGasModel:FundamentalAssumptions</h3><ul><li><p>Gasdensitylow→meanseparation(≫)moleculardiameter.</p></li><li><p>Intermolecularforcesnegligibleexceptperfectlyelasticpoint−likecollisions.</p></li><li><p>Undertheselimits,allrealgasesconvergetowardidealbehavior.</p></li></ul><h3collapsed="false"seolevelmigrated="true">EmpiricalBuildingBlocks</h3><ul><li><p>Atfixedatoms.</p></li></ul></li></ul><h3 collapsed="false" seolevelmigrated="true">Ideal Gas Model: Fundamental Assumptions</h3><ul><li><p>Gas density low → mean separation ( \gg ) molecular diameter.</p></li><li><p>Intermolecular forces negligible except perfectly elastic point-like collisions.</p></li><li><p>Under these limits, all real gases converge toward ideal behavior.</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Empirical Building Blocks</h3><ul><li><p>At fixedV,n::P \propto T(originofabsolutetemperaturescale;extrapolatesto(origin of absolute temperature scale; extrapolates toT=0atatP=0).</p></li><li><p>Atfixed).</p></li><li><p>At fixedV,T::P \propto n(doublingmoleculesdoublespressure).</p></li><li><p>Atfixed(doubling molecules doubles pressure).</p></li><li><p>At fixedn,T:Boyle’slaw: Boyle’s lawP \propto \frac{1}{V}⇒⇒PV=\text{constant}.</p></li></ul><h3collapsed="false"seolevelmigrated="true">Synthesis→IdealGasLaw</h3><ul><li><p>Combinethethreeproportionalities:</p><ul><li><p>.</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Synthesis → Ideal Gas Law</h3><ul><li><p>Combine the three proportionalities:</p><ul><li><p>P \propto \frac{nT}{V} \;\Rightarrow\; PV = nRT</p></li><li><p></p></li><li><p>R=8.31\,\text{J K}^{-1}\text{mol}^{-1}(universalgasconstant).</p></li></ul></li><li><p>Microscopicform</p><ul><li><p>Using(universal gas constant).</p></li></ul></li><li><p>Microscopic form</p><ul><li><p>Usingn=\tfrac{N}{NA}andandk = \tfrac{R}{NA}=1.38\times10^{-23}\,\text{J K}^{-1}(Boltzmann’sconstant):</p></li><li><p>(Boltzmann’s constant):</p></li><li><p>PV = NkT

Units & Conversions Quick-Reference

  • T–Kelvin(K).0°C=273.15K.</p></li><li><p>– Kelvin (K). 0 °C = 273.15 K.</p></li><li><p>V–cubicmetre.– cubic metre.1\,\text{L}=10^{-3}\,\text{m}^3.</p></li><li><p>.</p></li><li><p>P–pascal(Pa)=– pascal (Pa) =\text{N m}^{-2};1atm=; 1 atm =1.013\times10^{5}\,\text{Pa}.</p></li><li><p>.</p></li><li><p>n–mole(mol).– mole (mol).N–purecount(dimensionless).</p></li></ul><h3collapsed="false"seolevelmigrated="true">ClassicSub−LawsDerivedfromPV=nRT</h3><ul><li><p>Boyle(isothermal,– pure count (dimensionless).</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Classic Sub-Laws Derived from PV = nRT</h3><ul><li><p>Boyle (isothermal,T,nfixed):fixed):PiVi=PfVf;curved“isotherms”on; curved “isotherms” onP––Vdiagram.Eachisothermlabelledbyitsconstantdiagram. Each isotherm labelled by its constantT.</p></li><li><p>Charles(isobaric,.</p></li><li><p>Charles (isobaric,P,nfixed):fixed):\dfrac{V}{T}=\text{constant}⇒⇒\frac{Vi}{Ti}=\frac{Vf}{Tf}.</p></li><li><p>(Gay−Lussac/Amontons,isochoric)notexplicitlyinslidesbutconsistent:.</p></li><li><p>(Gay-Lussac / Amontons, isochoric) not explicitly in slides but consistent:\dfrac{P}{T}=\text{constant}forconstantfor constantV,n.</p></li></ul><h3collapsed="false"seolevelmigrated="true">VisualizingIsotherms</h3><ul><li><p>Foragiven.</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Visualizing Isotherms</h3><ul><li><p>For a givenT(e.g.,100Kvs300K)the(e.g., 100 K vs 300 K) theP––Vcurveisahyperbola.</p><ul><li><p>Highercurve is a hyperbola.</p><ul><li><p>HigherT→curveliesfurtherfromorigin(larger→ curve lies further from origin (largerPforsamefor sameV).</p></li><li><p>Anycompression).</p></li><li><p>Any compressionVi\to Vfalongisothermraisespressurealong isotherm raises pressurePi\to Pfinversely.</p></li></ul></li></ul><h3collapsed="false"seolevelmigrated="true">AdditionalConceptQuestions(Slides22–24)</h3><ul><li><p>Q4–Identicalcylinders,sameinversely.</p></li></ul></li></ul><h3 collapsed="false" seolevelmigrated="true">Additional Concept Questions (Slides 22–24)</h3><ul><li><p>Q4 – Identical cylinders, sameT,Ahas3×moleculesofB.Whichhashigher, A has 3× molecules of B. Which has higherP?</p><ul><li><p>?</p><ul><li><p>P\propto n(fixed(fixedV,T) ⇒ Cylinder A pressure is 3× higher.

  • Q5 – Two cylinders, same gas & T;Bhas; B hasVB=2VAandandnB=\tfrac12nA.</p><ul><li><p>Idealgas:.</p><ul><li><p>Ideal gas:P=\frac{nRT}{V}⇒⇒\frac{PB}{PA}=\frac{(\tfrac12 nA)}{2VA}\big/\frac{nA}{VA}=\tfrac{1}{4}⇒⇒PB=\tfrac14PA(OptionC).</p></li></ul></li><li><p>Q6–Sealedwarmedsodabottlemovedtofridge(constant(Option C).</p></li></ul></li><li><p>Q6 – Sealed warmed soda bottle moved to fridge (constantn,V;lower; lowerT).</p><ul><li><p>).</p><ul><li><p>P \propto Tatfixedat fixedV.Coolinglowersinternal. Cooling lowers internalP below atmospheric, external air pushes sides inward ⇒ bottle contracts (Option C).

    • Broader Connections & Relevance

    • Kinetic theory underlies thermodynamic temperature concept; kTisaveragetranslationalkineticenergyscale.</p></li><li><p>Ideal−gasequationbridgeschemistry(moles,molarmasses)andphysics(force,energy,work),enablingcalorimetry,atmosphericmodels,enginecycles.</p></li><li><p>Departurefromidealbehaviorathighdensity/lowis average translational kinetic energy scale.</p></li><li><p>Ideal-gas equation bridges chemistry (moles, molar masses) and physics (force, energy, work), enabling calorimetry, atmospheric models, engine cycles.</p></li><li><p>Departure from ideal behavior at high density / lowT$$ motivates van der Waals corrections—foundation for real-gas studies, critical phenomena, materials science.

    • Ethical/practical lens: Accurate molar-mass statements crucial in pharmaceutical dosing, environmental monitoring (greenhouse gas inventories), and forensic gemology (e.g., verifying Hope diamond provenance).