The properties of gasesThe state of a gas depends on: P, V, T, nPressure: P = F/A SI units = N/m2= Pa (Pascals)Other commonly used units are:1 atm = 101.3 kPa = 1.013 bar = 14.696 psi = 760.0 Torrevacuated tubehliquidPatmSchematic diagram of a barometerExercise: Calculate the atmospheric pressure in kPa measured by awater barometer having a column height of 410.25 in.hPatmFPgasMeasurement of gas pressure with a U tubeP = ρgh + PatmThis week’s lab experimentLF = mgπ/2θβαFllFFll= F sin αθ + β + π/2 = πθ = π/2 − ββ + α = π/2β = π/2 − αθ = π/2 − π/2 + α = αhenceFll= F sin θand soP = L sin θairHgBoyles law: PV = constantDependence of volume on pressureT (o K)0 100 200 300 400T (oC)-250 -200 -150 -100 -50 0 50 100 150V (L)05101520253035V = aT(oK)Charles law: V/T = constantThe absolute temperature scale:oK= oC + 273.15Dependence of volume on temperatureV = aT + bIdeal gas lawVn,T= (constant) /PVn,P= (constant) TVP,T= (constant) nPV = nRTR = ideal gas constantR = 0.08206 L atm/mole-K = 8.314 J/mole-KTherefore V = (constant) nT/PorPVm=V/nTThe PVT surfaceEvery point on this surfacerepresents a set of P,Vm, andT values that satisfies theideal gas law.vacuumvacuumReality checkvacuumvacuumvacuumExercises:1. A mass of O2occupies 5.00 L at a pressure of 740 torr. Determine the volume of the same mass of gas at a pressure of 760 torr.2. A steel tank contains CO2at 27oC and 12.0 atm. Calculate the internalpressure when the tank is heated to 100oC.3. To what pressure must 1.0 L of gas measured at 1.0 atm and -20oC besubjected to be compressed to a volume of 0.50 L when the temperatureis 40o C?Alternate forms of the ideal gas law4. Calculate the density of acetylene (C2H2) in a gas cylinder at a pressureOf 2000 psi and a temperature of 25oC.molarity: M = n/V so P = MRTmolecular weight: n = m/W so P = mRT/(WV)density: ρ = m/V = nW/Vn/V = ρ/W so P = ρRT/W1. Two containers are connected by a valve. One container holds 5 L of gas at 9 atm and the other container holds 10 L of gas at 6 atm. What is the total pressure in the two containers after the valve is opened?Dalton’s law of partial pressures: In a mixture of gases, each gas has the same pressure it would have if it were the only gas In the container.2. A very thin 200 mL glass bulb filled with gas at a pressure of 900 torr isplaced inside a 1.00 L flask which is then filled with 60 torr of gas and sealed. Next, the thin bulb is broken by shaking the flask. What is the final pressure in the flask? You may neglect the volume of the glassfragments.3. The reaction PCl5(g)→ PCl3(g)+ Cl2(g)is carried out by placing 2.69 gof solid PCl5in a 1.00 L flask. The flask is then evacuated and sealed. Next, the PCl5is completely vaporized at 250oC and allowed to reach equilibrium. If the final pressure in the flask is 1.00 atm, what fraction of the PCl5has decomposed?Kinetic theory of gasesAssumptions:1. Gas molecules are extremely small compared to the distances between them and can be treated as volumeless hard spheres.2. Gas molecules are in a state of constant random motion. Their directions of motion are changed only by collisions with other gas molecules and with the walls of the container.3. Gas molecules exert no forces on one another. Their collisions are perfectly elastic (i.e., the total kinetic energy of the colliding moleculesis conserved).A collision with a wallVxiVxiVyiVyi∆pxi= (pxi)f–(pxi)i= – mvxi–mvxi= –2mvxiProcedure for deriving the pressure:1. Calculate the force fxiimparted to the yz wall by molecules having vxi.darea of yz wall = Ani/2vxid = vxiΔtα = fraction of molecules that hit wall in time Δt = Va/Vα = vxiΔtA/VVaarea of yz wall = Ani/2vxiVafxi= Δpxi/Δt = (A/V)nimvxi22. Calculate the total force imparted to the yz wall by adding up the forces due to molecules having all values of vx.molecule)(perkT23ε =mole)(perRT23E =final resultsTemperature is a measure of average kinetic energy2. Calculate the total force imparted to the yz wall by adding up the forces due to molecules having all values of vx.Exercises:1. Calculate the rms speed of an O2molecule at room temperature (25oC).2. Calculate the average speed and the rms speed from the following data:Number of moleculesSpeed (m/s)4 4822 3553 5231 299nivinivi21928 929296719 2520501569 820587299 894014506 2091334sm/6.450104506vnN1viiiave=∑==sm/3.457102091334vnN1v2/12/1i2iirms=⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛∑=before afterGaseous diffusion2/1HClNHNHHCl2/1NHNH2/1HClHClNHHCNHHClMTMTMTMTvv333333⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=><><=lRREffusion3. Compare the rates of effusion of equal molar quantities of H2and O2present in the same container.Distribution of molecular speedsdvevkT2m4F(v)dv/2kT2mv23/2−⎟⎠⎞⎜⎝⎛=ππ∫==∞01 F(v)dv Fdvv4f(v)dv2π∝kT/2-mveP(v)2∝dv)vC(4vCP(v)f(v)dF(v)dvkT/2-mv2 2eπ==velocity diagramWhat fraction of molecules have speeds between v and v + dv?Comparison of speed distributionsdependence on massdependence on temperature/2kT2mv23/2evkT2m4F(v)−⎟⎠⎞⎜⎝⎛=ππRms speed:Most probable speed:Average speed:Quantities obtained using the speed distribution functionAll depend on (T/M)1/2v (m/s)0 250 500 750 1000 1250 1500 1750 2000F(v) (m/s)-10.00000.00020.00040.00060.00080.00100.00120.00140.0016vPvavevrmsN2T = 500 KDeviations from ideal behaviorAssumptions of the kinetic theory that cause problems:Gas molecules are extremely small compared to the distances between them and can be treated as volumeless hard spheres.Gas molecules exert no forces on one another. Their collisions are perfectly elastic (i.e., the total kinetic energy of the colliding moleculesis conserved).Not true at high pressures where molecules are forcedto be very close together.Not true: weak intermolecular forces, called van der Waals forces, cause molecules to attract each other at intermediate distances and repel each other when they are very close together. They become important at high pressures and low temperatures.Potential energy due to intermolecular forcesattractiverepulsiveR (Angstrom)0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Energy (eV)0123456HClMolecular potential energy curveCompressibility factorRTPVZm=repulsive forcesattractive forcesExercise: Calculate the pressure of 5 moles of CO2in a 2.46 L containerat 300 K using the ideal gas law and the van der Waals equation. (The van der Waals constants for CO2are a = 3.640 L2-atm/mol2and
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