Presentation on theme: "Chapter 17: Thermal Behavior of Matter"— Presentation transcript:
1 Chapter 17: Thermal Behavior of Matter Equations of stateState variables and equations of stateVariables that describes the state of the material are called statevariables: pressure, volume, temperature, amount of substanceThe volume V of a substance is usually determined by its pressurep, temperature T, and amount of substance, described by the massmIn a few cases the relationship among p, V, T and m (or n) is simpleenough to be expressed by an equation called the equation of stateFor complicated cases, we can use graphs or numerical tables.
2 Equations of state (cont’d) Ideal gasAn ideal gas is a collection of atoms or molecules that moverandomly and exert no long-range forces on each other. Eachparticle of the ideal gas is individually point-like, occupying anegligible volume.Low-density/low-pressure gases behave like ideal gases.Most gases at room temperature and atmospheric pressurecan be approximately treated as ideal gases.
3 Equations of state (cont’d) Definition of a moleOne mole (mol) of any substance is that amount of the substancethat contains as many particles (atoms, or other particles) as thereare atoms in 12 g of the isotope carbon-12 12C. This number is calledAvogadro’s number and is equal to 6.02 x 1023.One atomic mass unit (u) is equal to 1.66x10-24 g.The mass m of an Avogadro’s number of carbon-12 atoms is :The mass per atom for a given element is:1.66x10-24=1/6.02x1023
4 Equations of state (cont’d) Definition of a mole (cont’d)The same number of particles is found in a mole of a substance.Atomic mass of hydrogen 1H is 1 u, and that of carbon 12C is 12 u.12 g of 12C consists of exactly NA atoms of 12C. The molecular massof molecular hydrogen H2 is 2u, and NA molecules are in 2 g of H2 gas.Molar mass of a substanceThe molar mass of a substance is defined as the mass of one moleof that substance, usually expressed in grams per mole.Number of molesThe number of moles of a substances n is:m : mass of the substance
5 Equation of state (cont’d) Ideal gas equation (Equation of state for ideal gas)Boyle’s lawWhen a gas is kept at a constant temperature, its pressure isinversely proportional to its volume.Charles’s lawWhen the pressure of a gas is kept constant, its volume isdirectly proportional to the temperature.Gay-Lussac’s lawWhen the volume of a gas is kept constant, its pressure isdirectly proportional to the temperature.p : pressure, V : volume, T : temperature in KR : universal gas constant 8.31 J/(mole K)L atm/(mol K)1 L (litre) = 103 cm3 = 10-3 m3Ideal gas equation:The volume occupied by 1 mol of an ideal gas at atmosphericpressure and at 0oC is 22.4 L
6 Equations of state (cont’d) The ideal gas equationtotal mass = # of moles times molar mass# of molesgas const.Ideal-gas equation:An ideal gas is one for whichthe above equation holdsprecisely for all pressure andtemperatures.densityconstant for constant mass
7 Equations of state (cont’d) The ideal gas equation (cont’d)The condition called standard temperature and pressure (STP)for a gas is defined to be a temperature of 0 oC= K anda pressure of 1 atm = x 105 Pa.Example 18.1If you want to keep a mole of an ideal gas in your room at STP,how big a container do you need?
8 Equations of state (cont’d) The van der Waals equationThe ideal gas equation can be derived from a simple molecularmodel that ignores:the volumes of the molecules themselvesthe attractive forces between themThe van der Waals equation makes corrections for these short-comings.reduces volume of gas due tofinite size of moleculesThe attractive intermolecular forces reduces the pressure by gasA constant that depends on the attractive intermolecular forces.A constant that represents the size of a gas molecule
9 Equations of state (cont’d) pV diagram and phasesNon-ideal gasses behave differentlybecause the attractive forces betweenmolecules become comparable to orgreater than the kinetic energy of motionideal gasbehaviorpV=constThe molecules are pulled closer togetherand the volume V is less than for an idealgasconst.temp.c is called the critical pointFor T<Tc a gas will liquify underincreased pressureFor T>Tc a gas will not liquify underpressure
10 Molecular properties of matter Molecules and phases of matterAll familiar matter is made up of molecules. The smallest size ofa molecule is about of order of m. Larger molecules may havea size of 10-6 m.The force between molecules in a gas varies with distance r betweenmolecules. The major source of the force in this case is electro-magnetic interaction between them.A molecule that has energy E>E0 (see below) can move around asdepicted in the figure.A molecule with E>0 (see below) can escapeas in gaseous phase of matter.U(r)rIn solids molecules vibrate around fixed pointmore or less.rE<0E0A molecule in liquid has slightly higher energythan in solid.
11 Kinetic-molecular model of an ideal gas AssumptionsMolecules are featureless points, occupy negligible volume.Total number of molecules (N) is very large.Molecules follow Newton’s laws of motion.Molecules move independently making elastic collisions.No potential energy of interaction (no bonding).
12 Kinetic-molecular model of an ideal gas Molecule of mass m moving with speed vx in negative-x direction in a cube of side l collides elastically with the wall.-vxvxvxElastic collision:y-component ofvelocity does notchange before andafter the collision.Impulse:Time between collisionswith the same wall:Force exerted by molecule on the wall:Force exerted by N molecules with different speeds:
13 Kinetic-molecular model of an ideal gas (cont’d) Mean square velocityThe mean-square velocity is defined as follows:The total force exerted on the wall by N particles:
14 Kinetic-molecular model of an ideal gas (cont’d) Velocities in random directionsVelocities in random directions:(true for any vector)If the velocities have random directions:The total force on the wall:
15 Kinetic-molecular model of an ideal gas (cont’d) Ideal gas law and mean velocityPressure on the wall due to molecular impactIdeal Gas LawkB = Boltzmann’s ConstantMean translational kinetic energy is proportional to T
16 Kinetic-molecular model of an ideal gas (cont’d) The mean kinetic energy and ideal gasCompare with ideal gas equation:average translationalkinetic energykBAvogadro’snumberIdeal Gas LawkB = Boltzmann’s ConstantMean translational kinetic energy is proportional to T
17 Kinetic-molecular model of an ideal gas (cont’d) RMS speedMolecular speed (root-mean-square speed)At a given temperature, gas molecules of different mass m have theSAME average kinetic energy but DIFFERENT rms speeds.
18 Kinetic-molecular model of an ideal gas (cont’d) Collision between moleculesConsider N spherical moleculeswith radius r, and suppose onlyone molecule is moving.rr2rvrWhen it collides with another moleculethe distance between centers is 2r.rThe moving molecule collides with anyother molecule whose center inside acylinder of radius 2r.vdtIn a short time dt a molecule with speed v travels a distance vdt,during which time it collides with any molecule that is in the cylindricalvolume of radius 2r and length vdt.The volume of the cylinder is 4pr2vdt.There are N/V molecules per unit volume.
19 Kinetic-molecular model of an ideal gas (cont’d) Collisions between molecules (cont’d)The number of molecules dN with centersin the cylinder:rThe number of collisions per unit time:r2rvrrIf there are more than one molecule moving,it can be shown that:vdtThe average time between collision (the mean free time):The average distance traveled between collision (the mean free path):
20 Heat capacities Heat capacities of ideal gases (~ monatomic gasses) Change of translational kinetic energy due to change in temperature:Heat input needed for a change in temperature:heat capacity (specific heat) at const.volumeFrom dK=dQ,For an ideal gas
21 Heat capacities (cont’d) Theorem of equipartition of energyEach quadratic term in the expression of the average total energyof a particle in thermal equilibrium with its surrounding contributeson the average (1/2)kT to the total energy.OREach degree of freedom contributes an average energy of(1/2)kT.Translational kinetic energy comprises 3 terms (degree of freedom):Rotational energy of diatomic molecule has 2 degree of freedom:
22 Heat capacities (cont’d) Heat capacities of gases in general
23 Heat capacities (cont’d) Heat capacities of gases in general (cont’d)However, which degree of freedom is available depends on thetemperature. Furthermore, in case of diatomic molecules, forexample, two more degrees of freedom are possible from twopossible modes of vibration.diatomic molecule
24 Heat capacities (cont’d) Heat capacities of solidsConsider a crystalline solidconsisting of N identical atoms(monatomic solid).Each atom is bound to an equi-librium position by interatomic forces.Each atom has three degrees offreedom, corresponding to its threecomponents of velocity.In addition each atom acts as 3Dharmonic oscillator because of thepotential created by interatomicforces – three more degrees of freedom.6 degrees of freedom
25 Phase Diagram for Water Phases of matterPhase equilibriumA transition from one phase to another ordinarily takes placeunder conditions of phase equilibrium between two phases, andfor a given pressure this occurs at only one specific temperature.Phase diagramPhase Diagram for WaterAll three phases exist inequilibrium at the triplepoint. ( K and 610Pa for water)
26 Phases of matter (cont’d) Phase diagram (cont’d)Water has an Unusual PropertyMost substances contract when transforming from a liquid to a solid (e.g. carbon dioxide).Water is unusual in that it expands upon freezing (solid-liquid interface curve has a negative slope).Ice floats on liquid water.waterCarbon dioxide
27 Phases of matter (cont’d) Phase diagram (cont’d)
28 ExercisesProblem 1A hot-air balloon stays aloft because hot air at atmospheric pressureis less dense than cooler air at the same pressure. If the volume of theballoon is 500 m3 and the surrounding air is at 15.0oC, what must thetemperature of the air in the balloon be for it to lift a total load 290 kg(in addition to the mass of the hot air)? The density of air at 15.0oC andatmospheric pressure is 1.23 kg/m3.SolutionThe density of the hot air must be where is the densityof the ambient air and m is the load. The density is inversely proportionalto the temperature, so
29 ExercisesProblem 2The vapor pressure is the pressure of the vapor phase of a substancewhen it is in equilibrium with the solid or liquid phase of the substance.The relative humidity is the partial pressure of water vapor in the airdivided by the vapor pressure of water at that same temperature,expressed as a percentage. The air is saturated when the humidity is100%. a) The vapor pressure of water at 20.0oC is 2.34 x 103 Pa. If theair temperature is 20.0oC and the relative humidity is 60%, what is thepartial pressure of water vapor in the atmosphere ( the pressure due towater vapor alone)? b) Under the conditions of part (a), what is the massof water in 1.00 m3 of air? (The molar mass of water is 18g/mol.)Solution(a)(b)
30 ExercisesProblem 3Modern vacuum pumps make it easy to attain pressures of order atmin the laboratory. At a pressure of 9.00 x atm and an ordinarytemperature (say T=300 K), how many molecules are present in a volumeof 1.00 cm3?Solution
31 Exercises Problem 4 Solution A balloon whose volume is 750 m3 is to be filled with hydrogen atatmospheric pressure (1.01 x 105 Pa). a) If the hydrogen is storedin cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20 x106 Pa, how many cylinders are required? b) What is the total weight(in addition to the weight of gas) that can be supported by the balloonif the gas in the balloon and the surrounding air are both at 15.0oC?The molar mass of hydrogen (H2) is 2.02 g/mol. The density of air at15.0oC and atmospheric pressure is 1.23 kg/m3.SolutionThe absolute pressure of the gas in a cylinder is:At atmospheric pressure, the volume of hydrogen will increase by afactor of so the number of cylinders is:(b) The difference between the weight of the air displaced and the weightof hydrogen is: