chapter 17: thermal behavior of matter equations of state state variables and equations of state...

Chapter 17: Thermal Behavior of Matter

Equations of state State variables and equations of state

• Variables that describes the state of the material are called state variables: pressure, volume, temperature, amount of substance• The volume V of a substance is usually determined by its pressure p, temperature T, and amount of substance, described by the mass m• In a few cases the relationship among p, V, T and m (or n) is simple enough to be expressed by an equation called the equation of state• For complicated cases, we can use graphs or numerical tables.

Ideal gas • An ideal gas is a collection of atoms or molecules that move randomly and exert no long-range forces on each other. Each particle of the ideal gas is individually point-like, occupying a negligible volume.

• Low-density/low-pressure gases behave like ideal gases.

• Most gases at room temperature and atmospheric pressure can be approximately treated as ideal gases.

Equations of state (cont’d)

Definition of a mole • One mole (mol) of any substance is that amount of the substance that contains as many particles (atoms, or other particles) as there are atoms in 12 g of the isotope carbon-12 12C. This number is called Avogadro’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 :

g 0.12u

g 1066.1)u 12(1002.6)u 12(

2423

ANm

• The mass per atom for a given element is:

Aatom N

mmassmolar

g/atom1064.6atoms/mol 1002.6

g/mol 00.4 2423

Hem

1.66x10-24=1/6.02x1023


• 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 mass of molecular hydrogen H2 is 2u, and NA molecules are in 2 g of H2 gas.

Molar mass of a substance • The molar mass of a substance is defined as the mass of one mole of that substance, usually expressed in grams per mole.

Number of moles

• The number of moles of a substances n is:

massmolar

mn m : mass of the substance

Equations of state (cont’d) Definition of a mole (cont’d)

Equation of state (cont’d)

Ideal gas equation (Equation of state for ideal gas) • Boyle’s law

When a gas is kept at a constant temperature, its pressure isinversely proportional to its volume.

• Charles’s law

When 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.

Ideal gas equation:

nRTpV

p : pressure, V : volume, T : temperature in KR : universal gas constant 8.31 J/(mole K) 0.0821 L atm/(mol K) 1 L (litre) = 103 cm3 = 10-3 m3

The volume occupied by 1 mol of an ideal gas at atmosphericpressure and at 0oC is 22.4 L


The ideal gas equation

nMmtot

RTM

mnRTpV tot K)J/(mol)15(314472.8; R

RT

pM

nRT

pV

total mass = # of moles times molar mass

Ideal-gas equation: gas const.

density

constant for constant mass

An ideal gas is one for whichthe above equation holds precisely for all pressure andtemperatures.

# of moles


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=273.15 K anda pressure of 1 atm = 1.013 x 105 Pa.

Example 18.1

If you want to keep a mole of an ideal gas in your room at STP,how big a container do you need?

.4.22m0224.0Pa 10013.1

)K 15.273(K)mol/(J 314.8)(mol 1( 35

p

nRTV


The van der Waals equation

The ideal gas equation can be derived from a simple molecularmodel that ignores:• the volumes of the molecules themselves• the attractive forces between them

The van der Waals equation makes corrections for these short-comings.

nRTnbVV

anp ))((

2

2

:a A constant that depends on the attractive intermolecular forces.

The attractive intermolecular forces reduces the pressure by gas

reduces volume of gas due tofinite size of molecules

:b A constant that represents the size of a gas molecule


pV diagram and phases

DCBA TTTT

• Non-ideal gasses behave differently because the attractive forces between molecules become comparable to or greater than the kinetic energy of motion

• The molecules are pulled closer together and the volume V is less than for an ideal gas

ideal gasbehaviorpV=const

• c is called the critical point• For T<Tc a gas will liquify under increased pressure• For T>Tc a gas will not liquify under pressure

const.temp.

Molecular properties of matter

U(r)

r

Molecules and phases of matter• All familiar matter is made up of molecules. The smallest size of a molecule is about of order of 10-10 m. Larger molecules may have a size of 10-6 m.• The force between molecules in a gas varies with distance r between molecules. The major source of the force in this case is electro- magnetic interaction between them.

rE<0

• A molecule that has energy E>E0 (see below) can move around as depicted in the figure.

E0

• A molecule with E>0 (see below) can escape as in gaseous phase of matter.

• In solids molecules vibrate around fixed point more or less.

• A molecule in liquid has slightly higher energy than in solid.

Kinetic-molecular model of an ideal gas

Assumptions

Molecules 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).

Kinetic-molecular model

Molecule of mass m moving with speed vx in negative-x direction in a cube of side l collides elastically with the wall.vx

-vx

vx

xxxif mvmvmvppp 2)( Impulse:

Time between collisionswith the same wall: xv

lt

2

Force exerted by molecule on the wall:

molecule)(per /2

2 2

l

mv

vl

mv

t

pF x

x

x

Force exerted by N molecules with different speeds:

)( 2222

321 Nxxxx vvvvl

mF

xx vv 1

xx vv 2

yy vv 1

yy vv 2

Elastic collision:y-component ofvelocity does notchange before andafter the collision.

Kinetic-molecular model of an ideal gas

Mean square velocity

The mean-square velocity is defined as follows:

N

vvvv Nxxx

x

2222 21

The total force exerted on the wall by N particles:

axis)- xalong (for 2 xvNl

mF

Kinetic-molecular model of an ideal gas (cont’d)

Velocities in random directions

Velocities in random directions:

(true for any vector)

If the velocities have random directions:

The total force on the wall:

2222zyx vvvv

22222

222

3 xxxx

zyx

vvvvv

vvv

3

2

vN

l

mF


Ideal gas law and mean velocityPressure on the wall due to molecular impact

22

32 3

1

3vm

V

NvN

l

m

l

F

A

Fp

3 since

2

vN

l

mF

nRTTN

RNTNkvmNvNmpV

AB )

2

1(

3

2

3

1 22

Tkvm B2

3

2

1 2

Mean translational kinetic energy is proportional to T

Ideal Gas LawkB = Boltzmann’s Constant


The mean kinetic energy and ideal gas

trB KTNkvmNvNmpV3

2)

2

1(

3

2

3

1 22

TN

RTkvm

AB )(

2

3

2

3

2

1 2

Mean translational kinetic energy is proportional to T

Ideal Gas LawkB = Boltzmann’s Constant

average translationalkinetic energy

Compare with ideal gas equation:

nRTpV

TN

R

N

nRTvm

N

KnRTK

A

trtr )(

2

3

2

3

2

1;

2

3 2

AA NNnnNN /1/

kB

:AN Avogadro’snumber

molmolecules /

10022.6 23


RMS speed

Molecular speed (root-mean-square speed)

M

RT

m

Tkvv B

rms

332

RTTkNvMvmN BAA 2

3

2

3

2

1

2

1 22

At a given temperature, gas molecules of different mass m have the SAME average kinetic energy but DIFFERENT rms speeds.


r

Collision between molecules

r

r

r

v 2r

vdt

• In 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 cylindrical volume of radius 2r and length vdt.

• Consider N spherical molecules with radius r, and suppose only one molecule is moving.

• When it collides with another molecule the distance between centers is 2r.

• The moving molecule collides with any other molecule whose center inside a cylinder of radius 2r.

• The volume of the cylinder is 4r2vdt.• There are N/V molecules per unit volume.


r

Collisions between molecules (cont’d)

r

r

r

v 2r

vdt

• The number of molecules dN with centers in the cylinder:

VvdtNrdN /4 2• The number of collisions per unit time:

V

vNr

dt

dN 24

• If there are more than one molecule moving, it can be shown that:

V

vNr

dt

dN 224

• The average time between collision (the mean free time):

vNr

Vtmean 224

• The average distance traveled between collision (the mean free path):

)(2424 22

TNkpVpr

Tk

Nr

Vvt B

Bmean


Heat capacities of ideal gasesChange of translational kinetic energy due to change in temperature:

Heat capacities

nRdTdKnRTK trtr 2

3

2

3

Heat input needed for a change in temperature:

:; VV CdTnCdQ heat capacity (specific heat) at const.volume

From dK=dQ,

K)J/(mol47.122

3

2

3 RCnRdTdTnC VV For an ideal gas

(~ monatomic gasses)

Theorem of equipartition of energy

Heat capacities (cont’d)

Each quadratic term in the expression of the average total energyof a particle in thermal equilibrium with its surrounding contributes on the average (1/2)kT to the total energy.

OR

Each degree of freedom contributes an average energy of(1/2)kT.

• Translational kinetic energy comprises 3 terms (degree of freedom):

)2

1(3

2

1,

2

1,

2

1 222 kTvmvmvm zyx

• Rotational energy of diatomic molecule has 2 degree of freedom:

)2

1(2

2

1,

2

1 222

211 kTII

Heat capacities of gases in general


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

Heat capacities of solids


• Consider a crystalline solid consisting of N identical atoms (monatomic solid).• Each atom is bound to an equilibrium position by interatomic forces.• Each atom has three degrees of freedom, corresponding to its three components of velocity.• In addition each atom acts as 3D harmonic oscillator because of the potential created by interatomic forces – three more degrees of freedom.

6 degrees of freedom

nRTTNkE Btot 33

)/(9.243 KmolJRCV

Phase equilibrium

Phases of matter

A 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 diagram

Phase Diagram for Water

All three phases exist inequilibrium at the triplepoint. (273.16 K and 610Pa for water)

Phases of matter (cont’d) Phase diagram (cont’d)

Water has an Unusual Property

water

Carbon dioxide

• Most 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.

Phases of matter (cont’d) Phase diagram (cont’d)

A 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.

The 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

Exercises

' Vm /'

Problem 1

Solution

.272545))500)(/23.1(

)290(1)(15.288(

)1()/('

'

133

1

CKmmkg

kgK

V

mT

VmTT

The 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.)

Exercises

PaPa 33 1040.1)1034.2)(60.0(

gKKmolJ

mPamolkg

RT

MpVm 10

)15.293))(/(3145.8(

)00.1)(1040.1)(/100.18( 333

Problem 2

Solution

(a)

(b)

ExercisesProblem 3

Solution

Modern vacuum pumps make it easy to attain pressures of order 10-13 atmin the laboratory. At a pressure of 9.00 x 10-14 atm and an ordinarytemperature (say T=300 K), how many molecules are present in a volumeof 1.00 cm3?

molecules

molmoleculesKKmolJ

mPaN

RT

pVnNN AA

6

23369

1020.2

)/10023.6()300))(/(3145.8(

)1000.1)(10119.9(

ExercisesProblem 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.

(a) The absolute pressure of the gas in a cylinder is:

At atmospheric pressure, the volume of hydrogen will increase by a factor of so the number of cylinders is:

.1030.1)10013.11020.1( 656 PaPa

56 1001.1/1030.1

.31)]1001.1/1030.1)(90.1/[(750 5633 mm(b) The difference between the weight of the air displaced and the weight of hydrogen is:

.1042.8

)750)(/80.9]()15.288))(/(3145.8(

)/1002.2)(1001.1()/23.1[(

)()(

3

3235

3

2

2

N

msmKKmolJ

molkgPamkg

VgRT

pMVg H

airHair

chapter 17: thermal behavior of matter equations of state state variables and equations of state...

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