Lecture 3Lecture 3

Turning up the heat: Kinetic molecular theory & et c o ecu a t eo y &

thermal expansion

Gas in an oven: at the “hot” of materials i science

Here, the size of helium atoms relative to their spacing is shown to scale under 1950 atm of pressure. The atoms have a certain, average scale under 1950 atm of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.

Kinetic Molecular Theory

• The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. g gy

• Experimental evidence for kinetic theory is generally perceived as the fist demonstration of the existence of atoms and molecules

Kinetic Molecular Theory

Main idea:

D i th f th t i ll i Derive the pressure of a gas on the container walls, using Newtonian mechanics. Then, compare the expression with the

ideal gas law.

Ch i M t f M l lChange in Momentum of a Molecule

p = 2mvxp x

p = change in momentum, m = mass of the molecule, vx = l i i h di ivelocity in the x direction

Rate of change of momentum:

F pt

2mvx

(2a / v )

mvx2

at (2a / vx ) aF = force exerted by the molecule, Δp = change in momentum, Δt = change in time m = mass of the molecule v = velocity in Δt = change in time, m = mass of the molecule, vx = velocity in

the x direction, a = side length of cubic container

T t l t d b N l l

PmNvx

2

Total pressure exerted by N molecules:

P x

VP = total pressure, m = mass of the molecule, p

= mean square velocity along x, V = volume of the cubic container

2xv

Relating Gas Pressure to Energy

P mNvx

2

V

Mean square velocity:

V

vx2 vy

2 vz2

Mean square velocities in the x, y, and z directions are the same

2 2 2 2 3 2

Total mean square velocity for a molecule:

v2 vx2 vy

2 vz2 3vx

2

Gas Pressure in the Kinetic Theory

22

1== vvNmP

P = gas pressure N = number of molecules m = mass of the

33v

V

P = gas pressure, N = number of molecules, m = mass of the gas molecule, v = velocity, V = volume, = density.

Compare with the ideal gas law:

N b f l l R T

PV = (N/NA)RT

N = number of molecules, R = gas constant, T = temperature, P = gas pressure, V = volume, NA = Avogadro’s number

Mean Kinetic Energy for a Molecule

KE =12

mv2 = 32

kT

k = Boltzmann constant, T = temperature2 2

1. The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms measure of the average kinetic energy of its atoms.

2. When heat is added to a gas, it’s internal energy and therefore it’s temperature will increasetherefore it s temperature will increase.

3. The rise in internal energy per unit temperature is the heat capacityheat capacity

Internal Energy per Mole for a Monatomic Gas

U = NA12

mv2

=

32

NAkT

U = total internal energy per mole, NA = Avogadro’s number, m = mass of the gas molecule k = Boltzmann constant T =

2 2

M l H t C it t C t t V l

m = mass of the gas molecule, k = Boltzmann constant, T = temperature

Molar Heat Capacity at Constant Volume

Cm = dUdT

= 32

NAk = 32

RdT 2 2

Cm = heat capacity per mole at constant volume (J K-1 mole-1), U t t l i t l l R t tU = total internal energy per mole, R = gas constant

Maxwell’s theorem: Equipartition of Energy

Translation Rotation

Possible translational and rotational motions of a diatomic molecule. Vibrational motions are neglected.

Thermal Expansion

The potential energy PE curve has a minimum when the atoms in the solid attain the interatomic separation r = r0. Due to thermal energy, the atoms will be vibrating

and will have vibrational kinetic energy. At T = T1, the atoms will be vibrating in such a way that the bond will be stretched and compressed by an amount such a way that the bond will be stretched and compressed by an amount

corresponding to the KE of the atoms. A pair of atoms will be vibrating between Band C. This average separation will be at A and greater than r0.

Thermal Expansion

Vibrations of atoms in the solid. We consider, for simplicity a pair of atoms. Total d h f f b lenergy E = PE + KE and this is constant for a pair of vibrating atoms executing simple

harmonic Motion. At B and C KE is zero (atoms are stationary and about to reverse direction of oscillation) and PE is maximum.

Definition of Thermal Expansion Coefficient

L 1

= thermal coefficient of linear expansion or thermal expansion

TLo p p

coefficient, Lo = original length, L = length at temperature T

Thermal Expansion

L Lo[1 (T To )]L l th t t t T L l th t t t TL = length at temperature T, Lo = length at temperature To

Dependence of the linear thermal expansion coefficient (K-1) on temperature T p p ( ) p(K) on a log-log plot. HDPE, high density polyethylene; PMMA,

Polymethylmethacrylate (acrylic); PC, polycarbonate; PET, polyethylene terepthalate (polyester); fused silica, SiO2; alumina, Al2O3.

Example: Expansion of a Si chip

Assume we have a 1mm long Si chip. How much will it expand upon heating to 320oC?expand upon heating to 320 C?

1 1 1 μm1 mm

Negative coefficient of Thermal Expansion

Some materials contract with increasing temperature. Why?

Quartz Zirconium Tungstate

Water

Molecular velocity & energy distribution

Schematic diagram of a “Stern type experiment” for determining the distribution of molecular velocities

Maxwell-Boltzmann Distribution for Molecular Speeds

Maxwell-Boltzmann Distribution for Molecular Speeds

m

3 / 2 mv2

nv 4N

m2kT

v2 exp

mv2kT

nv = velocity density function, N = total number of molecules, m = molecular mass, k = Boltzmann constant, T = temperature,

v = velocity

Maxwell-Boltzmann Distribution for Translational Kinetic Energies

n =2

N1

3 / 2

E1/ 2 exp E

nE

NkT E exp

kT

nE = number of atoms per unit volume per unit energy at an energy E, N = total number of molecules per unit volume, k = gy , p ,

Boltzmann constant, T = temperature.

Maxwell-Boltzmann Distribution for Translational Kinetic Energies

Energy distribution of gas molecules at two different temperatures. The number of molecules that have energies greater than EA is the shaded area. This

area depends strongly on the temperature as exp(-EA/kT)

Boltzmann Energy Distribution

nE C exp E

k

N

pkT

nE = number of atoms per unit volume per unit energy at an energy E, N = total number of atoms per unit volume in the system, C = a constant that depends on the specific system

(weak energy dependence), k = Boltzmann constant, T = temperaturetemperature

Thermal Fluctuations

Solid in equilibrium in air. During collisions between the gas and solid atoms, kinetic energy is exchanged.

Electrical Noise

Random motion of conduction electrons in a conductor results in electrical noise.

Electrical Noise

Ch i d di h i f i b d d h d Charging and discharging of a capacitor by a conductor due to the random thermal motions of the conduction electrons.

Root Mean Square Noise Voltage Across a R i tResistance

kTRBv 4rms R = resistance, B = bandwidth of the electrical system in which y

noise is being measured, vrms = root mean square noise voltage, k = Boltzmann constant, T = temperature