Turning up the heat: thermal expansion

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Clement Hopkins
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1 Lecture 3 Turning up the heat: Kinetic molecular theory & thermal expansion

2 Gas in an oven: at the hot of materials 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 speed, slowed down here two trillion fold from room temperature.

3 Kinetic Molecular Theory The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. Experimental evidence for kinetic theory is generally perceived as the fist demonstration of the existence of atoms and molecules

4 Kinetic Molecular Theory Main idea: Derive the pressure of a gas on the container walls, using Newtonian mechanics. Then, compare the expression with the ideal gas law. Change in Momentum of a Molecule l p p = 2mv x p = change in momentum, m = mass of the molecule, v x = velocity in the x directioni

5 Rate of change of momentum: F p 2mv x mv 2 x t (2a/ v x ) a F = force exerted by the molecule, Δp = change in momentum, Δt = change in time, m = mass of the molecule, v x = velocity in the x direction, a = side length of cubic container Total pressure exerted by N molecules: l P mnv 2 x V P = total pressure, m = mass of the molecule, 2 v x = mean square velocity along x, V = volume of the cubic container

6 Relating Gas Pressure to Energy P mnv 2 x V Mean square velocity: v x 2 v y 2 v z 2 Mean square velocities in the x, y, and z directions are the same Total mean square velocity for a molecule: v 2 v 2 v 2 v 2 3v 2 x y z x

7 Gas Pressure in the Kinetic Theory P 2 Nmv 1 = = v 3V 3 2 P = gas pressure, N = number of molecules, m = mass of the gas molecule, v = velocity, V = volume, = density. Compare with the ideal gas law: PV = (N/N A )RT N = number of molecules, l R = gas constant, T = temperature, P = gas pressure, V = volume, N A = Avogadro s number

8 Mean Kinetic Energy for a Molecule KE = 1 mv2 = kt k = Boltzmann constant, T = temperature 1. The temperature of an ideal monatomic gas is a 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 increase. 3. The rise in internal energy per unit temperature is the heat capacity

9 Internal Energy per Mole for a Monatomic Gas 1 U = N A 2 mv2 = 3 N kt A 2 2 U = total internal energy per mole, N A = Avogadro s number, m = mass of the gas molecule, k = Boltzmann constant, T = temperature Molar Heat Capacity at Constant t Volume C m = du = 3 N Ak = 3 dt 2 2 R C m = heat capacity per mole at constant volume (J K -1 mole -1 ), U = total t internal energy per mole, R = gas constant t

10 Maxwell s theorem: Equipartition of Energy Translation Rotation Possible translational and rotational motions of a diatomic molecule. Vibrational motions are neglected.

11 Thermal Expansion The potential energy PE curve has a minimum when the atoms in the solid attain the interatomic separation r = r 0. Due to thermal energy, the atoms will be vibrating and will have vibrational kinetic energy. At T = T 1, the atoms will be vibrating in 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 B and C. This average separation will be at A and greater than r 0.

12 Thermal Expansion Vibrations of atoms in the solid. We consider, for simplicity a pair of atom. Total energy E = PE + KE and this is constant for a pair of vibrating atoms executing simple harmonic Motion. At B and CKEis zero (atoms are stationary and about to reverse direction of oscillation) and PE is maximum.

13 Definition of Thermal Expansion Coefficient L 1 L o T = thermal coefficient of linear expansion or thermal expansion coefficient, L o = original length, L = length at temperature T Thermal Expansion L L o [1 (T T o )] L = length at temperature t T, L o = length at temperature t T o

14 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, SiO 2 ; alumina, Al 2 O 3.

15 Example: Expansion of a Si chip Assume we have a 1mm long Si chip. How much will it expand upon heating to 320 o C? 1 mm 1 μm

16 Negative coefficient of Thermal Expansion Some materials contract with increasing temperature. Why? Quartz Zirconium Tungstate Water

17 Molecular velocity & energy distribution Schematic diagram of a Stern type experiment for determining the distribution of molecular velocities

18 Maxwell-Boltzmann Distribution for Molecular Speeds 3/2 2 n 4 N m v 2 exp mv v 2 kt 2kT n v = velocity density function, N = total number of molecules, m = molecular mass, k = Boltzmann constant, T = temperature, v = velocity

19 Maxwell-Boltzmann Distribution for Molecular Speeds

20 Maxwell-Boltzmann Distribution for Translational Kinetic Energies = 2 n 1 E N kt 3 / 2 1/ 2 E exp E kt n E = number of atoms per unit volume per unit energy at an energy E,, N = total number of molecules per unit volume, k = Boltzmann constant, T = temperature.

The number of molecules that have energies

21 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 E A is the shaded area. This area depends strongly on the temperature as exp(-e A /kt)

22 Boltzmann Energy Distribution n E N C exp E kt n E = 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 = temperature

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

24 Each atom in a solid: like a mass on a spring

25 Root Mean Square Fluctuations of a Body Attached to a Spring of Stiffness K x rms kt K K = spring constant, T = temperature, ( x) rms = rms value of K spring constant, T temperature, ( x) rms rms value of the fluctuations of the mass about its equilibrium position.

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

27 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.

28 Root Mean Square Noise Voltage Across a Resistance v 4kTRB rms R = resistance, B = bandwidth of the electrical system in which y noise is being measured, v rms = root mean square noise voltage, k = Boltzmann constant, T = temperature

Atoms, electrons and Solids

Atoms, electrons and Solids Shell model of an atom negative electron orbiting a positive nucleus QM tells that to minimize total energy the electrons fill up shells. Each orbit in a shell has a specific