Lecture 3 Turning up the heat: Kinetic molecular theory & thermal expansion
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.
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
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
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
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
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
Mean Kinetic Energy for a Molecule KE = 1 mv2 = 3 2 2 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
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
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 = 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.
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.
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
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.
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
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 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
Maxwell-Boltzmann Distribution for Molecular Speeds
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.
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)
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
Thermal Fluctuations Solid in equilibrium in air. During collisions between the gas and solid atoms, kinetic energy is exchanged.
Each atom in a solid: like a mass on a spring
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.
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 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