Kinetic Molecular Theor of Matter Heat capacit of gases and metals Pressure of gas Average speed of electrons in semiconductors Electron noise in resistors Positive metal ion cores Free valence electrons forming an electron gas Fig..7: In metallic bonding the valence electrons from the metal atoms form a "cloud of electrons" which fills the space between the metal ions and "glues" the ions together through the coulombic attraction between the electron gas and positive metal ions. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
Ideal gas approimation. Pressure Square Container Microscopic model Face B v Area A Face A a Pressure (P) is caused b the collisions of molecules with walls Gas atoms v a Definition: P (pressure) force per unit area a Fig..5: The gas molecules in the container are in random motion. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca Our task is to show that Temperature (T) is related to average speed of molecules Ideal Gas Assumptions:. molecules are in constant and random motions, all directions are equivalent. no interactions between molecules but collisions 3. all collisions are elastic 4. molecule sie is negligible 5. Newtonian mechanics is valid
Connecting the pressure with the average speed of gas molecules N Step. Change in Momentum of a Molecule Δp change in momentum, Δp m mass of the molecule, v velocit in the direction Step. Force produced b unique molecule P Δp F Δt Total _ force... 3 a a N a / v mn 3 a v a Step3. Calculating pressure v... v N N mnv V Step4. Averaging speed v v v v 3v P N 3V ρv 3 Ν total number of molecules ρ densit of gas v velocit of molecules
Pressure Temperature Volume Ideal Gas Equation PV N N A RT P is the pressure, i.e. force per unit area N is the total number of molecules in volume V R is the gas constant (8.344 J mol - K - ) N A is the Avogadro s number (6.0 0 3 mol - ) Eperimental law macroscopic point of view
Gas Pressure in the Kinetic Theor P ρ v 3 3 N m v Ideal Gas Equation PV N N A RT P gas pressure, N number of molecules, m mass of the gas molecule, v velocit, V volume, ρ densit. Mean Kinetic Energ per Atom KE 3 kt k R/N A Boltmann constant, T absolute temperature 0 K -73 0 C
Gas constant R 8.344 J mol - K - Boltmann s constant k 8.6 0-5 ev K - R k N A R macroscopic (technical, thermodnamical etc.) calculations k microscopic (atomic) calculations Mole (a gram molecule) a quantit of a substance equal to the molecular weight of a substance epressed in grams Eample: Carbon has atomic mass of a mole of carbon is grams Mole contains 6.00 0 3 species (atoms, molecules, etc.) N A 6.00 03 - an Avogadro's number
KE Internal Energ per Mole for a 3 kt Monatomic Gas U N A 3 N kt A U total internal energ per mole, N A Avogadro s number, m mass of the gas molecule, k Boltmann constant, T temperature Molar Heat Capacit at Constant Volume C m du dt 3 N Ak 3 R C m specific heat per mole at constant volume (J K - mole - ), U total internal energ per mole, R gas constant Definition: Heat capacit is the rise of internal energ per unit temperature
Degree of freedom (DF) the method to absorb energ Eamples: Monoatomic gas has 3 DF Diatomic gas has 5 DF Mawell s principle of equipartition of energ : Each DF has an average energ ½ kt Monoatomic gas has 3 DF. E3/ kt C m 3/ R Diatomic gas has 5 DF. E5/ kt C m 5/ R Solids have 6 DF. E3kT C m 3 R (Dulong - Petit rule)
TRANSLATIONAL MOTION ROTATIONAL MOTION I 0 v v I I v ais out of paper Monoatomic gas: Diatomic gas: Fig..6: Possible translational and rotational motions of a diatomic molecule. Vibrational motions are neglected. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca E E I ω I ω KE U 3 5 kt kt I, momenta of inertia, ω, angular velocities
(a) E K K K K,, spring constant,,, etensions of springs (b) Fig..7 (a) The ball-and-spring model of solids in which the springs represent the interatomic bonds. Each ball (atom) is linked to its neighbors b springs. Atomic vibrations in a solid involve 3 dimensions. (b) An atom vibrating about its equilibrium position stretches and compresses its springs to the neighbors and has both kinetic and potential energ. 6 U kt 3kT From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
Dulong - Petit Rule U 3kT R N a k C m du dt 3R 5 J K- mol - C m specific heat per mole at constant volume (J K - mole - ), U total internal energ per mole, T temperature, R gas constant
What is the distribution of molecule speeds? Relative number of molecules per unit velocit (s/km).5.5 0.5 0 v* vav v rms 98 K (5 C) v* vav vrms 000 K (77 C) 0 500 000 500 000 Speed (m/s) 3 KE kt Fig..: Mawell-Boltmann distribution of molecular speeds in nitrogen gas at two temperatures. The ordinate is dn/(ndv),the fractional number of molecules per unit speed interval in (km/s) - From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
How to determine the speed of molecule?
What is the influence of temperature? Relative number of molecules per unit velocit (s/km).5.5 0.5 0 v* vav v rms 98 K (5 C) v* vav vrms 000 K (77 C) 0 500 000 500 000 Speed (m/s) Fig..: Mawell-Boltmann distribution of molecular speeds in nitrogen gas at two temperatures. The ordinate is dn/(ndv),the fractional number of molecules per unit speed interval in (km/s) - From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
Mawell-Boltmann Distribution for Molecular Speeds n v 4πN m πkt 3/v ep kt n v the velocit densit function, N total number of molecules, m molecular mass, k Boltmann constant, T temperature, v velocit
Mawell-Boltmann Distribution for Molecular Speeds n v 4πN m πkt 3/v ep kt E½ n v the velocit densit function, N total number of molecules, m molecular mass, k Boltmann constant, T temperature, v velocit
Mawell-Boltmann Distribution for Translational Kinetic Energies n E π N kt 3/ E / ep E kt n E number of atoms per unit volume per unit energ at an energ E, N total number of molecules per unit volume, k Boltmann constant, T temperature.
Boltmann Energ Distribution n E N E C ep kt n E number of atoms per unit volume per unit energ at an energ E, N total number of atoms per unit volume in the sstem, C a constant that depends on the specific sstem, k Boltmann constant, T temperature
Where lies the average kinetic energ? Number of atoms per unit energ, n E Average KE at T. T Average KE at T T > T E A Energ, E Fig..: Energ 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 strongl on the temperature as ep(-e A /kt). From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
What is thermal equilibrium? E K K K SOLID GAS E I ω I ω KE3/ kt M V v m Gas Atom KE3/ kt Fig..3: Solid in equilibrium in air. During collisions between the gas and solid atoms, kinetic energ is echanged. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca
What is thermal equilibrium? SOLID KE3/ kt GAS M V v m Gas Atom KE3/ kt Heat amount of energ transferred from kinetic energ of the atoms in solid to the kinetic energ of gas molecules In equilibrium heat transfer 0
Mechanical noise Equilibrium Compression m m Δ 0 Δ < 0 Compression Etension Δ Mean displacement Δ 0 Instantaneous potential energ PE(t)½K(Δ) Mean potential energ Etension Δ> 0 ½K(Δ) ½kT m Fig..4: Fluctuations of a mass attached to a spring due to random bombardment b air molecules From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca t ( Δ ) rms kt K K spring constant, T temperature, (Δ) rms rms value of the fluctuations of the mass about its equilibrium position.
Electrical noise A B A B v 0 V v -3 μv A B Voltage, v(t) v 5 μv Time Fig..5: Random motion of conduction electrons in a conductor results in electrical noise. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca Average value 0 Average power?
Root Mean Square Noise Voltage Across a Resistance R Electron Flow Electron Flow Current C Current Fig..6: Charging and discharging of a capacitor b a conductor due to the random thermal motions of the conduction electrons. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 00) http://materials.usask.ca v(t) Time. Instantaneous energ E(t)½ C V(t). Mean energ E CV ( t) kt 3. RMS voltage kt V ( t) C 4. Bandwidth B πrc V rms [4kTRB] / Johnson resistor noise equation R resistance, B bandwidth, V rms root mean square noise voltage, k Boltmann constant, T temperature
Root Mean Square Noise Voltage Across a Resistance E(t)½ C V(t) E CV ( t) V ( t) B kt C πrc kt Instantaneous energ Mean energ and equipartition principle RMS voltage Bandwidth V rms [4kTRB] / Johnson resistor noise equation R resistance, B bandwidth, V rms root mean square noise voltage, k Boltmann constant, T temperature