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Thermal Expansion Solid Liquid Gas Thermal Expansion: Linear Linear Expansion 0 T 0 (1 T ) coefficien t of linear expansion 5 Typical values range from 10 10 C -6 -1 Thermal Expansion: Volume Volume Expansion DV » 3aV0 DT = bV0 DT (follows from linear expansion and neglecting higher order terms) b = 3a º coefficient of volume expansion Typical values range from 10-6 -10 -5 C°-1 for solids from 10-4 -10 -3 C°-1 for liquids around 3.4x10-3C°-1 for most gasses at atmospheric pressure Thermal Expansion When a material with a hole is warmed, does the hole expand or contract? A. Expand B. Contract C. Neither Thermal Expansion: Exceptions Water Polymers (rubber band) Kinetic Theory of Gasses The average translati onal kinetic energy of molecules in random motion in an ideal gas is directly proportion al to the absolute temperatu re of the gas. K 12 mv 2 32 kT v rms 3kT m translational degrees of freedom 2 kT Maxwell Distribution of Speeds m f (v) 4N 2kT Consider limits : lim v0 0 lim v 0 Discuss : v P , v and v rms 3/ 2 2 2 v e 12 mv kT Calorimetry and Conservation of Energy If a system (or set of subsystems ) is isolated, then energy is conserved. This implies that the heat flowing to or from various subsystems must sum to zero. Qi 0 Calorimetry and Conservation of Energy For solids or liquids, the heat flowing into or out of a system is given by Q mcT where m is the mass, c is the specific heat and ΔT T f Ti is the change in tempera ture. Calorimetry and Conservation of Energy Phase transitions between liquid and gas Q = ±mLV where m is the mass, LV is the latent heat of vaporization. Phase transitions between liquid and solid Q = ±mL f where m is the mass, L f is the latent heat of fusion. If the transition is from a higher energy state of matter to a lower energy state (e.g. gas to liquid or liquid to solid), then energy is being released and the sign is negative. If the transition is from a lower energy state to a higher one, energy is being gained by the system and the sign is positive. Calorimetry and Conservation of Energy For a gas, the heat flowing into or out of a system depends on the PROCESS. There are two special processes with expression s for heat that are analagous to those of solids/liq uids Avogadro’s Hypothesis Avogadro's Hypothesis: equal volumes of gas at the same pressure and temperature contain equal numbers of molecules. specific heat for gas is process dependent and gas dependent. However, AH implies that the specific heat is the same for all (monatomic) gasses for a given process. This is why we use molar specific heat for ideal gasses. R=“universal” gas constant Ideal Gas Law Assumptions: • • • Gas molecules are non-interacting Collisions are elastic Dimension of molecules<<average intermolecular distance PV=nRT R=“universal” gas constant Calorimetry and Conservation of Energy For an isochoric process Q nCV T where m is the mass, C V is the molar specific heat at constant v olume and ΔT T f Ti is the change in tempera ture Calorimetry and Conservation of Energy For an isochoric process Q nCV T where m is the mass, C V is the molar specific heat at constant v olume and ΔT T f Ti is the change in tempera ture. For an isobarric process Q nC P T where m is the mass, C P is the molar specific heat at constant pressure and ΔT T f Ti is the change in tempera ture. Molar Specific Heats For an isochoric process Q nCV T degrees of freedom where CV R 2 R is the Gas Constant 8.314 J/mol - K For an isobarric process Q nC P T where where C P CV R First Law of Thermodynamics U Q Wby Change in internal energy of a system is equal to the heat added to a system minus the work done by the system. Heat (Q) is the energy transferred to or from a system due to a temperature gradient. Work on a system (Won) is the energy transferred to or from a system via a force across a distance OR a pressure over a volume change. Wby=-Won W PdV Example of calculated work: Isothermal Process for Ideal Gas W P (V ) dV nRT dV V dV nRT V Vf nRT ln Vi First Law of Thermodynamics and Simple Processes for an Ideal Gas First Law U=Q-Wby Process Meaning Internal Energy U=nCVT U Ideal Gas Law PV=nRT=NAkBT Q Wby First Law of Thermodynamics and Simple Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT Process Meaning U Q Isothermal T=0 0 Vf Wby nrT ln Vo Wby Vf nrT ln Vo First Law of Thermodynamics and Simple Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT Process Meaning U Q Isothermal T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T Wby U nCV T Vf nrT ln Vo 0 First Law of Thermodynamics and Simple Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT Process Meaning U Q Isothermal T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T Isobaric P=0 nCV T nCP T Wby Vf nrT ln Vo 0 PV or nRT First Law of Thermodynamics and Simple Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT Process Meaning U Q Isothermal T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T Isobaric P=0 nCV T nCP T Adiabatic Q=0 nCV T 0 Wby Vf nrT ln Vo 0 PV or nRT U nCV T More about Adiabatic Processes For a quasistati c adiabatic process with an ideal gas : PV constant CP where γ 1 CV More about Adiabatic Processes For a quasistati c adiabatic process with an ideal gas : PV constant CP where γ 1 CV Monatomic gas : 5/3 More about Adiabatic Processes For a quasistati c adiabatic process with an ideal gas : PV constant CP where γ 1 CV Monatomic gas : 5/3 Diatomic gas : 7/5 Equipartition Theorem http://www.ux1.eiu.edu/~cfadd/1360/21KineticTheory/Equipart.html Free Expansion Gas is allowed to expand in volume adiabatically without doing any work. Q=0, Wby=0 Therefore, U=0 and T=0 The internal energy of an ideal gas does not change during a free expansion. Heat Engine/Efficiency need an example for closed cycle. Carnot Efficiency In general : QL W e 1 QH QH Carnot effiency : TL e 1 TH Entropy: Second Law of Thermodynamics Heat can flow spontaneou sly from a hot object to a cold object; heat will not flow spontaneou sly from a cold object to a hot object. No device is possible whose sole effect is to transform a given amount of heat completely into work. All reversible engines operating between th e same two constant temperatur es TH and TL have the same efficiency . Any irreversib le engine operating between th e same two fixed temperatu res will have an efficiency less than this . Entropy: dQ S T dQ S 0 T First Law of Thermodynamics and Simple (quasi-static or reversible) Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT S Process Meaning U Q Isotherm al T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T 0 Isobaric P=0 nCV T nCP T PV or Q=0 nCV T Adiabatic 0 Wby Vf nrT ln Vo nRT U nCV T Q/T First Law of Thermodynamics and Simple (quasi-static or reversible) Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT S Process Meaning U Q Isotherm al T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T 0 Isobaric P=0 nCV T nCP T PV or Q=0 nCV T Adiabatic 0 Wby Vf nrT ln Vo Q/T nRT U nCV T 0 First Law of Thermodynamics and Simple (quasi-static or reversible) Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT S Process Meaning U Q Isotherm al T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T 0 Isobaric P=0 nCV T nCP T PV or Q=0 nCV T Adiabatic 0 Wby Vf nrT ln Vo Q/T Tf nCV ln Ti nRT U nCV T 0 First Law of Thermodynamics and Simple (quasi-static or reversible) Processes for an Ideal Gas First Law U=Q-Wby Internal Energy U=nCVT Ideal Gas Law PV=nRT=NAkBT S Process Meaning U Q Isotherm al T=0 0 Vf Wby nrT ln Vo Isochoric V=0 nCV T U nCV T 0 Isobaric P=0 nCV T nCP T PV or Q=0 nCV T Adiabatic 0 Wby Vf nrT ln Vo Q/T nRT Tf nCV ln Ti Tf nC P ln Ti U 0 nCV T Entropy: Second Law of Thermodynamics The entropy of an isolated system never decreases. It either stays the constant (reversibl e processes) or increases (irreversi ble processes) . The total change in entropy of any system plus that of the environmen t increases or stays the same. S S environment Ssystem 0 Entropy: Second Law of Thermodynamics Statistical Description S k ln( W ) W number of micro states k Boltzmann' s constant Wf S k ln Wi Black-Body Radiation Rate at which an object radiates energy is observed to be proportion al to temperatu re raised to the fourth power. ΔQ 4 AT Δt emissivity with a value between 0 and 1 Perfrect emitter is a " black body". Perfect emitter is a perfect absorber. Black-Body Radiation A black body at tempera ture T1 in an environmen t of temperatu re T2 ΔQ 4 4 A(T1 T2 ) Δt Planck’s radiation formula Resolving the ultraviole t catastroph e, Planck derived the following by assuming that blackbody radiation was quantized. 2hc I ( , T ) hc / kT e 1 2 5 Wein’s Law Wien' s Law, first found empiricall y, can be derived by finding the maximum of Planck' s Radiation Formula. P T 2.90 x 10 3 m - K where P is the wavelengt h of peak intensity T is the absolute temperatu re Boltzmann Distribution Consider t wo singly degenerate energy levels with energies E 1and E 0 E1 When the system is in theral contact with a reservoir, Boltzmann statistics will describe the probabilit y of level occupation . P e E / kT E0 Boltzmann Distribution P e E / kT E1 Relative population of levels can be found by taking ratio of probabilit ies P1 ( E1 E0 ) / kT E / kT e e P0 E0 Boltzmann Distribution Consider limits lim T 0 E1 P1 E / kT e e 0 P0 E0 lim T P1 e E / kT e 0 1 P0 Boltzmann Distribution: Partition Function P e E / kT What is the proportion ality factor? E1 The factor must scale the Boltzmann factor so that the sum of all probabilit ies adds to 1. The sum of all Boltzmann factors is called the partition factor (Z) Z e Ei / kT states Thus the proportion ality factor is 1 Z E0 Average values If all states are equally probable X X X i all states N 1 X i Pi X i all states N all states g X i all X values N i gi X i P( X i ) X i all X values N all X values Average values If the system obeys Boltzmann statistics e X Pi X i Z all states all states Ei / kT X i e Ei / kT all states Z Xi Average values If the system obeys Boltzmann statistics X Pi X i all states e Z all states Ei / kT X i e Ei / kT Xi all states e Ei / kT X P ( X i ) X i g i Z all X values all X values Z X i g e Ei / kT i all X values Z Xi Average values If the system obeys Boltzmann statistics e Ei / kT X Pi X i Z all states all states X i Ei / kT e Xi all states e Ei / kT X P ( X i ) X i g i Z all X values all X values Z X i Ei / kT g e Xi i all X values where Z is called the PARTITION FUNCTION and is given by the sum of all the Boltzmann factors Z Ei / kT e all states Ei / kT g e i all energies Z Example: Average Energy E E e i all states Ei / kT Ei / kT g E e i i all Energies Z Z where g is the multiplici ty Example: Average Energy E E e Ei / kT i all states Ei / kT g E e i i all Energies Z Z where g is the multiplici ty and since Z Ei / kT e all states (ln( Z )) E kT T 2 Ei / kT g e i all energies