PH2105 Summary PDF

PH2105 Summary 1 Temperature and Heat 1.1 Temperature and temperature scales Two bodies in thermal equilibrium must have the same temperature. A conducting material between two bodies permits them to interact and come to thermal equilibrium; an insulating material impedes...

PH2105 Summary 1 Temperature and Heat 1.1 Temperature and temperature scales Two bodies in thermal equilibrium must have the same temperature. A conducting material between two bodies permits them to interact and come to thermal equilibrium; an insulating material impedes this interaction. The Kelvin scale has its zero at the extrapolated zero-pressure temperature for a gas thermometer, -273.15◦C = 0 K: TK = TC + 273.15 In the gas-thermometer scale, the ratio of two temperatures T1 and T2 is defined to be equal to the ratio of the two corresponding gas-thermometer pressures p1 and p2 : T2 p2 = T1 p1 1.2 Thermal expansion A temperature change ∆T causes a change in any linear dimension L0 of a solid body. 1 The change ∆L is approximately proportional to L0 and ∆T : ∆L = αL0 ∆T Similarly, a temperature change causes a change ∆V in the volume V0 of any solid or liquid; ∆V is approximately proportional to V0 and ∆T. ∆V = βV0 ∆T The quantities α and β are the coefficients of linear expansion and volume expansion, respec- tively. For solids, β = 3α. 1.3 Conduction, convection, and radiation Conduction is the transfer of heat within materials without bulk motion of the materials. The heat current H depends on the area A through which the heat flows, the length L of the heat-flow path, the temperature difference (TH − TC ), and the thermal conductivity k of the material: dQ TH − TC H= = kA dt L 2 Convection is a complex heat-transfer process that involves mass motion from one region to another. Radiation is energy transfer through electromagnetic radiation. The radiation heat current H depends on the surface area A, the emissivity  of the surface (a pure number between 0 and 1), and the Kelvin temperature T : H = AσT 4 Here σ is the Stefan-Boltzmann constant σ = 5.67 × 10−8 kg s−3 K−4. The net radiation heat current Hnet from a body at temperature T to its surroundings at temperature Ts depends on both T and Ts : Hnet = Aσ(T 4 − Ts4 ) 2 Thermal Properties of Matter 2.1 Equations of state The pressure p, volume V , and absolute temperature T of a given quantity of a substance are related by an equation of state. This relationship applies only for equilibrium states, in which p and T are uniform throughout the system. The ideal-gas equation of state involves the number of moles n and a constant R that is the same for all gases: pV = nRT 3 2.2 Molecular properties of matter The molar mass M of a pure substance is the mass per mole. The mass mtotal of a quantity of substance equals M multiplied by the number of moles n. mtotal = nM Avogadros number NA is the number of molecules in a mole. The mass m of an individual molecule is M divided by NA. M = NA m 2.3 Kinetic-molecular model of an ideal gas In an ideal gas, the total translational kinetic energy of the gas as a whole 3 Ktr = nRT 2 The average translational kinetic energy per molecule 1 3 m(v 2 )av = kT 2 2 The root-mean-square speed of molecules is s q 3kT vrms = (v 2 )av = m These expressions involve the Boltzmann constant k = R/N A. The mean free path λ of molecules in an ideal gas depends on the number of molecules per volume (N/V ) and the molecular radius r: V λ = vtmean = √ 4π 2r2 N 4 2.4 Heat capacities The molar heat capacity at constant volume CV is a simple multiple of the gas constant R for certain idealized cases: an ideal monatomic gas: 3 CV = R 2 An ideal diatomic gas including rotational energy: 5 CV = R 2 An ideal monatomic solid: CV = 3R Many real systems are approximated well by these idealizations. 5 2.5 Molecular speeds The speeds of molecules in an ideal gas are distributed according to the Maxwell Boltzmann distribution f (v): 3 m  2 2 f (v) = 4π v 2 e−mv /2kT 2πkT The quantity f (v)dv describes what fraction of the molecules have speeds between v and v +dv. 6 3 The First Law of Thermodynamics 3.1 Heat and work in thermodynamic processes A thermodynamic system has the potential to exchange energy with its surroundings by heat transfer or by mechanical work. When a system at pressure p changes volume from V1 to V2 , it does an amount of work W given by: Z V2 W = pdV V1 If the pressure is constant, the work done is W = p(V2 − V1 ) A negative value of W means that work is done on the system. In any thermodynamic process, the heat added to the system and the work done by the system depend not only on the initial and final states, but also on the path (the series of intermediate states through which the system passes). 7 3.2 The first law of thermodynamics The first law of thermodynamics states that when heat Q is added to a system while the system does work W , the internal energy U changes by ∆U = Q − W This law can also be expressed for an infinitesimal process dU = dQ − dW The internal energy of any thermodynamic system depends only on its state. The change in internal energy in any process depends only on the initial and final states, not on the path. The internal energy of an isolated system is constant. 3.3 Important kinds of thermodynamic processes Adiabatic process: No heat transfer into or out of a system; Q = 0. Isochoric process: Constant volume; W = 0. Isobaric process: Constant pressure; W = p(V2 − V1 ). Isothermal process: Constant temperature. 8 3.4 Thermodynamics of ideal gases The internal energy of an ideal gas depends only on its temperature, not on its pressure or volume. For other substances the internal energy generally depends on both pressure and temperature. The molar heat capacities CV and Cp of an ideal gas differ by R, the ideal-gas constant Cp = CV + R The dimensionless ratio of heat capacities, Cp /CV , is denoted by γ Cp γ= CV 9 3.5 Adiabatic processes in ideal gases For an adiabatic process for an ideal gas, the quantities T V γ−1 and pV γ−1 are constant. The work done by an ideal gas during an adiabatic expansion can be expressed in terms of the initial and final values of temperature, or in terms of the initial and final values of pressure and volume: CV 1 W = nCV (T1 − T 2) = (p1 V1 − p2 V2 ) = (p1 V1 − p2 V2 ) R γ−1 4 The Second Law of Thermodynamics 4.1 Reversible and irreversible processes A reversible process is one whose direction can be reversed by an infinitesimal change in the con- ditions of the process, and in which the system is always in or very close to thermal equilibrium. All other thermodynamic processes are irreversible. 10 4.2 Heat engines A heat engine takes heat QH from a source, converts part of it to work W , and discards the remainder |QC | at a lower temperature. The thermal efficiency e of a heat engine measures how much of the absorbed heat is converted to work W QC QC e= =1+ =1− QH QH QH 4.3 The Otto cycle A gasoline engine operating on the Otto cycle has a theoretical maximum thermal efficiency e that depends on the compression ratio r and the ratio of heat capacities γ of the working substance: 1 e = 1 − γ−1 r 11 4.4 Refrigerators A refrigerator takes heat QC from a colder place, has a work input |W |, and discards heat |QH| at a warmer place. The effectiveness of the refrigerator is given by its coefficient of performance K: |QC | |QC | K= = W |QH | − |QC | 4.5 The second law of thermodynamics The second law of thermodynamics describes the directionality of natural thermodynamic pro- cesses. It can be stated in several equivalent forms. The engine statement is that no cyclic process can convert heat completely into work. The refrigerator statement is that no cyclic process can transfer heat from a colder place to a hotter place with no input of mechanical work. 4.6 The Carnot cycle The Carnot cycle operates between two heat reservoirs at temperatures TH and TC and uses only reversible processes. Its thermal efficiency depends only on TH and TC. An additional equivalent statement of the second law is that no engine operating between the same two temperatures can be more efficient than a Carnot engine: TC TH − TC eCarnot = 1 − = TH TH A Carnot engine run backward is a Carnot refrigerator. Its coefficient of performance depends on only TH and TC. Another form of the second law states that no refrigerator operating 12 between the same two temperatures can have a larger coefficient of performance than a Carnot refrigerator: TC KCarnot = TH − TC 4.7 Entropy Entropy is a quantitative measure of the randomness of a system. The entropy change in any reversible process depends on the amount of heat flow and the absolute temperature T : Z 2 dQ ∆S = reversible process 1 T Entropy depends only on the state of the system, and the change in entropy between given initial and final states is the same for all processes leading from one state to the other. This fact can be used to find the entropy change in an irreversible process. 13 An important statement of the second law of thermodynamics is that the entropy of an isolated system may increase but can never decrease. When a system interacts with its surroundings, the total entropy change of system and surroundings can never decrease. When the interaction involves only reversible processes, the total entropy is constant and ∆S = 0; when there is any irreversible process, the total entropy increases and ∆S > 0. 14