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UF PHY 4523 - The First Law

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Lecture 4 – The First Law (Ch. 1)Lecture 4 – The First Law (Ch. 1)Monday January 14Monday January 14thth•Finish previous class: functions of state•Reversible work•Enthalpy and specific heat•Adiabatic processesReading: Reading: All of chapter 1 (pages 1 - 23)All of chapter 1 (pages 1 - 23)1st homework set due next Friday 1st homework set due next Friday (18th).(18th).Homework assignment available on web Homework assignment available on web page.page.Assigned problems: 2, 6, 8, 10, 12Assigned problems: 2, 6, 8, 10, 12How to know if quantity is a function of How to know if quantity is a function of statestateThere is a mathematical basis.....There is a mathematical basis.....Consider the function F = f(x,y):yxf fdF dx dyx y� �� �� �= +� �� �� �� �� �zyxdSdrHow to know if quantity is a function of How to know if quantity is a function of statestateU1U2đW is path dependent�( )UD =�đQ + đWdoes not depend on pathIn general, F is a state function if the differential dF is ‘exact’. dF dF ((= Adx = Adx  Bdy Bdy) is exact if:1.2. 03. is independent of pathbaA By xdFdF� �=� �=���See also: See also: •Appendix EAppendix E•PHY3513 notesPHY3513 notes•Appendix A in Carter bookAppendix A in Carter book•In thermodynamics, all state variables are by definition exact. However, differential work and heat are not.How to know if quantity is a function of How to know if quantity is a function of statestateThere is a mathematical basis.....There is a mathematical basis.....Consider the function F = f(x,y):yxf fdF dx dyx y� �� �� �= +� �� �� �� �� �Differentials satisfying the following condition are said to be ‘exact’:0dF =��This condition also guarantees that any integration of dF will not depend on the path of integration, i.e. only the limits of integration matter.This is by no means true for any function!If integration does depend on path, then the differential is said to be ‘inexact’, i.e. it cannot be integrated unless a path is also specified. An example is the following:đF = ydx  xdy.Note: is a differential đ F is inexact, this implies that it cannot be integrated to yield a function F.How to know if quantity is a function of How to know if quantity is a function of statestateCalculation of work for a reversible Calculation of work for a reversible processprocessarea under curve;W PdV=- =�( )UD =�đQ + đWPV(1)(2)(3)(4)1. Isobaric (P = const)2. Isothermal (PV = const)3. Adiabatic (PV = const)4. Isochoric (V = const)•For a given reversible path, there is some associated For a given reversible path, there is some associated physics.physics.Heat CapacityHeat CapacityThe heat capacity C of a system is defined as the limiting ratio of the heat Q added to a system (causing it to change from one equilibrium state to another) divided by the accompanying temperature increase:0limQđQCdqq qD �� �� =� �D� �•Note that this is a rather awkward definition, because the differential đQ is inexact.The specific heat capacity c of a system, often abbreviated to “specific heat”, is the heat capacity per unit mass (or per mole, or per kilomole)1 đQ đqcn d dq q� �� =� �� �Heat CapacityHeat CapacityBecause the differential đQ is inexact, we have to specify under what conditions heat is added. Or, more precisely, which parameters are held constant. This leads to two important cases:•the heat capacity at constant volume, CV •the heat capacity at constant pressure, CpandV PV PđQ đQC Cd dq q� � � �� �� � � �� � � �More on heat capacityMore on heat capacityUsing the first law, it is easily shown that:•Finding a similarly straightforward expression for CP is not as easy, and requires knowledge of the state equation.VV VđQ UCdq q�� � � �� =� � � ��� � � �( )00 0 andV V VduC U U U C d Cdqqq q qq� D = - = = -�•U is a function of state, so it does not actually matter how we add the heat!•For an idea gas, it can be shown that the internal energy depends only on the temperature of the gas . Therefore,Always trueAlways trueEnthalpy and heat capacityEnthalpy and heat capacity•Enthalpy, H = U + PV, turns out to be a useful quantity for calculating the heat capacity at constant pressurePP PđQ HCdq q�� � � �� =� � � ��� � � �Always trueAlways true( )00 0 and P P PdHC H H H C d CdTqqq q q� D = - = = -�•For an idea gas, it can be shown that the enthalpy depends only on the temperature of the gas . Therefore, dH = dU + PdV + VdP = đQ + VdPConfiguration Work and ideal gasesConfiguration Work and ideal gases( )0 IsochoricIsobaricln Isothermalfif iVfViW PdVW P dV P V VVdVW PdV nR nRV Vq q=- ==- =- -� �=- =- =-� �� ���� �Note: for an ideal gas, U = U(), so W = Q for isothermal processes.It is also always true that, for an ideal gas, ( ) ( )andV f i P f iU C H Cq q q qD = - D = -Adiabatic processes: đQ = 0, so W = U, also PV = constant.( ) ( )11V f i f f i iW C P V PVq qg� = - = --3 5 5 5 7 7Monatomic: ; ; Diatomic: ; ;2 2 3 2 2 5P PV P V PV VR R c R R cc c c cc cg g� �= = = = = = = =� ��


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UF PHY 4523 - The First Law

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