Lecture 4 Lecture 4 ––The First Law (Ch. 1)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 All of chapter 1 (pages 1 --23)23)1st homework set due next Friday (18th).1st homework set due next Friday (18th).Homework assignment available on web page.Homework assignment available on web page.Assigned problems: 2, 6, 8, 10, 12Assigned problems: 2, 6, 8, 10, 12How to know if quantity is a function of stateHow to know if quantity is a function of stateThere is a mathematical basis.....There is a mathematical basis.....Consider the function F = f(x,y):yxffdF dx dyxy⎛⎞∂∂⎛⎞=+⎜⎟⎜⎟∂∂⎝⎠⎝⎠zyxdSdrHow to know if quantity is a function of stateHow to know if quantity is a function of stateU1U2đW is path dependent∫()UΔ=∫đQ + đWdoes not depend on pathIn general, F is a state function if the differential dF is ‘exact’. dFdF((= = AdxAdx++BdyBdy) is exact if:1.2. 03. is independent of pathbaAByxdFdF∂∂=∂∂=∫∫vSee 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 stateHow to know if quantity is a function of stateThere is a mathematical basis.....There is a mathematical basis.....Consider the function F = f(x,y):yxffdF dx dyxy⎛⎞∂∂⎛⎞=+⎜⎟⎜⎟∂∂⎝⎠⎝⎠Differentials satisfying the following condition are said to be ‘exact’:0dF=∫vThis condition also guarantees that any integration of dFwill 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 đF is inexact, this implies that it cannot be integrated to yield a function F.How to know if quantity is a function of stateHow to know if quantity is a function of stateCalculation of work for a reversible processCalculation of work for a reversible processarea under curve;WPdV=− =∫()UΔ=∫đQ + đWPV(1)(2)(3)(4)1. Isobaric (P = const)2. Isothermal (PV = const)3. Adiabatic (PVγ= const)4. Isochoric (V = const)••For a given reversible path, there is some associated physics.For a given reversible path, there is some associated 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:0limQ đQCdθθθΔ→⎛⎞≡=⎜⎟Δ⎝⎠•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 đqcnd dθθ⎛⎞≡=⎜⎟⎝⎠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, CpandVPVPđQ đQCCddθθ⎛⎞ ⎛⎞≡≡⎜⎟ ⎜⎟⎝⎠ ⎝⎠More on heat capacityMore on heat capacityUsing the first law, it is easily shown that:•Finding a similarly straightforward expression for CPis not as easy, and requires knowledge of the state equation.VVVđQUCdθθ∂⎛⎞⎛⎞≡=⎜⎟⎜⎟∂⎝⎠⎝⎠()000 andVVVduCUUUCdCdθθθθθθ≡Δ=−==−∫•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 θ. 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 pressurePPPđQHCdθθ∂⎛⎞⎛⎞≡=⎜⎟⎜⎟∂⎝⎠⎝⎠Always trueAlways true()000 and PPPdHCHHHCdCdTθθθθθ≡Δ=−==−∫•For an idea gas, it can be shown that the enthalpy depends only on the temperature of the gas θ. Therefore, dH = dU + PdV + VdP = đQ + VdPConfiguration Work and ideal gasesConfiguration Work and ideal gases()0IsochoricIsobaricln IsothermalfifiVfViWPdVWPdVPVVVdVW PdV nR nRVVθθ=− ==− =− −⎛⎞=− =− =−⎜⎟⎝⎠∫∫∫∫Note: for an ideal gas, U = U(θ), so W = −Q for isothermal processes.It is also always true that, for an ideal gas, ()()andVf i Pf iUC HCθθθθΔ= − Δ= −Adiabatic processes: đQ = 0, so W = ΔU, also PVγ= constant.() ()11Vf i ff iiWC PV PVθθγ⇒=−= −−35 5 57 7Monatomic: ; ; Diatomic: ; ;22 3 22 5PPVP VPVVRRc RRccc ccccγγ⎡ ⎤==== ====⎢ ⎥⎣