Chapter 3 Work, heat and thefirst law of thermodynamics 3.1 Mechanical work Mechanical work is defined as an energy transfer to the system through the change of an external parameter. Work is the only energy which is transferred to the system through external macroscopic forces. Example: consider the mechanical work performed on a gas due to an infinitesimal volume change (reversible transfor- mation) dV= adx , wherea is the active area of the piston. In equilibrium, the external forceF is related to pressureP as F= P a . − For an infinitesimal process, the change of the position of the wall by dx results in per- forming workδW: δW= F dx= P dV,δW= P adx . (3.1) − − For a transformation of the system along afinite reversible path in the equation-of-state space (viz for a process withfinite change of volume), the total work performed is V2 ΔW= P dV. − �V1 Note: Mechanical work is positive when it is performed on the system. • δW is not an exact differential, i.e.,W(P, V ) does not define any state property. • ΔW depends on the path connectingA(V ) andB(V ). • 1 2 21 22 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS Cyclic process. During a cyclic process the path in the equation-of-state space is a closed loop; the work done is along a closed cycle on the equation-of-state surfacef(P, V, T ) = 0: W= P dV. − � 3.2 Heat Energy is transferred in a system in the from of heat when no mechanical work is exerted, viz whenδW= P dV vanishes. Compare ( 3.1). Other forms of energy (magnetic, electric, gravitational,− ...) are also considered to be constant. Heat transfer is a thermodynamic process representing the transfer of energy in the form of thermal agitation of the constituent particles. In practice one needs heating elements to do the job, f.i. aflame. As an example of a process where only heat is transferred, we consider two isolated systems with temperaturesT A andT B such thatT A > TB. The two systems are brought together without moving the wall between them. Due to the temperature difference, the energy is transferred through the static wall without any change of the systems’ volume (no work is done). Under such conditions, the transferred energy fromA toB is heat. Heat capacity. If a system absorbs an amount of heatΔQ, its temperature rises pro- portionally by an amountΔT: ΔQ=CΔT . (3.2) The proportionality constantC is the heat capacity of the substance (W¨armekapazit¨at). It is an extensive quantity. Specific heat. The intensive heat capacityc may take various forms: per particle : C/N per mole : C/n per unit volume : C/V The unit of heat is calorie or, equivalently, Joule 1 cal 4.184 J. ≡ 3.3. EXACT DIFFERENTIALS 23 Thermodynamic processes. Heat may be absorbed by a body retaining one of its defining variables constant. The various possible processes are: isothermal :T = const. isobaric :P = const. isochoric :V = const. adiabatic :ΔQ = 0 (no heat is transferred) A corresponding subscript is used to distinguish the various types of paths. For example, CV – for the heat capacity at constant volume, CP – for the heat capacity at constant pressure. Thermodynamic response coefficients. Examples of other thermodynamic coeffi- cients measuring the linear response of the system to an external source are 1 ΔV compressibility :κ= − V ΔP 1 ΔV the coefficient of thermal expansion :α= . V ΔT Sign convention. We remind that the convention is that δQ>0 when heat is transferred to the system δW >0 when work is done on the system, withδW= P dV. − 3.3 Exact differentials A functionf=f(x, y) of two variables has the differential ∂f ∂f df= Adx+ Bdy, A= ,B= . ∂x ∂y Reversely one says that Adx+ Bdy is an exact differential if ∂A ∂B ∂2f ∂2f = , = . ∂y ∂x ∂x∂y ∂y∂x An example from classical example is the potential energyΦ(x, y). For all exact differen- tialsdΦ the path of integration is irrelevant, (x2,y2) Φ(x2, y2) =Φ(x 1, y1) + dΦ. �(x1,y1) 24 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS Any path connecting (x1, y1) to (x2, y2) results in the sameΦ(x 2, y2). Stirling cycle. Heat and work are both –not– exact differentials. This is an experimental fact which can be illustrated by any reversible cyclic process. As an example we consider here the Stirling cycle, which consist of four sub-processes. (1) Isothermal expansion. The heatQ 1 delivered to the gas makes it expand at constant temperature. (2) Isochoric cooling. The volume of the piston is kept constant while the gas cools down. The transferred heat and work areQ 2 andW 2. (3) Isothermal compression. The heatQ 3 removed makes the gas contract at constant temperature. (4) Isochoric heating. The volume of the piston is kept constant while is heated up. The transferred heat and work areQ 4 andW 4. The experimental fact that the Stirling cycle can be used either as an engine (W2 +W 4 < 0), or as an heat pump (W2 +W 4 > 0), proves that work and heat cannot be exact differentials, viz that δQ= 0. � � 3.4 First law of thermodynamics – internal energy Thefirst law of thermodynamics expresses that energy is conserved, when all forms of energy, including heat, are taken into account. Definition 1. For a closed thermodynamic system, there exists a function of state, the internal energyU, whose changeΔU in any thermodynamic transformation is given by ΔU=ΔQ+ΔW+..., (3.3) whereΔQ is heat transferred to the system andΔW is mechanical work performed on the system. If present, other forms of energy transfer processes need to taken into account on the RHS of (3.3). ΔU is independent of the path of transformation, althoughΔQ andΔW are path- dependent. Correspondingly, in a reversible infinitesimal transformation, the infinitesimal 3.4. FIRST LAW OF THERMODYNAMICS – INTERNAL ENERGY 25 δQ andδW are not exact differentials (in the sense that they do not represent the changes of definite functions of state), but dU=δQ+δW , dU=C dT P dV (3.4) V − is an exact differential. For the second part of (3.4) we have used (3.2) and (3.1), namely thatδQ=C dT (when the volumeV is constant) and thatδW= P dV. V − Definition 2. Energy cannot be created out of nothing in a closed cycle process: dU=0 ΔQ= ΔW. ⇒ − � Statistical mechanics. The iternal energyU is a key quantity in statistical mechanics, as it is given microscopically by the sum of kinetic and potential energy of the constituent particles of the system 1 N N 1 N U=E= p2 + φ(�r) + V(�r �r), 2m i i 2 i − j i=1 i=1 i=j � � �� whereφ(�ri) is the external potential andV(�r i �rj) is the potential of the interaction between particles (f.i. the Coulomb interaction potential− between charged particles). 3.4.1 Internal energy of an ideal gas We considerN molecules, i.e.n= N/N A moles, in a cubic box of sideL and volumeV=L 3. A particle hitting a given wall changes its momentum by y Δpx = 2mvx,Δt=2L/v x L wherem is the mass,v x the velocity inx direction andΔt the average time between collisions. The− momentum hence changes on the average as 2 2 Δpx 2mvx mvx mv 2 x = = = = Ekin , Δt 2L/vx L 3L 3L 2 z whereE kin =mv /2 is the kinetic energy of the molecule 2 2 2 2 andv =v x +v y +v z . Newton’s law. Newton’s law,dp/dt=F, tells us that the total forceF tot on the wall is 2NEkin/(3L). We then obtain for the pressure F 2 N m P= tot = E ,E = v2 . L2 3 L3 kin kin 2 Assuming the ideal gas relation (1.3) wefind consequently N 2 N 3 P= k T= E ,E = k T (3.5) V B 3 L3 kin kin 2 B 26 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS for the kinetic energyE kin of a single molecule. The internal energy is then, withφ(�ri) = V(�r �r) = 0, i − j 3 3 3 U= Nk T= nRT= PV , (3.6) 2 B 2 2 whereR=8.314 J/(mol K) is the gas constant. 3.5 Energy and heat capacity for various processes In this section we shall analyze various heat transfer processes and derive the correspond- ing heat capacities with the help of thefirst law of thermodynamics. 3.5.1 Isochoric process An isochoric process is a constant volume pro- cess. We have hence dV=0,δW=0, dU =δQ , | V | V and δQ dU = =C (3.7) dT dT V � �V � �V for the heat capacity at constant volumeC V . Ideal gas. An ideal gas containingn moles is defined by its equation of state, PV=Nk BT , NkB = nR . (3.8) Using the internal energy (3.6), 3 3 3 U= Nk T= nRT, dU= nRdT , 2 B 2 2 we obtain 3 3 PV C = nR= (3.9) V 2 2 T for the heat capacity of the ideal gas at constant volume. 3.5.2 Isobaric process 3.5. ENERGY AND HEAT CAPACITY FOR VARIOUS PROCESSES 27 An isobaric process is a constant pressure pro- cess. In order to evaluateC P we consider δQ = dU + P dV , |P | P | P which, under an infinitesimal increment of tem- perature, is written as ∂U ∂V δQ = dT+P dT |P ∂T ∂T � �P � �P C dT.