Chapter 3

Work, and theﬁrst law of

3.1 Mechanical

Mechanical work is deﬁned as an transfer to the through the change of an external parameter. Work is the only energy which is transferred to the system through external macroscopic . Example: consider the mechanical work performed on a due to an inﬁnitesimal change (reversible transfor- mation) dV= adx , wherea is the active of the piston. In equilibrium, the external forceF is related to pressureP as

F= P a . − For an inﬁnitesimal process, the change of the 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 (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. is a representing the transfer of energy in the form of thermal agitation of the constituent . 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 with temperaturesT A andT B such thatT A > TB. The two systems are brought together without moving the wall between them. Due to the 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 of the substance (W¨armekapazit¨at). It is an extensive quantity. Speciﬁc heat. The intensive heat capacityc may take various forms: per : C/N per : C/n per unit volume :C/V The unit of heat is or, equivalently,

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 .

Thermodynamic response coeﬃcients. Examples of other thermodynamic coeﬃ- cients measuring the linear response of the system to an external source are 1 ΔV :κ= − V ΔP 1 ΔV the coeﬃcient of :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). . 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 (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 –

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 , 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 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 . The iternal energyU is a key quantity in , as it is given microscopically by the sum of kinetic and 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

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 by y

Δpx = 2mvx,Δt=2L/v x L

wherem is the ,v x the inx direction andΔt the average 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 of the molecule 2 2 2 2 andv =v x +v y +v z .

Newton’s law. ’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 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 ,

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 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. ≡ P The speciﬁc heat at constant pressure,

∂U ∂V C = +P , (3.10) P ∂T ∂T � �P � �P reduces then for an ideal gas, for whichU=3nRT/2 andPV= nRT , to

3 5 C = nR+ nR= nR=C +Nk . (3.11) P 2 2 V B

Mayer’s relation betweenC P andC V . In order to evaluate the partial derivative (∂U/∂T) P entering the deﬁnition (3.10) of the speciﬁc heat at constant pressure we note that the equation of statef(P,V,T ) = 0 determines the interrelation betweenP,V and T . A constant pressureP deﬁnes hence a functional dependence betweenV andT . We therefore have ∂U ∂U ∂T ∂U ∂V = + ∂T ∂T ∂T ∂V ∂T � �P � �V � �P � �T � �P =C V = 1 and hence with � �� � � �� � ∂U ∂V C =C + P+ (3.12) P V ∂V ∂T � � �T � � �P the Mayer relation. In Sect. 4.5 we will connect the partial derivatives entering (3.12) with measurable quantities.

3.5.3 Isothermal processes for the ideal gas An takes place at constant temper- ature. The work performed

V2 ΔW= dV P − �V1 28 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS

is hence given by the area belowP=P(T,V) T . Using the equation-of-state relationPV= nRT of| the ideal gas we obtain

V2 dV V ΔW= nRT = nRT ln 2 − V − V �V1 1 = ΔQ, − where the last relation follow from theﬁrst law,ΔU=ΔQ+ΔW , and from the fact that the internal energyU=3nRT/2 of the ideal gas remains constant during the isothermal process.ΔW > 0 forV 1 > V2, viz when the gas is compressed. Note. Heat cannot be transformed in work forever, as we will discuss in the next chapter.

3.5.4 Free expansion of an ideal gas

A classical experiment, as performedﬁrst by Joule, consist of allowing a thermally isolated ideal gas to expand freely into an isolated chamber, which had been initially empty. After a new equilibrium state was established, in which the gasﬁlls both compartments, the ﬁnal temperature of the gas is found to be identical to the initial temperature.

The expansion process is overall isolated. Neither heat nor work is transferred into the system, ΔW=0,ΔQ=0,ΔU=0, and internal energyU stay constant

Ideal gas. The internal energy 3 U= nRT, 2 of the ideal gas with a contant numbern of mols depends only on the temperatureT , and not on the volumeV . The kinetic energyE kin = 3kBT/2 of the constitutent particles is not contingent on the enclosing volume. We hence have

∂T = 0,T =T . ∂V 2 1 � �U Which means that for the ideal gas the free expansion is an isothermal expansion, in agreement with Joule’sﬁndings. 3.6. 29

3.5.5 Adiabatic processes for the ideal gas An happens without heat transfer: δQ=0, dU= P dV , − where the second relation follows from theﬁrst law of thermodynamics, dU=δQ+δW. For the ideal gas we havePV= nRT,U=3nRT/2 = 3PV/2 and hence

3 5 dV 3 dP dU= P dV+ V dP = P dV, = , 2 − 2 V − 2 P � � which can be solved as V P γ log = log 0 , PV γ = const. ,γ=5/3. V P � 0 � � � Using the ideal gas equation of state,PV= nRT , we may write equivalently

γ 1 TV − = const. . Sinceγ> 1, an adiabatic path has a steeper than an isotherm in aP V diagram. − 3.6 Enthalpy

The internal energy is a continuous diﬀerentiable for which the relations ∂U dU=δQ P dV,δQ=C dT, C = − V V ∂T � �V hold. One can deﬁne equivalently with H U+PV , (3.13) ≡ a state functionH, denote the enthalpy, which obeys ∂H dH=δQ+ V dP,δQ=C dT, C = . (3.14) P P ∂T � �P Note thatP,V andT determine each others in pairs via the equation-of-state function f(P,V,T ) = 0. Derivation. We have dH= dU+ P dV+ V dP=δQ P dV+ P dV+ V dP − =δQ+ V dP , in accordance with and (3.14), and ∂H ∂(U+PV) ∂U ∂V = = +P =C ∂T ∂T ∂T ∂T P � �P � �P � �P � �P in agreement with the deﬁnition (3.10) of the speciﬁc heatC P at constant pressure. 30 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS 3.7 Magnetic systems

We now discuss how the concepts developed hitherto for a mono-atomic and non-magnetic gas can be generalized to for which either a magnetizationM and/or a magneticﬁeld is present. H - : magneticﬁeld (intensive, generated by external currents) H -M: magnetization (extensive, produced by ordered local moments)

Magnetic work. The magnetic work done on the system is dM, as derived in electro- . The modiﬁedﬁrst law of thermodynamics then takesH the form, dU=δQ+δW,δW= dM , (3.15) H when the volumeV is assumed to be constant. All results previously for thePVT system can be written intoHMT variables when using P,M V. H↔− ↔ Susceptibility. A magneticﬁeld induces in general a magnetization densityM/V, which is given for a paramagneticH substance by Curie’s law

M c =χ(T) ,χ(t) = 0 . V H T

χ=χ(T ) is denoted the magnetic susceptibility.

Phase transitions. Ferromagnetic systems order spontaneously below the Curie temper- atureT c, becoming such a permanent with aﬁnite magnetizationM. The transition is washed-out for anyﬁniteﬁeld = 0, which induces aﬁnite magnetization at all . H� Hysteresis. 3.7. MAGNETIC SYSTEMS 31

Impurities and lattice imperfections induce magnetic do- mains, which are then stabilized by minimizing the mag- netic energy of the surfaceﬁelds. The resulting domain walls may move in response to the change of . This is a dissipative process which leads to hysteresis. H 32 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS