The gas occupies a constant volume Heat is then added to the gas until the temperature reaches 400 K This process is shown on the P-V diagram in Figure 15 8,
It means any heat transfer that increases the energy of a system is positive, and b) n =1, the pressure volume relationship is PV = constant
This means that the heat capacity at constant pressure measures the rate of enthalpy increase with temperature during and isobaric process Over ranges of
The heat transferred c The change of enthalpy d The average specific heat at constant pressure [ ] kJ
consists solely in the transfer of heat from one (a) Cooling at constant pressure followed by heating at constant volume ? (b) Heating at constant
isobaric pressure isothermal temperature isochoric volume isentropic entropy Heat transfer for constant pressure process {[ ] = ( ? )}
at constant volume CV , because when heat is added at constant pressure, the There can be no process whose only final result is to transfer thermal
tic" means involving the transfer of heat The term "diabatic" would be If heat is added to a material at constant pressure, so that the specific volume
Thermal Engineering is the science that deals with the energy transfer to practical Define specific heat capacity at constant pressure
quantity of energy is constant, and when energy consists solely in the transfer of heat from one (a) Cooling at constant pressure followed by heating at
system is positive, and heat transfer that decreases the energy of a system is c) For n = 0, the pressure-volume relation reduces to P=constant (isobaric
Heat transfer is a thermodynamic process representing the transfer of energy in the form of thermal agitation of CP – for the heat capacity at constant pressure
isothermal: T = constant - isochoric: V = constant - isobaric: P = constant Amount of heat transferred also depends on the initial, final, and intermediate
The heat capacity at constant pressure CP is greater than the heat capacity There can be no process whose only final result is to transfer thermal energy from
Derive expressions for heat and work transfer in important thermodynamic processes such as: a) Isochoric process (Section 5 3) b) Isobaric process ( Section 5 3)
Pressure: p Heat transfer depends on the initial final states, also on the path capacity at constant pressure Isochoric: W=0, Q=∆U=nC V ∆T Isobaric:
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Chapter 3
Work, heat and the
fi rst law of thermodynamics
3.1 Mechanical work
Mechanical work
is de fi ned 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 in fi nitesimal volume change (reversible transfor- mation) dV = adx , where a is the active area of the piston. In equilibrium, the external force F is related to pressure P as F = - Pa .
For an in
fi nitesimal process, the change of the position of the wall by dx results in per- forming work δ W : δ W = Fdx = - PdV, δ W = -
Padx .
(3.1)
For a transformation of the system along a
fi nite reversible path in the equation-of-state space (viz for a process with fi nite change of volume), the total work performed is Δ W = - � V 2 V 1 PdV. Note : • Mechanical work is positive when it is performed on the system. • δ W is not an exact di ff erential, i.e., W ( P,V ) does not de fi ne any state property. • Δ W depends on the path connecting A ( V 1 ) and B ( V 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 surface f ( P,V,T ) = 0: W = - � PdV.
3.2 Heat
Energy is transferred in a system in the from of
heat when no mechanical work is exerted, viz when δ W = - PdV 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. a fl ame. As an example of a process where only heat is transferred, we consider two isolated systems with temperatures T A and T B such that T A > T B . The two systems are brought together without moving the wall between them. Due to the temperature di ff erence, 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 from A to B is heat .
Heat capacity.
If a system absorbs an amount of heat Δ Q , its temperature rises pro- portionally by an amount Δ T : Δ Q = C Δ
T.(3.2)
The proportionality constant
C is the heat capacity of the substance (W¨armekapazit¨at).
It is an extensive quantity.
Speci fi c heat. The intensive heat capacity c 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 DIFFERENTIALS23
Thermodynamic processes.
Heat may be absorbed by a body retaining one of its de fi ning variables constant. The various possible processes are: isothermal : T = const. isobaric : P = const. isochoric : V = const. adiabatic : Δ Q = 0 (no heat is transferred) A corresponding subscript is used to distinguish the various types of paths. For example, C V - for the heat capacity at constant volume, C P - for the heat capacity at constant pressure.
Thermodynamic response coe
ffi cients. Examples of other thermodynamic coe ffi - cients measuring the linear response of the system to an external source are compressibility : κ = - 1
VΔVΔP
the coe ffi cient of thermal expansion : α = 1
VΔVΔT.
Sign convention.
We remind that the convention is that δ Q > 0 when heat is transferred to the system δ W > 0 when work is done on the system, with δ W = - PdV .
3.3 Exact di
ff erentials
A function
f = f ( x,y ) of two variables has the di ff erential df = Adx + Bdy, A = ∂ f ∂ x, B =∂f∂y.
Reversely one says that
Adx + Bdy is an exact di ff erential if ∂ A ∂ y=∂B∂x ,∂ 2 f ∂ x ∂ y=∂ 2 f ∂ y ∂ x. An example from classical example is the potential energy Φ ( x,y ). For all exact di ff eren- tials d Φ the path of integration is irrelevant, Φ ( x 2 ,y 2 ) = Φ ( x 1 ,y 1 ) +� ( x 2 ,y 2 ) ( x 1 ,y 1 ) d Φ .
24 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS
Any path connecting (
x 1 ,y 1 ) to ( x 2 ,y 2 ) results in the same Φ ( x 2 ,y 2 ).
Stirling cycle.
Heat and work are both -not- exact di ff erentials. 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 heat Q 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 are Q 2 and W 2 . (3) Isothermal compression. The heat Q 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 are
Q 4 and W 4 . The experimental fact that the Stirling cycle can be used either as an engine ( W 2 + W 4 <
0), or as an heat pump (
W 2 + W 4 > 0), proves that work and heat cannot be exact di ff erentials, viz that� δ Q � = 0 .
3.4 First law of thermodynamics - internal energy
The fi rst law of thermodynamics expresses that energy is conserved, when all forms of energy, including heat, are taken into account. De fi nition 1. For a closed thermodynamic system, there exists a function of state , the internal energy U , whose change Δ U in any thermodynamic transformation is given by Δ U = Δ Q + Δ W + ... ,(3.3) where Δ Q is heat transferred to the system and Δ W 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). Δ U is independent of the path of transformation, although Δ Q and Δ W are path- dependent. Correspondingly, in a reversible in fi nitesimal transformation, the in fi nitesimal
3.4. FIRST LAW OF THERMODYNAMICS - INTERNAL ENERGY 25
δ Q and δ W are not exact di ff erentials (in the sense that they do not represent the changes of de fi nite functions of state), but dU = δ Q + δ
W, dU = C
V dT - PdV (3.4) is an exact di ff erential. For the second part of (
3.4) we have used (3.2) and (3.1), namely
that δ Q = C V dT (when the volume V is constant) and that δ W = - PdV . De fi nition 2. Energy cannot be created out of nothing in a closed cycle process: � dU = 0 ⇒ Δ Q = - Δ W .
Statistical mechanics.
The iternal energy U 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 U = E = 1 2 m N � i =1 p 2i + N � i =1 φ ( � r i ) +1 2 N � i � = j V ( � r i - � r j ) , where φ ( � r i ) is the external potential and V ( � r i - � r j ) 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
y x z L
We consider
N molecules, i.e. n = N/N A moles, in a cubic box of side L and volume V = L 3 . A particle hitting a given wall changes its momentum by Δ p x = 2 mv x , Δ t = 2 L/v x where m is the mass, v x the velocity in x - direction and Δ t the average time between collisions. The momentum hence changes on the average as Δ p x Δ t=2mv x 2 L/v x =mv 2x L=mv 2 3
L=23LE
kin , where E kin = m v 2 /
2 is the kinetic energy of the molecule
and v 2 = v 2x + v 2y + v 2z .
Newton's law.
Newton's law, d p /dt = F , tells us that the total force F tot on the wall is 2 NE kin / (3 L ). We then obtain for the pressure P = F tot L 2 =2 3NL 3 E kin , E kin =m 2v 2 .
Assuming the ideal gas relation (
1.3) we find consequently
P = N Vk B T =2 3NL 3 E kin , E kin =3 2k B T (3.5)
26 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS
for the kinetic energy E kin of a single molecule. The internal energy is then, with φ ( � r i ) = V ( � r i - � r j ) = 0, U =3 2Nk B T =3
2nRT =32PV
, (3.6) where R = 8 .
314J/(molK) 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 the fi rst law of thermodynamics.
3.5.1 Isochoric process
An isochoric process is a constant volume pro- cess. We have hence dV = 0 , δ W = 0 , dU | V = δ Q | V , and �δQ dT� V =�dU dT� V = C V (3.7) for the heat capacity at constant volume C V .
Ideal gas.
An ideal gas containing n moles is de fi ned by its equation of state, PV = Nk B T, Nk B = nR . (3.8)
Using the internal energy (
3.6), U = 3 2Nk B T =3
2nRT, dU =32nRdT ,
we obtain C V =3
2nR =32PVT
(3.9) 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 evaluate C P we consider δ Q | P = dU | P + PdV | P , which, under an in fi nitesimal increment of tem- perature, is written as δ Q | P =�∂U ∂
T�
P dT + P�∂V ∂
T�
P dT ≡ C P dT.
The speci
fi c heat at constant pressure, C P =�∂U ∂
T�
P + P�∂V ∂
T�
P , (3.10) reduces then for an ideal gas, for which U = 3 nRT/ 2 and PV = nRT , to C P =3
2nR + nR =52nR = C
V + Nk B . (3.11)
Mayer's relation between
C P and C V . In order to evaluate the partial derivative ( ∂ U/ ∂ T ) P entering the de fi nition (3.10) of the specific heat at constant pressure we note that the equation of state f ( P,V,T ) = 0 determines the interrelation between P , V and T . A constant pressure P de fi nes hence a functional dependence between V and T . We therefore have �∂U ∂
T�
P =�∂U ∂
T�
V ���� = C V �∂T ∂
T�
P ���� = 1+ �∂U ∂
V�
T �∂V ∂
T�
P and hence with C P = C V +� P +�∂U ∂
V�
T ��∂V ∂
T�
P (3.12) the Mayer relation . In Sect.
4.5we will connect the partial derivatives entering (3.12)
with measurable quantities.
3.5.3 Isothermal processes for the ideal gas
An isothermal process takes place at constant temper- ature. The work performed Δ W = - � V 2 V 1 dV P
28 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS
is hence given by the area below P = P ( T,V ) | T . Using the equation-of-state relation PV = nRT of the ideal gas we obtain Δ W = - nRT � V 2 V 1 dV
V= -nRT lnV
2 V 1 = - Δ Q , where the last relation follow from the fi rst law, Δ U = Δ Q + Δ W , and from the fact that the internal energy U = 3 nRT/
2 of the ideal gas remains constant during the isothermal
process. Δ W > 0 for V 1 > V 2 , 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 performed
fi rst 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 gas fi lls both compartments, the fi nal 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, Δ W = 0 , Δ Q = 0 , Δ U = 0 , and internal energy U stay constant
Ideal gas.
The internal energy U = 3 2nRT, of the ideal gas with a contant number n of mols depends only on the temperature T , and not on the volume V . The kinetic energy E kin = 3 k B T/
2 of the constitutent particles is
not contingent on the enclosing volume. We hence have �∂T ∂
V�
U = 0 , T 2 = T 1 . Which means that for the ideal gas the free expansion is an isothermal expansion, in agreement with Joule's fi ndings.
3.6. ENTHALPY29
3.5.5 Adiabatic processes for the ideal gas
An adiabatic process happens without heat transfer: δ Q = 0 , dU = - PdV , where the second relation follows from the fi rst law of thermodynamics, dU = δ Q + δ W .
For the ideal gas we have
PV = nRT , U = 3 nRT/ 2 = 3 PV/
2 and hence
dU = 3
2�
PdV + V dP� = - PdV, 5
2dVV= -32dPP
, which can be solved as γ log�V V 0 � = log�P 0
P�
, PV γ = const., γ = 5/3 .
Using the ideal gas equation of state,
PV = nRT , we may write equivalently TV γ - 1 = const.. Since γ > 1, an adiabatic path has a steeper slope than an isotherm in a P - V diagram.
3.6 Enthalpy
The internal energy is a continuous di
ff erentiable state function for which the relations dU = δ Q - PdV, δ Q = C V dT, C V =�∂U ∂
T�
V hold. One can de fi ne equivalently with H ≡ U + PV,(3.13) a state function H , denote the enthalpy , which obeys dH = δ Q + V dP, δ Q = C P dT, C P =�∂H ∂
T�
P . (3.14)
Note that
P , V and T determine each others in pairs via the equation-of-state function f ( P,V,T ) = 0.
Derivation.
We have dH = dU + PdV + V dP = δ Q - PdV + PdV + V dP = δ Q + V dP , in accordance with and (
3.14), and�∂H
∂
T�
P =�∂(U + PV ) ∂
T�
P =�∂U ∂
T�
P + P�∂V ∂
T�
P = C P in agreement with the de fi nition (3.10) of the specific heat C 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 magnetization M and/or a magnetic fi eld H is present. - H : magnetic fi eld (intensive, generated by external currents) - M : magnetization (extensive, produced by ordered local moments)
Magnetic work.
The magnetic work done on the system is H dM , as derived in electro- dynamics. The modi fi ed fi rst law of thermodynamics then takes the form, dU = δ Q + δ W, δ W = H dM , (3.15) when the volume V is assumed to be constant. All results previously for the PV T system can be written into HMT variables when using H ↔ - P, M ↔ V .
Susceptibility.
A magnetic fi eld H induces in general a magnetization density M/V , which is given for a paramagnetic substance by Curie's law M
V= χ(T)H
, χ ( t ) =c 0 T. χ = χ ( T ) is denoted the magnetic susceptibility .
Phase transitions.
Ferromagnetic systems order spontaneously below the Curie temper- ature T c , becoming such a permanent magnet with a fi nite magnetization M . The phase transition is washed-out for any fi nite fi eld H � = 0, which induces a fi nite magnetization at all temperatures.
Hysteresis.
3.7. MAGNETIC SYSTEMS31
Impurities and lattice imperfections induce magnetic do- mains, which are then stabilized by minimizing the mag- netic energy of the surface fi elds. The resulting domain walls may move in response to the change of H . This is a dissipative process which leads to hysteresis.
32 CHAPTER 3. WORK, HEAT AND THE FIRST LAW OF THERMODYNAMICS
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