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Chapter 3
Work, heat and the first 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 ,
where a is the active area of the piston. In equilibrium, the external force F is related to pressure P as
F = −P a.
For an infinitesimal process, the change of the position of the wall by dx results in per- forming work δW :
δW = F dx = −P dV, δW = −P adx. (3.1)
For a transformation of the system along a finite reversible path in the equation-of-state space (viz for a process with finite change of volume), the total work performed is
ΔW = −
� V 2
V (^1)
P dV.
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 connecting A (V 1 ) and B (V 2 ).
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 = −
P dV.
3.2 Heat
Energy is transferred in a system in the from of heat when no mechanical work is exerted, viz when δW = −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. a flame.
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 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 from A to B is heat.
Heat capacity. If a system absorbs an amount of heat ΔQ, its temperature rises pro- portionally by an amount ΔT :
ΔQ = CΔT. (3.2)
The proportionality constant C is the heat capacity of the substance (W¨armekapazit¨at). It is an extensive quantity.
Specific 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.
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 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 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 differentials, viz that (^) �
δQ �= 0.
3.4 First law of thermodynamics – internal energy
The first 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 energy U , whose change ΔU in any thermodynamic transformation is given by
ΔU = ΔQ + ΔW +... , (3.3)
where ΔQ is heat transferred to the system and ΔW 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, although ΔQ and ΔW are path- dependent. Correspondingly, in a reversible infinitesimal transformation, the infinitesimal
3.4. FIRST LAW OF THERMODYNAMICS – INTERNAL ENERGY 25
δQ and δW 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 (^) V dT − P dV (3.4)
is an exact differential. For the second part of (3.4) we have used (3.2) and (3.1), namely that δQ = CV dT (when the volume V is constant) and that δW = −P dV.
Definition 2. Energy cannot be created out of nothing in a closed cycle process: � dU = 0 ⇒ ΔQ = −ΔW.
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 =
2 m
�^ N
i=
p (^2) i +
�^ N
i=
φ(�r (^) i ) +
�^ N
i�=j
V (�r (^) i − �rj ) ,
where φ(�r (^) i ) is the external potential and V (�ri − �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
y
x
z
L
We consider N molecules, i.e. n = N/NA moles, in a cubic box of side L and volume V = L^3. A particle hitting a given wall changes its momentum by
Δp (^) x = 2mv (^) x , Δt = 2L/v (^) x
where m is the mass, vx the velocity in x−direction and Δt the average time between collisions. The momentum hence changes on the average as
Δp (^) x Δt
2 mv (^) x 2 L/v (^) x
mv (^) x^2 L
mv 2 3 L
3 L
E (^) kin ,
where E (^) kin = mv 2 /2 is the kinetic energy of the molecule and v 2 = v (^) x^2 + v (^) y^2 + v (^2) z.
Newton’s law. Newton’s law, dp/dt = F, tells us that the total force Ftot on the wall is 2 N E (^) kin /(3L). We then obtain for the pressure
P =
F (^) tot L 2
N
L 3
E (^) kin , E (^) kin =
m 2
v 2.
Assuming the ideal gas relation (1.3) we find consequently
P =
N
V
k (^) B T =
N
L 3
E (^) kin , E (^) kin =
k (^) B T (3.5)
3.5. ENERGY AND HEAT CAPACITY FOR VARIOUS PROCESSES 27
An isobaric process is a constant pressure pro- cess. In order to evaluate CP we consider
δQ| (^) P = dU | (^) P + P dV | (^) P ,
which, under an infinitesimal increment of tem- perature, is written as
δQ| (^) P =
∂U
∂T
P
dT + P
∂V
∂T
P
dT
≡ C (^) P dT.
The specific heat at constant pressure,
C P =
∂U
∂T
P
+ P
∂V
∂T
P
reduces then for an ideal gas, for which U = 3nRT /2 and P V = nRT , to
C P =
nR + nR =
nR = CV + N k (^) B. (3.11)
Mayer’s relation between CP and C (^) 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 state f (P, V, T ) = 0 determines the interrelation between P , V and T. A constant pressure P defines hence a functional dependence between V and T. We therefore have (^) � ∂U ∂T
P
∂U
∂T
� �� V�
= C V
∂T
∂T
� �� P�
∂U
∂V
T
∂V
∂T
P
and hence with
C (^) P = C (^) V +
P +
∂U
∂V
T
∂V
∂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 isothermal process takes place at constant temper- ature. The work performed
ΔW = −
� 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 P V = nRT of the ideal gas we obtain
ΔW = −nRT
� V 2
V (^1)
dV V
= −nRT ln
V 2
V 1
= −ΔQ ,
where the last relation follow from the first law, ΔU = ΔQ + ΔW , and from the fact that the internal energy U = 3nRT /2 of the ideal gas remains constant during the isothermal process. ΔW > 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 first 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 fills 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 energy U stay constant
Ideal gas. The internal energy
U =
nRT,
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 Ekin = 3k (^) 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 findings.
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 field H is present.
- H: magnetic field (intensive, generated by external currents)
- M : magnetization (extensive, produced by ordered local moments)
Magnetic work. The magnetic work done on the system is HdM , as derived in electro- dynamics. The modified first law of thermodynamics then takes the form,
dU = δQ + δW, δW = HdM , (3.15)
when the volume V is assumed to be constant. All results previously for the P V T system can be written into HM T variables when using
H ↔ −P, M ↔ V.
Susceptibility. A magnetic field 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 finite magnetization M. The phase transition is washed-out for any finite field H �= 0, which induces a finite magnetization at all temperatures.
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 surface fields. The resulting domain walls may move in response to the change of H. This is a dissipative process which leads to hysteresis.
