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Chapter 3: The Second Law
The Direction of Spontaneous Change
•Spontaneous: the direction of change that does not require work to bring it about.
•Second Law of Thermodynamics: no process is possible in which the sole result is the
absorption of heat from a reservoir and its complete conversion into work (some
energy is discarded and not converted into work).
•During a spontaneous change in an isolated system, the total energy is dispersed
into random thermal motion of the particles in the system.
•The First Law uses the internal energy to identify permissible changes while the
Second Law uses the entropy to identify the spontaneous changes among those
•Entropy: a measure of the energy dispersed in a process; acts as a signpost of
oThe entropy of an isolated system increase in the course of a spontaneous
change: ∆Stot > 0.
oThermodynamically irreversible processes are spontaneous processes.
odS = dqrev/T, where qrev is the heat supplied reversibly.
oEntropy change of a perfect gas when it expands isothermally from Vi to Vf: ∆S
oEntropy Change of the Surroundings: ∆Ssurr = qsurr/Tsurr
oFor any adiabatic change, qsurr = 0, so ∆Ssurr = 0.
oBoltzmann Formula for the Entropy: S = klnW, where k = 1.381 x 10-23 J/K and
W is the number of microstates, ways in which the molecules of a system can
be arranged while keeping the total energy constant (more disorderly
distribution, more microstates).
Entropy is related to dispersal of energy.
The more microstates or more dispersal, the higher the entropy.
Heating will increase the number of accessible energy levels and W will
The change in entropy should be inversely proportional to the
temperature at which the transfer takes place (high temperature,
oEntropy is a state function so ʃ(dqrev/Tsurr) = 0 around a close path.
Reversible isothermal expansion from A to B at Th; the entropy change
is qh/Th, where qh is the energy supplied to the system as heat from the
Reversible adiabatic expansion from B to C. No energy leaves the
system as heat, so the change in entropy is zero. In the course of
expansion, the temperature falls from Th to Tc, the temperature of the
Reversible isothermal compression from C to D at Tc. Energy is
released as heat to a cold sink; change in entropy of the system is
qc/Tc; in this expression, qc is negative.
Reversible adiabatic compression from D to A. No energy enters the
system as heat, so the change in entropy is zero. The temperature
rises from Tc to Th.
oEfficiency (ɳ) = work performed/heat absorbed from hot source = |w|/|qh|
The greater the work output for a given supply of heat from the hot
reservoir, the grater is the efficiency of the engine.
Carnot Efficiency: ɳ = 1 – Tc/Th
All reversible engines have the same efficiency regardless of their
•Thermodynamic Temperature Scale: constructing an engine in which the hot source
is at a known temperature and the cold sink is the object of interest (T = (1 - ɳ)Th).
•Kelvin Scale: using water at its triple point as the notional hot source and defining
that temperature as 273.16 exactly.
•Clausius Inequality: dS ≥ dq/T
oIf the system is isolated from its surroundings, so that dq = 0, the Clausius
inequality implies that dS ≥ 0.
oIn an isolated system, the entropy cannot decrease when a spontaneous
•Entropy Change for the Isothermal Expansion of a Perfect Gas: ∆S = nR ln(Vf/Vi) and it
applies whether the change of state occurs reversibly or irreversibly.
oThe entropy of a perfect gas increases when it expands isothermally.
o∆Stot = 0 for a reversible process and ∆Stot > 0 for an irreversible process.
•The change in entropy of a substance accompanying a change of state at its
transition temperature is calculated from its enthalpy of transition.
oNormal Transition Temperature (Ttrs): the temperature at which two phases
are in equilibrium at 1 atm.
oEntropy of Phase Transition: ∆trsS = ∆trsH/Ttrs
If the phase transition is exothermic (∆trsH < 0), then the entropy
change of the system is negative (increasing order from solid to liquid
or from liquid to gas).
If the phase transition is endothermic (∆trsH > 0), then the entropy
change of the system is positive, which is consistent with dispersal of
matter in the system.
oTrouton’s Rule: a wide range of liquids give approximately the same standard
entropy of vaporization of about 85 J/K mol because a comparable change in
volume occurs when any liquid evaporates and becomes a gas.
•The increase in entropy when a substance is heated is expressed in terms of its heat
oEntropy Variation with Temperature: S(Tf) = S(Ti) + ʃTfTi (CpdT/T) = S(Ti) + Cp
ln(Tf/Ti) when Cp is independent of temperature in the range of interest.
•The entropy of a substance at a given temperature is determined from
measurements of its heat capacity from T = 0 up to the temperature of interest,
allowing for phase transitions in that range.
•Third Law of Thermodynamics: the entropy of all perfect crystalline substances is
zero at T = 0.