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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

permissible changes.

•Entropy: a measure of the energy dispersed in a process; acts as a signpost of

spontaneous change.

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

= nRlnVf/Vi

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

increase.

The change in entropy should be inversely proportional to the

temperature at which the transfer takes place (high temperature,

higher entropy).

oEntropy is a state function so ʃ(dqrev/Tsurr) = 0 around a close path.

oCarnot Cycle

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

hot source.

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

cold sink.

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

construction.

•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

change occurs.

•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

capacity.

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.