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CHAPTER 5Entropy and the Second and Third Laws of Thermodynamics

Key Points

• Entropy, S, is a state function that predicts the direction of natural, or spontaneous, change.

• Entropy increases for a spontaneous change in an isolated system.

• Variation of S with P, V, and T: conditions for spontaneity

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The Universe Has a Natural Direction of Change• First law of thermodynamics, U = q + w, dictates energy

conservation of a process. It does not specify the direction of the change.

• Spontaneous process: natural transformation that will occur with a high probability, and is consistent with common sense

• Entropy is a measure of the number (W) of quantum states accessible to a macroscopic system with a given energy, S = k lnW (with k being the Boltzmann’s constant)

• Entropy of an isolated system is maximized at equilibrium

Heat Engines and the Second Law of Thermodynamics• The concept of S arose as 19th century scientists

attempted to maximize the work output of an engine.

• Such a heat engine is represented below involving an ideal gas confined in a piston and cylinder assembly that is in contact with a hot reservoir (Thot) and a cold sink (Tcold)

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Conversion Asymmetry

• Work (e.g., electrical work) may be converted to heat at 100% efficiency. What is the maximum efficiency of heat converted into work?

• Maximum work output in an isothermal expansion occurs in a reversible process. Thus, the efficiency of a reversible engine is an upper bound to the efficiency of a real engine.

• The reversible engine converts heat into work by exploiting the spontaneous tendency of heat to flow from a hot reservoir to a cold sink.

Cycle of a Reversible Heat Engine• Two isothermal processes + two adiabatic processes

• Heat is taken up by the engine in the first step and released to the surroundings in the third step

• Work is done on the surroundings in the first two steps and on the system in the last two steps

Carnot cycle

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Nicolas Léonard Sadi Carnot

• Carnot (1 June 1796 – 24 August 1832) was a French military engineer and physicist, often described as the "father of thermodynamics".

Carnot Cycle

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Carnot Cycle

• -qab = wcycle + qcd. That is, not all heat withdrawn from the higher temperature reservoir is converted to work done by the system (engine) on the surroundings

• The efficiency of the reversible Carnot engine

Second Law of Thermodynamics

• Kevin-Planck formulation: it is impossible for a system to undergo a cyclic process whose sole effects are the flow of heat into the system from a heat reservoir and the performance of an equal amount of work by the system on the surroundings

Cannot be constructed

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Indicator Diagram

• For an engine to produce work, the area of the cycle in a P-V diagram must be greater than zero

Carnot Cycle

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Introduction of Entropy, S

• Because the cyclic integral of is zero, this quantity must be the exact differential of a state function, which is defined as entropy (S)

Change of Entropy

• S is a state function

• S must be calculated along a reversible path

• For an irreversible path, S must be calculated for a reversible process that proceeds between the same initial and final states corresponding to the irreversible process

• For a reversible isothermal expansion/compression

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Ideal Gas undergoes a reversible change of T

Constant V Constant P

Vi, Ti → Vf, Tf Pi, Ti → Pf, Tf

S for Phase Change

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S for an Arbitrary Process- real gases, solids, and liquids

Vi, Ti → Vf, Tf

Pi, Ti → Pf, Tf

Using S to Calculate the Natural Direction of a Process in an Isolated System

• Because T1 > T2, if qP > 0, S < 0

• If qP < 0, S > 0

Heat gained by the left = lost by the right

The process in which S increases is the direction of natural change in an isolated system

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Direction of Change

• For any irreversible process in an isolated system, there is a unique direction of spontaneous change: • S > 0 for the spontaneous process

• S < 0 for the opposite or nonspontaneous direction of change

• S = 0 only for a reversible process.

• In a quasi-static reversible process, there is no direction of spontaneous change because the system is proceeding along a path, each step of which corresponds to an equilibrium state

S > 0 is a criterion for spontaneous change only if the system does not exchange energy in the form of heat or work with its surroundings

S and U

• In an isolated system, U cannot be created or destroyed(energy conservation)

• In an isolated system, if any process occurs, it is by definition spontaneous and therefore S can be created (S > 0), but not destroyed.

• The universe is expanding…

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Clausius Inequality

• Consider the first law in which only P-V work is involved,

which is valid for both reversible and irreversible processes

• If the process is reversible

Which is valid for both reversible and irreversible processes because U is a state function, as long as there are no phase transformations or chemical reactions, and only P-V work involved.

Clausius Inequality

• If P – Pexternal > 0, the system will spontaneously expand, and dV > 0

• If P – Pexternal < 0, the system will spontaneously contract, and dV > 0

• In both cases, (P – Pexternal)dV > 0 The equal sign hold only for a

reversible process

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Clausius Inequality

• For any irreversible process in an isolated system, S > 0

• This will be used to define two new state functions, the Gibbs free energy (G) and Helmhotz free energy (A), which predict the direction of change in process for which the system interacts with the surroundings (Chapter 6)

• S is a state function, so

Change of Entropy in the Surroundings

• The statement that “a process is spontaneous if S for the system is positive” is true only for an isolated system

• In general, a system is interacting only with the part of the universe that is very close. • One can combine the system with this part of the universe into an

interacting composite system that is isolated from the rest of the universe

• This part of the surroundings is a thermal reservoir at a fixed T (because of its large mass)

• The criterion for spontaneous change is Stotal = S + Ssurroundings > 0, which defines a unique direction of time

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Change of Entropy in the Surroundings• If the system is surrounded by adiabatic walls, qsurroundings = 0

• If the surroundings are at constant V, qsurroundings = Usurroundings

• If the surroundings are at constant P, qsurroundings = Hsurroundings

• The amount of heat entering the surroundings is independent of the path; and is the same whether the transfer occurs reversibly or irreversibly

Absolute Entropy and Third Law of Thermodynamics

• Third Law of Thermodynamics: the entropy of a pure, perfectly crystalline substance (element or compound) is zero at zero Kelvin

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Third Law of Thermodynamics

• In a perfect crystal, individual atoms are indistinguishable, exchanging the positions of two atoms does not lead to a new state. That is, there is only one state at zero Kelvin, and S = klnW = kln1 = 0

• This allows the calculation of absolute entropy at temperature T

Absolute Entropy

• Sm(gas) > Sm(liquid) > Sm(solid)

• Sm increases with the size of the molecule because the number of degrees of freedom increases with the number of atoms

• Sm(solid) is larger for weekly bound solids than for strongly bound solids at low and moderate temperatures

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Standard State in Entropy Calculation

• For an ideal gas at constant T,

• Choose Pi = P = 1 bar,

Entropy Change in Chemical Reactions

• ∆ is equal to the difference in the entropies of products and reactants

• Temperature dependence