doPhysics · Thermal Physics 2
Heat Engines, the Second Law and Entropy
Engines that turn heat into work in a cycle and must reject heat to a sink. The second law in the forms of Kelvin-Planck and Clausius, reversible and irreversible processes, the Carnot cycle and its limit \(\eta = 1 - T_C/T_H\), refrigerators, air conditioners and heat pumps, entropy and its growth, and the T-S diagram.
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Why this chapter
Why this chapter exists
A coal power station burns coal to make electricity, yet it throws away nearly two thirds of the heat through its cooling towers. A refrigerator makes food cold and warms the kitchen. Hot chai cools, and never warms itself again. The first law allows all of these to run either way. This chapter meets the law that decides the direction: the second law of thermodynamics. It sets a hard ceiling on every engine and a price on every refrigerator. And it leads to entropy, the one quantity that grows in every change that cannot be undone.
You will learn
By the end of this chapter you will
- Describe a heat engine by its source, working substance and sink, and find its efficiency, \(\eta = W/Q_H = 1 - Q_C/Q_H\).
- State the second law in the Kelvin-Planck and Clausius forms, show that they are equivalent, and tell reversible processes from irreversible ones.
- Follow a Carnot cycle leg by leg, derive \(\eta = 1 - T_C/T_H\), and use Carnot's theorem to judge any engine.
- Find the coefficient of performance of a refrigerator, an air conditioner and a heat pump, and link it to the efficiency of the same cycle.
- Define entropy by \(dS = dQ_{\text{rev}}/T\) and find its change for melting, heating and any process of an ideal gas.
- Show that the entropy of the universe grows in heat flow, free expansion and mixing, and state the Clausius inequality.
- Read heat and work from a T-S diagram, and find the Carnot efficiency from entropy.
Before you begin
Before you begin
From the chapter on the First Law you should know the first law, \(Q = \Delta U + W\), and the work \(nRT\ln(V_2/V_1)\) of an isothermal expansion. You should know \(TV^{\gamma - 1} = \text{constant}\) on an adiabat, and how to read a cycle on a p-V diagram. From kinetic theory you should know \(U = nC_vT\) for an ideal gas and \(C_p = C_v + R\). From calorimetry you should know \(Q = mc\,\Delta T\) and \(Q = mL\). You should be able to integrate \(1/x\).
Lessons
1
Heat Engines and Their Efficiency
A heat engine takes heat from a hot source, turns part of it into work in a cycle, and must reject the rest to a cold sink. Its efficiency is \(W/Q_H = 1 - Q_C/Q_H\). A coal station reaches about 35 %, while the water turbines at Bhakra reach 90 %, because they are not heat engines at all.
What a heat engine needsThermal efficiencyReal engines and the water turbine
2
The Second Law of Thermodynamics
No engine working in a cycle turns all its heat into work (Kelvin-Planck), and heat never flows from cold to hot by itself (Clausius). The two statements stand or fall together. Friction, free expansion and heat across a temperature gap make every real process irreversible.
Kelvin planck statementClausius statementThe two statements agreeReversible and irreversible processes
3
The Carnot Cycle
Two isotherms and two adiabats make the ideal, reversible engine. For it the heats are in the ratio of the absolute temperatures, so \(\eta = 1 - T_C/T_H\), whatever the working substance. Carnot's theorem makes this the ceiling for every engine between two temperatures.
Four legs of the carnot cycleCarnot efficiencyCarnots theorem
4
Refrigerators and Heat Pumps
A heat engine run backwards uses work to carry heat from cold to hot. Its coefficient of performance is \(K = Q_C/W\), at best \(T_C/(T_H - T_C)\). The same machine warms a house as a heat pump, with a heating coefficient \(Q_H/W = K + 1\).
The refrigeratorCoefficient of performanceHeat pumps and air conditioners
5
Entropy
Entropy changes by \(dS = dQ_{\text{rev}}/T\), so heat taken in at a low temperature carries more entropy than the same heat at a high one. Entropy is a state function. Melting gives \(\Delta S = mL/T\), heating gives \(mc\ln(T_2/T_1)\), and an ideal gas has \(\Delta S = nC_v\ln(T_2/T_1) + nR\ln(V_2/V_1)\).
Entropy definedEntropy is a state functionEntropy of melting and heatingEntropy change of an ideal gas
6
Entropy, Irreversibility and T-S Diagrams
In every irreversible change the entropy of the universe grows; in a reversible one it stays the same. Heat flow, free expansion and mixing all raise it. On a T-S diagram the area under a path is the heat, the Carnot cycle is a rectangle, and its efficiency follows in two lines.
Entropy grows in irreversible changeClausius inequalityT s diagramsCarnot efficiency from entropy
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- Carnot cycle
- The reversible cycle of two isotherms and two adiabats. It is an isothermal expansion at \(T_H\), an adiabatic expansion to \(T_C\), an isothermal compression at \(T_C\) and an adiabatic compression back to \(T_H\).
- Carnot engine
- An engine running on the Carnot cycle. Its efficiency is \(1 - T_C/T_H\), the highest possible between two reservoirs.
- Carnot's theorem
- No engine between two reservoirs is more efficient than a reversible one. All reversible engines between the same two reservoirs have the same efficiency, \(1 - T_C/T_H\), whatever the working substance.
- Clausius inequality
- For any cycle \(\oint dQ/T \le 0\), with equality only for a reversible cycle.
- Clausius statement
- No device working in a cycle can move heat from a colder body to a hotter body with no other effect.
- coefficient of performance
- The measure of a refrigerator, \(K = Q_C/W\), also written \(\beta\). For a Carnot refrigerator \(K = T_C/(T_H - T_C)\).
- cyclic process
- A series of processes that brings a system back to its starting state. Over a cycle \(\Delta U = 0\), so the net heat taken in equals the net work done.
- entropy
- A property of a system whose change is \(dS = dQ_\text{rev}/T\), measured in J/K. It is a state function, and the entropy of an isolated system never decreases.
- heat engine
- A device that takes a working substance round a cycle. In each cycle it takes heat \(Q_H\) from a hot reservoir, does work \(W\) and gives heat \(Q_C\) to a cold reservoir, with \(Q_H = W + Q_C\).
- heat pump
- A refrigerator used to warm a space by delivering the heat \(Q_H\) to it. Its heating coefficient is \(Q_H/W = K + 1\).
- irreversible process
- A process that cannot be undone without leaving a change somewhere, such as friction, a free expansion or heat flowing across a finite temperature difference. Every real process is irreversible.
- isentropic process
- A process at constant entropy. A reversible adiabatic process is isentropic, and it is a vertical line on a T-S diagram.
- Kelvin-Planck statement
- No engine working in a cycle can take heat from a single reservoir and turn all of it into work.
- perpetual motion machine of the second kind
- An imagined engine that takes heat from a single reservoir, such as the sea, and turns all of it into work. It would conserve energy but break the second law, so it cannot be built.
- refrigerator
- A heat engine run backwards: work \(W\) carries heat \(Q_C\) out of a cold space and delivers \(Q_H = Q_C + W\) to warmer surroundings.
- reversible process
- A process that can be run backwards so that the system and its surroundings both return exactly to their first states. It must be quasi-static and free of friction, with heat crossing only vanishingly small temperature differences.
- second law of thermodynamics
- The law that fixes the direction of natural processes. No cyclic engine turns all its heat into work (Kelvin-Planck), and heat never flows from cold to hot by itself (Clausius). The entropy of an isolated system never decreases.
- state function
- A property that depends only on the present state of a system, not on how it got there, such as \(U\), \(T\) or \(S\). Its change round any cycle is zero.
- T-S diagram
- A graph of temperature against entropy. The area under a reversible path is the heat taken in. An isotherm is horizontal, a reversible adiabat is vertical, and the Carnot cycle is a rectangle.
- thermal efficiency
- The fraction of the heat taken in by an engine that becomes work, \(\eta = W/Q_H = 1 - Q_C/Q_H\). It is always less than 1.
- thermal reservoir
- A body so large that taking heat from it or giving heat to it does not change its temperature. A furnace kept burning, a river and the air are examples. It serves as the source or the sink of an engine.
- working substance
- The material, usually a gas or steam, that an engine takes round its cycle and that does the work.