Second Law of Thermodynamics. The Second Law of Thermodynamics Introduction Heat Engines Reversible...

Second Law of Thermodynamics

• The Second Law of Thermodynamics—Introduction

• Heat Engines

• Reversible and Irreversible Processes; the Carnot Engine

• Refrigerators, Air Conditioners, and Heat Pumps

• Entropy

• Entropy and the Second Law of Thermodynamics

• Order to Disorder

• Unavailability of Energy; Heat Death

• Statistical Interpretation of Entropy and the Second Law

• Thermodynamic Temperature; Third Law of Thermodynamics

• Thermal Pollution, Global Warming, and Energy Resources

The first law of thermodynamics tells us that energy is conserved. However, the absence of the process illustrated above indicates that conservation of energy is not the whole story. If it were, movies run backwards would look perfectly normal to us!

The Second Law of Thermodynamics—Introduction

The second law of thermodynamics is a statement about which processes occur and which do not. There are many ways to state the second law; here is one:

Heat can flow spontaneously from a hot object to a cold object; it will not flow spontaneously from a cold object to a hot object.

The Second Law of Thermodynamics—Introduction

It is easy to produce thermal energy using work, but how does one produce work using thermal energy?

This is a heat engine; mechanical energy can be obtained from thermal energy only when heat can flow from a higher temperature to a lower temperature.

Heat Engines

We will discuss only engines that run in a repeating cycle; the change in internal energy over a cycle is zero, as the system returns to its initial state.

The high-temperature reservoir transfers an amount of heat QH to the engine, where part of it is transformed into work W and the rest, QL, is exhausted to the lower temperature reservoir. Note that all three of these quantities are positive.

Heat Engines

A steam engine is one type of heat engine.

Heat Engines

The internal combustion engine is a type of heat engine as well.

Heat Engines

Why does a heat engine need a temperature difference?

Otherwise the work done on the system in one part of the cycle would be equal to the work done by the system in another part, and the net work would be zero.

Heat Engines

The efficiency of the heat engine is the ratio of the work done to the heat input:

Using conservation of energy to eliminate W, we find:

Heat Engines

Heat Engines

Car efficiency.

An automobile engine has an efficiency of 20% and produces an average of 23,000 J of mechanical work per second during operation.

(a) How much heat input is required, and

(b) How much heat is discharged as waste heat from this engine, per second?

Heat EnginesNo heat engine can have an efficiency of 100%. This is another way of writing the second law of thermodynamics:

No device is possible whose sole effect is to transform a given amount of heat completely into work.

The Carnot engine was created to examine the efficiency of a heat engine. It is idealized, as it has no friction. Each leg of its cycle is reversible.

The Carnot cycle consists of:

• Isothermal expansion

• Adiabatic expansion

• Isothermal compression

• Adiabatic compression

Reversible and Irreversible Processes; the Carnot Engine

From this we see that 100% efficiency can be achieved only if the cold reservoir is at absolute zero, which is impossible.

Real engines have some frictional losses; the best achieve 60–80% of the Carnot value of efficiency.

For an ideal reversible engine, the efficiency can be written in terms of the temperature:

Reversible and Irreversible Processes; the Carnot Engine

Reversible and Irreversible Processes; the Carnot Engine

A phony claim?

An engine manufacturer makes the following claims: An engine’s heat input per second is 9.0 kJ at 435 K. The heat output per second is 4.0 kJ at 285 K. Do you believe these claims?

Reversible and Irreversible Processes; the Carnot Engine

Automobiles run on the Otto cycle, shown here, which is two adiabatic paths alternating with two constant-volume paths. The gas enters the engine at point a and is ignited at point b. Curve cd is the power stroke, and da is the exhaust.

Reversible and Irreversible Processes; the Carnot Engine

The Otto cycle.

(a) Show that for an ideal gas as working substance, the efficiency of an Otto cycle engine is

e = 1 – (Va/Vb)1-γ

where γ is the ratio of specific heats (γ = CP/CV) and Va/Vb is the compression ratio.

(b) Calculate the efficiency for a compression ratio Va/Vb = 8.0 assuming a diatomic gas like O2 and N2.

These appliances are essentially heat engines operating in reverse.

By doing work, heat is extracted from the cold reservoir and exhausted to the hot reservoir.

Refrigerators, Air Conditioners, and Heat Pumps

Refrigerators, Air Conditioners, and Heat Pumps

This figure shows more details of a typical refrigerator.

Refrigerator performance is measured by the coefficient of performance (COP):

Refrigerators, Air Conditioners, and Heat Pumps

Substituting:

For an ideal refrigerator,

Refrigerators, Air Conditioners, and Heat Pumps

Making ice.

A freezer has a COP of 3.8 and uses 200 W of power. How long would it take this otherwise empty freezer to freeze an ice-cube tray that contains 600 g of water at 0°C?

A heat pump can heat a house in the winter:

Refrigerators, Air Conditioners, and Heat Pumps

Refrigerators, Air Conditioners, and Heat Pumps

Heat pump.

A heat pump has a coefficient of performance of 3.0 and is rated to do work at 1500 W.

(a) How much heat can it add to a room per second?

(b) If the heat pump were turned around to act as an air conditioner in the summer, what would you expect its coefficient of performance to be, assuming all else stays the same?

Entropy

Definition of the change in entropy S when an amount of heat Q is added:

if the process is reversible and the temperature is constant. Otherwise,

Entropy

Any reversible cycle can be written as a succession of Carnot cycles; therefore, what is true for a Carnot cycle is true of all reversible cycles.

EntropySince for any Carnot cycle QH/TH + QL/TL = 0, if we approximate any reversible cycle as an infinite sum of Carnot cycles, we see that the integral of dQ/T around a closed path is zero. This means that entropy is a state variable—the change in its value depends only on the initial and final states.

Entropy and the Second Law of Thermodynamics

Entropy change when mixing water.

A sample of 50.0 kg of water at 20.00°C is mixed with 50.0 kg of water at 24.00°C. Estimate the change in entropy.

Entropy and the Second Law of Thermodynamics

The total entropy always increases when heat flows from a warmer object to a colder one in an isolated two-body system. The heat transferred is the same, and the cooler object is at a lower average temperature than the warmer one, so the entropy gained by the cooler one is always more than the entropy lost by the warmer one.

Entropy and the Second Law of Thermodynamics

Entropy changes in a free expansion.

Consider the isothermal expansion of n moles of an ideal gas from volume V1 to volume V2, where V2 > V1. Calculate the change in entropy

(a) of the gas and

(b) of the surrounding environment.

(c) Evaluate ΔS for 1.00 mole, with V2 = 2.00 V1.

Entropy and the Second Law of Thermodynamics

Heat transfer.

A red-hot 2.00-kg piece of iron at temperature T1 = 880 K is thrown into a huge lake whose temperature is T2 = 280 K. Assume the lake is so large that its temperature rise is insignificant. Determine the change in entropy

(a) of the iron and

(b) of the surrounding environment (the lake).

Entropy and the Second Law of Thermodynamics

The fact that after every interaction the entropy of the system plus the environment increases is another way of putting the second law of thermodynamics:

The entropy of an isolated system never decreases. It either stays constant (reversible processes) or increases (irreversible processes).

Entropy is a measure of the disorder of a system. This gives us yet another statement of the second law:

Natural processes tend to move toward a state of greater disorder.

Example: If you put milk and sugar in your coffee and stir it, you wind up with coffee that is uniformly milky and sweet. No amount of stirring will get the milk and sugar to come back out of solution.

Order to Disorder

Another example: When a tornado hits a building, there is major damage. You never see a tornado approach a pile of rubble and leave a building behind when it passes.

Thermal equilibrium is a similar process—the uniform final state has more disorder than the separate temperatures in the initial state.

Order to Disorder

Another consequence of the second law:

In any natural process, some energy becomes unavailable to do useful work.

If we look at the universe as a whole, it seems inevitable that, as more and more energy is converted to unavailable forms, the ability to do work anywhere will gradually vanish. This is called the heat death of the universe.

Unavailability of Energy; Heat Death

Statistical Interpretation of Entropy and the Second Law

Microstate: a particular configuration of atomsMacrostate: a particular set of macroscopic variablesThis example uses coin tosses:

Statistical Interpretation of Entropy and the Second Law

Similarly, the most probable distribution of velocities in a gas is Maxwellian:

The most probable state is the one with the greatest disorder, or the greatest entropy. With k being Boltzmann’s constant and W the number of microstates,

Statistical Interpretation of Entropy and the Second Law

Free expansion—statistical determination of entropy.

Determine the change in entropy for the adiabatic free expansion of one mole of a gas as its volume doubles. Assume W, the number of microstates for each macrostate, is the number of possible positions.

Statistical Interpretation of Entropy and the Second Law

In this form, the second law of thermodynamics does not forbid processes in which the total entropy decreases; it just makes them exceedingly unlikely.

Thermodynamic Temperature; Third Law of Thermodynamics

Since the ratio of heats exchanged between the hot and cold reservoirs in a Carnot engine is equal to the ratio of temperatures, we can define a temperature scale using the triple point of water:

T = (273.16K)(Q/Qtp).

Here, Q and Qtp are the heats exchanged by a Carnot engine with reservoirs at temperatures T and Ttp.

Thermodynamic Temperature; Third Law of Thermodynamics

there is no way to achieve a temperature of absolute zero. This is the third law of thermodynamics:

It is not possible to reach absolute zero in any finite number of processes.

Also, since the maximum efficiency of a heat engine is

Summary• Heat engine changes heat into useful work

• Efficiency: work/heat input

• Maximum efficiency: 1 – TL/TH

• Refrigerators and air conditioners are heat engines, reversed; COP = heat removed/work

• Heat pump: COP = heat delivered/work

• Second law of thermodynamics: Natural processes always tend to increase entropy

Summary

• Change in entropy gives direction to “arrow of time”

• As time goes on, energy becomes degraded.

• Heat engines cause thermal pollution.

• Entropy change in reversible process: