Ch. 12 Thermodynamics

 

Internal energy  all the energy of a stationary system (chemical, nuclear, kinetic, etc.)

Thermal energy  That part of the internal energy that changes T

 

Gases  translational, vibrational, rotational all contribute to thermal energy

            (monatomic gases have only translational)

 

Ways to change internal energy of gas system:

            Heat Q

            Work W

            Both!

 

State: the values of P, V, T, U of the system at a given time (PVTU  “state variables”)

Change of State: altering P, V, T, and/or U

 

12.1 Work in Thermodynamic Processes

 

Compress gas with (approximately) constant pressure P:

 

 

 

Work has been done on the gas to compress it to a smaller volume.

 

If compressed gas expands against the piston, work is done BY the gas (W  is then negative).

 

PV Diagrams & “PV Work”

In above case, P=constant  - “isobaric”

 

The amount of work involved is equal to the area under the curve. In compression,  is negative and the work done on the gas,  will be positive.

 

If the process went the other way,  is positive, and the work done ON the gas  is negative, i.e. work is actually done BY the gas.

 

Other possible cases:

 

Decrease V at constant P

Then increase P

Increase P at constant V

Then decrease V

P and V both change simultaneously

 

Note: the work in (b) is larger than in (a) because the change in V occurred at higher P in (b), so the force required was larger. W depends on the way the system goes from one state to another.

 

How to change P or V without doing W? Add/subtract Q!

 

Example: Problem #6

Sketch a PV diagram of the following processes:

(a)   A gas expands at constant pressure P1 from volume V1 to volume V2. It is then kept at constant volume while the pressure is reduced to P2.

(b)  A gas is reduced in pressure from P1 to P2 while its volume is held constant at V1. It is then expanded at constant pressure to a final volume V2.

(c)   In which process is more work done by the gas? Why?

 

12.2 The First law of Thermodynamics

 

If positive Q is energy transferred to system, positive W is work done on system, then:

 

 

 

Isolated System: where Q and W are ZERO. Here .

 

Cyclic Process:  

 

Note that a cyclic process only requires that the SUM of Q and W is zero, not both Q and W individually. (READ Tip 12.2 regarding the sign convention here).

 

Isothermal Process: another special case where . Here .

 

Example: ideal monatomic gas in cylinder in contact with heat reservoir:

 

Add heat carefully, keeping T constant and allowing V to increase and P to drop.

 

Since we added heat, Q is positive and W is negative. Work is done BY the gas on the outside world.

 

Adiabatic Process: .

 

Example: Ignition phase in internal combustion engine. Hot gas expands air against piston so quickly that little Q has time to be lost.

 

Example: Problem #14

 

A monatomic ideal gas undergoes the thermodynamic process shown in the PV diagram here. Determine whether each of the following values of ΔU, Q, and W for the gas is positive, negative, or zero.

Example: Problem #17

A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. If at the same time 220 J of work is done on the system. Find the energy transferred to or from it by heat.

 

Example: Problem #22

 

One mole of gas is initially at a pressure of 2.00 atm, volume of 0.300 L and has an internal energy equal to 91.0 J. It its final state the gas is at a pressure of 1.50 atm and a volume of 0.800 L and its internal energy equals 180 J. For the paths IAF, IBF, and IF in this figure, calculate: (a) the work done on the gas and (b) the net energy transferred to the gas by heat in the process.

 

12.3 The First Law and Human Metabolism

 

What is important here is not simply the total body heat Q and work W:  but the rate at which they are “performed”:

 

In order to radiate body heat and perform work on the outside world (note that Q and W will be negative, in general), the body’s internal energy must be re-supplied by food and oxygen.

 

Average rate of oxygen used in metabolizing food  1 liter for every 4.8 kcal (= 4.8 Cal = 20 J).

 

 

NOTE: Extreme activity produces a lot of Watts of power, but most individuals cannot do this for very long! You burn up almost as many Calories sleeping 8 hours as you would in 1 hour of heavy activity!

 

Example: Jog 1 mile = burn 120 Calories = 2 slices of bread

 

Efficiency

 

 

 

 

Not everybody can produce as much work for the same intake of energy. The more fit you are, the more efficient you will be at producing power.

 

          Efficiency  

 

 

 

 

 

 

Smooth activity, without a lot of stops & starts (when no W is done but Q is still lost) is more efficient as well.

 

 

12.4 Heat Engines and the Second law of Thermodynamics

 

During heat transfer from a hot reservoir to a cold reservoir, a portion of the randomized kinetic energy of a gas can be converted into directed non-randomized work. Example: coal-fired or gas-fired electrical power plants.

 

Heat Engine  A device that converts part of the internal energy into work.

 

Cyclic Process  Gas returned to its initial state, so that its . So the work done on the engine is , and so the work done by the engine is  

 

 

 

 

 

 

Engine absorbs net  which is converted to work done by the engine .

 

 

 

 

The work done by the engine for a cyclic process is the area enclosed by the cyclic curve in its PV diagram. Think of this as PV work.

 

Thermal Efficiency: the fraction of the heat flowing in that is converted to work in 1 cycle:

 

 

2nd Law of Thermodynamics: It is impossible to construct a heat engine that, operating in a cycle, produces no other effect than the absorption of energy from a reservoir and the performance of an equal amount of work.

 

In essence: it is always true that . No heat engine is 100% efficient.

 

 

Applications: power generators, etc.

 

Reverse process: Heat Pumps. Examples: refrigerators heat pumps for home heating/cooling.

 

 

 

12.5 Reversible & Irreversible Processes

 

Reversible  every state along the path is in equilibrium, and can return to initial state along same path. In practice, needs to be slow & have no “unwanted” losses (such as friction).

 

Irreversible  the real world.

 

 

 

 

 

12.6 The Carnot Engine

 

An idealized engine of maximum efficiency, working in reversible cycle. (Does not really exist).

 

 

 

 

 

 

 

 

 

 

 

 

All real engines are less efficient than the Carnot Engine because they operate irreversibly (due to friction) and because they complete a cycle in a brief time period (are never in a state of equilibrium).

 

Example: Problem#27

 

One of the most efficient engines ever built is a coal-fired steam turbine in the Ohio valley, driving an electric generator as it operates between 1870°C and 430°C.

(a)   What is its maximum theoretical efficiency?

(b) Its actual efficiency is 42%. How much mechanical power does the engine deliver if it absorbs 1.40x105 J of energy each second from the hot reservoir?

 

 

12.7 Entropy

 

For a reversible system, if  is the heat absorbed or expelled by the system:

 

Strictly speaking,  is for a reversible path. For real (irreversible) systems, we must model he process by a reversible one (with the same initial and final states, of course!).

 

The entropy of the Universe increases in all natural processes.

 

Perpetual Motion Machines  ain’t no such thing!

 

1st kind  violate the First Law of Thermodynamics by having  but  (puts out more energy than is put in).

 

2nd kind  violate the Second Law of Thermodynamics by having no heat loss from the system  (having  ).

 

12.8 Entropy and Disorder

 

Observation: A disorderly arrangement is much more probable than an orderly one if the laws of nature are allowed to operate without interference.

 

Isolated (“closed”) systems tend toward greater disorder and entropy is a measure of that disorder.

 

NOTE: within such an isolated system, some parts might experience a decrease in entropy, but only at the expense of an even greater increase in entropy by the rest of the system.

 

Entropy a la Boltzmann:   where  represents the probability of the system having that specific configuration.

 

Example: Drawing colored marbles from a bag with equal numbers of Red & Green:

 

The 2R2G result is most probable, having 6 ways to get that result. Also most disordered.

The second law of thermodynamics is really a statement of what is most probable, rather than of what must be. In terms of entropy & the bag of marbles, sometimes you really will pull 4 red ones in a row!

 

Implicit with our statement that entropy increases during a process is the definition of the “arrow of time”.  One often sees a dropped plate shatter on the floor. One never sees a broken plate self-assemble and jump off the floor into our hand!

 

Degradation of Energy

 

In all real processes, the energy available for doing work decreases. As time progresses, higher-grade energy (that which can do useful work) generally gets transformed into lower-grade energy (that which can do less useful work).

 

Example: Dropping a ball to do work. Mechanically, a ball can do mgh work. But if all the mgh were first transformed into heat, less work could be done, because no heat engine operates with 100% efficiency.

 

Example: Problem #37

 

A 70-kg log falls from a height of 25 m into a lake. If the log, the lake, and the air are all at 300 K, find the change in entropy of the universe for this process.

 

Example: Problem #39

 

The surface of the Sun is at approximately 5700 K, and the temperature of the Earth’s surface is approximately 290 K. What entropy change occurs when 1000 J of energy is transferred by heat from the Sun to Earth?

 

Question: If entropy continually increases, how is it that complex living organisms can form and grow out of simpler molecules on the Earth?