Chapter 12: Laws of Thermodynamics

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Chapter 12: Laws of Thermodynamics

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Title: Chapter 12: Laws of Thermodynamics

1
Chapter 12 Laws of Thermodynamics
Suggested homework assignment 4,21,33,39,48

Work in Thermodynamics Processes

Work done on a gas

Energy can be transferred to a system by
heat and by work done on the system.

In this chapter, all the systems we are
concerned with are volumes of gas and
they are in thermodynamic equilibrium
every part of the gas is at the same
temperature and pressure.

Consider a gas contained by a cylinder
with a movable piston and in equilibrium.
The gas occupies a volume V and exerts
a uniform pressure P on the cylinder wall.

Work in Thermodynamics Processes

Work done on a gas (contd)

The gas is compressed slowly enough so
that the system remains essentially in
thermodynamic equilibrium.

As the piston is pushed downward by an
external force F through a distance Dy, the
work done on gas is

under constant pressure

If the gas is compressed, DV is negative and the
work done on the
gas is positive. If the gas expands, the work
done on the gas is
negative.

Work in Thermodynamics Processes

PV diagram

PV diagram

isobaric process

In general, the area under the graph
in PV diagram is equal in magnitude
to the work done on the gas, whether
or not the process proceeds at
constant pressure. Note that Pa x m3
J.

First Law of Thermodynamics

First law of thermodynamics

The first law of thermodynamics is another
energy conservation law
that relates changes in internal energy the
energy associated with
the position and movement of all the molecules
of a system to
energy transfers due to heat and work.

Two mechanisms to transfer energy between a
system and its
environment a macroscopic displacement of an
object by a force
and by heat which occurs through random
molecular collisions.
Both mechanisms result in a change in internal
energy DU.

First law of thermodynamics
Ui initial internal energy Uf final internal
energy Q energy transferred to the system by
heat W work done on the system
5

First Law of Thermodynamics

First law of thermodynamics (contd)

The internal energy of any isolated system must
remain constant.

Molar specific heat at constant volume

Molar specific heat at constant volume Cv is the
amount of
heat Q needed to change the temperature by DT
per mole
at constant volume and is defined as

n number of mole

Internal energy of a monatomic ideal gas

We learned that the internal energy of a
monatomic ideal gas is

Therefore a change in U is proportional to a
change in T

First Law of Thermodynamics

Molar specific heat

From the first law of thermodynamics

At constant volume, W0. Therefore the first law
becomes

However, for an ideal gas
Therefore, whether its volume is constant or
not,

First Law of Thermodynamics

Molar specific heat (contd)

A gas with a larger specific heat requires more
energy to realize
a given temperature change. The size of molar
specific heat
depends on the structure of the gas molecule
and how many ways
it can store energy.

A monatomic gas (He etc.) can store energy as
motion in three
different directions (3-dimension). Degrees of
freedom 3

A diatomic gas (H2 etc.) can store energy as
motion in three
different directions (3-dimension), and can
tumble and rotate
in two different directions.
Degrees of freedom 5

Other molecules can store energy as vibrations,
which add more
degrees of freedom.

Each degree of freedom contribute to the molar
specific heat by (1/2)R.
So, for example, the molar specific heat of a
diatomic gas such as oxygen O2 is 5 x (1/2)R
(5/2)R.
8

First Law of Thermodynamics

Molar specific heat (contd)

See Table of molar specific heats

Isobaric process

An isobaric process is a process
where the pressure remains
constant.

Isobaric process and first law

First Law of Thermodynamics

Isobaric process (contd)

Ideal gas in isobaric process

For a monatomic ideal gas
For an isobaric process
Define molar specific heat at constant pressure
as
For an ideal gas in an isobaric process
This is valid for all ideal gasses
10

First Law of Thermodynamics

Adiabatic process

An adiabatic process is a process where no
energy enters or leaves
the system by heat the system is thermally
isolated.

For an adiabatic process,
First law of thermodynamics

It can also be shown that for an ideal gas in an
adiabatic process

First Law of Thermodynamics

Isovolumetric process

An isovolumetric process, also called an
isochoric process, is a
process where the volume is constant.

Since the volume does not change, there is no
work done.

For an ideal gas
12

First Law of Thermodynamics

Isothermal process

An isothermal process is a process where the
temperature is constant.

For an ideal gas, since the internal energy
depends only on the
temperature

First law of thermodynamics

It can be shown that the work
done on an ideal gas is given by

First Law of Thermodynamics

Examples

Example 12.4 Expanding gas

Suppose a system of monatomic ideal gas at
2.00x105 Pa and an initial temperature of 293 K
slowly expands at constant pressure from a volume
of 1.00 L to 2.50 L. (a) Find the work done on
the environment.
(b) Find the change in the internal energy of the
gas.
14

First Law of Thermodynamics

Examples

Example 12.4 Expanding gas (contd)

(c) Use the first law to obtain the energy
transferred by heat.
(d) Use the molar heat capacity at constant
pressure to obtain Q.
(e) How would the answers change for a diatomic
gas?
15

First Law of Thermodynamics

Examples

Example 12.5 Work and engine cylinder

In a car engine operating at 1.80x103 rev/min,
the expansion of hot, high-pressure gas against a
piston occurs in about 10 ms. because energy
transfer by heat typically takes a time on the
order of minutes or hours, it is safe to assume
that little energy leaves the hot gas during
expansion. Estimate the work done by the gas on
the piston during this adiabatic expansion by
assuming the engine cylinder contains 0.100 moles
of an ideal monatomic gas which goes
from 1.20x103 K to 4.00x102 K typical engine
temperatures, during the expansion.
16

First Law of Thermodynamics

Examples

Example 12.6 An adiabatic expansion

A monatomic ideal gas at a pressure 1.01x105 Pa
expands adiabatically from an initial volume of
1.50 m3, doubling its volume. (a) Find the new
pressure.
(b) Estimate the work done on the gas.
17

First Law of Thermodynamics

Examples

Example 12.7 An isovolumetric process

How much thermal energy must be added to 5.00
moles of monatomic ideal gas at 3.00x102 K and
with a constant volume of 1.50 L in order to
raise the temperature of the gas by 3.80x102 K?

Example 12.8 An isothermal expansion

A balloon contains 5.00 moles of monatomic ideal
gas. As energy is added to the system by heat,
the volume increases by 25 at a constant
temperature of 27.0oC. Find the work Wenv done by
the gas in expanding the balloon, the thermal
energy Q transferred to the gas, and the work W
done on the gas.
18

Second Law of Thermodynamics

Heat engines

A heat engine takes in energy by heat and
partially converts it to
other forms, such as electrical and mechanical
energy.

A heat engine, in general, carries some working
substance through
a cyclic process during which
(1) energy is transferred by heat from a source
at a high temperature
(2) work is done by the engine
(3) energy is expelled by the engine by heat to
a source at lower
temperature.

A steam engine, the working substance is water.
The water in the
engine is carried through a cycle in which it
first evaporates into
steam in a boiler and then expands against a
piston. After the steam
is condensed with cooling water, it returns to
the boiler, and the
process is repeated.

Second Law of Thermodynamics

Heat engines (contd)

The engine absorbs energy Qh from
the hot reservoir, does work Weng,
then gives up energy Qc to the cold
reservoir. Because the working sub-
stance goes through a cycle, always
returning to its initial thermodynamic
state the initial and final internal
energy is the same, so DU0.

The work Wenv done by a heat engine equals the
net energy absorbed by the engine.
20

Second Law of Thermodynamics

Heat engines (contd)

If the working substance is a gas,
the work done by the engine for a
cyclic process is the area enclosed
by the curve representing the process
on a PV diagram.

The thermal efficiency of a heat
engine is defined by

Second Law of Thermodynamics

Examples

Example 12.10 Efficiency of an engine

During one cycle, an engine extracts 2.00x103 J
of energy from a hot reservoir and transfers
1.50x103 J to a cold reservoir. (a) Find the
thermal efficiency of the engine.
(b) How much work does this engine do in one
cycle?
(b) How much power does the engine generate if it
goes through four cycles in 2.50 s?
22

Second Law of Thermodynamics

Examples

Example 12.11 Analyzing an engine cycle

A heat engine contains an ideal gas
confined to a cylinder by a movable
piston. The gas starts at A where
T3.00x102 K and B-gtC is an iso-
thermal process.
Find the number n of moles of
the gas and the temperature at B.

Second Law of Thermodynamics

Examples

Example 12.11 Analyzing an engine cycle
(contd)

(b) Find DU, Q, and W for iso- volumetric
process A-gtB.
24

Second Law of Thermodynamics

Examples

Example 12.11 Analyzing an engine cycle
(contd)

Second Law of Thermodynamics

Examples

Example 12.11 Analyzing an engine cycle
(contd)

(d) Find DU, Q, and W for iso- baric
process C-gtA.
26

Second Law of Thermodynamics

Examples

Example 12.11 Analyzing an engine cycle
(contd)

(e) Find the net change in internal energy
DUnet
(f) Find the energy input, Qh the energy
rejected, Qc the thermal efficiency and
the net work performed by the engine.
27

Second Law of Thermodynamics

Refrigeration and heat pump

A reversed heat engine is called refrigeration!

Energy is injected into the engine called heat
pump and that results in extraction of energy
from the cold reservoir to the hot reservoir.
Examples are refrigerator and air conditioner.

Coefficient of performance

For a heat pump in the cooling mode
For a heat pump in the heating mode
28

Second Law of Thermodynamics

Example 12.12 Cooling the leftovers

2.00 L of leftover soup at T323 K is placed in
a refrigerator. Assume
the specific heat of soup is the same as that
of water and the density
1.25x103 kg/m3. The refrigerator cools the soup
to 283 K.
(a) If the COP of the refrigerator is 5.00,
find the energy needed, in the
form of work, to cool the soup.

(b) If the compressor has a power rating 0.250 hp
find the time needed to cool the food.
29

Second Law of Thermodynamics

Second law of thermodynamics

There are limits to the efficiency of heat
engines.
An ideal engine which would convert all the
input energy into work
does not exist.

The Kelvin-Planck formulation of the second law
of thermodynamics

No heat engine operating in a cycle can absorb
energy from a reservoir and use it entirely for
the performance of an equal amount of work.

This means that the efficiency eWeng/Qh of
engines must always
be less than one. Some energy Qc must always
be lost to the
environment.

It is theoretically impossible to construct a
heat engine with an
efficiency 100.

We cannot get a greater amount of energy out of a
cyclic process that we put in.
30

Second Law of Thermodynamics

Reversible and irreversible processes

No engine can operate with 100 efficiency, but
different designs
yield different efficiencies.

One design called Carnot cycle (engine) delivers
the maximum
possible efficiency.

A reversible process is a process in which every
state along the same
path is an equilibrium state. In a reversible
process, the system can
return to its initial condition (state) by
going along the same path in
reverse direction.

An irreversible process is a process which does
not satisfy the
condition for a reversible process.

Most natural processes are irreversible, but
some of them are almost
reversible. If a real process occurs so slowly
that the system is virtually
always in equilibrium, the process can be
considered reversible.

Second Law of Thermodynamics

Carnot engine

Consider a heat engine operating in an ideal,
reversible cycle called
a Carnot cycle.

Second Law of Thermodynamics

Carnot engine (contd)

Consider a heat engine operating in an ideal,
reversible cycle called
a Carnot cycle.

A-gtB Isothermal expansion at Th. Qh
from hot reservoir. WAB done by gas.
B-gtC Adiabatic expansion Th-gtTc. No
heat goes out or comes in. WBC done
by gas.
C-gtD Isothermal compression at Tc.
Qc to cold reservoir. WCD done on gas.
D-gtA Adiabatic compression Tc-gtTh.
No heat goes out or comes in. WDA
done on gas.
33

Second Law of Thermodynamics

Carnot engine (contd)

Ratio of heat input to output vs. ratio of
temperatures

Thermal efficiency of a Carnot enegine

All Carnot engines operating reversibly between
the same two temperatures have the same
efficiency.

All real engines operate irreversibly, due to
friction and brevity of their
cycles, and are therefore less efficient that
the Carnot engine.

Second Law of Thermodynamics

Example 12.13 Steam engine

A steam engine has a boiler that operates at
5.00x102 K. The energy
from the boiler changes water to steam which
drives the piston. The
temperature of the exhaust is that of the
outside air, 300 K.
(a) What is the engines efficiency if it is an
ideal engine?

(b) If the 3.50x103 J of energy is supplied from
the boiler, find the work done by the engine
on its environment.
35

Entropy

Definition of entropy

Let Qr be the energy absorbed or expelled during
a reversible,
constant temperature process between two
equilibrium states.
Then the change in entropy during any constant
temperature
process connecting the two equilibrium states
id defined by

SI unit joules/kelvin (J/K)

A similar formula holds even when the
temperature is not constant.
Although calculation of DS during a transition
between two equilibrium
states requires finding a reversible path that
connects the states, the
entropy change calculated on that reversible
path is taken to be DS
for the actual path. This is valid logic as the
change in entropy DS
depends only on the initial and final states
and not on the path taken.

The entropy of the Universe increases in all
natural processes.

Entropy

Meaning of entropy

In nature a disorderly arrangement is much more
probable than an
orderly one if the laws of nature are allowed
act without interference.

Using statistical mechanics it can be concluded
that isolated systems
tend toward great disorder, and entropy is a
measure of that disorder.

kB Boltzman constant W a number proportional
to the probability that system has a
particular configuration.

The second law of thermodynamics is really a
statement of what is
most probable rather than of what must be.

The entropy of the Universe always increases

Entropy

Examples

Example 12.14 Melting a piece of lead

Find the change in entropy of 300 g of lead when
it melts at 327oC.
Lead has a latent heat of fusion of 2.45x104
J/kg.

(b) Suppose the same amount of energy is used to
melt part of a piece of silver, which is
already at its melting point of 961oC. Find the
change in the entropy of the sliver.
38

Entropy

Examples

Example 12.15 Ice, steam, and the entropy of
the Universe

A block of ice at 273 K is put in thermal contact
with a container at 373 K, converting 25.0 g of
ice to water at 273 K while condensing some of
the steam to water at 373 K. (a) Find the change
in entropy of the ice.
(b) Find the change in entropy of the steam.
Thermal energy lost by the steam is equal to the
thermal energy gained by the ice.
(c) Find the change in entropy of the Universe.
39

Entropy

Examples

Example 12.16 A falling boulder

A chunk of rock of mass 1.00x103 kg at 293 K
falls from a cliff of height 125 m into a large
lake, also at 293 K. Find the change in entropy
of the lake, assuming that all of the rocks
kinetic energy upon entering the lake converts to
thermal energy absorbed by the lake.
The rocks kinetic energy at the time of entrance
to the lake.
40