Download Chapter 20 - UCF Physics

Chapter 20
Heat and the
First Law of Thermodynamics
Mechanical Equivalent of Heat
•
•
•
Joule found that it took approximately 4.18 J of
mechanical energy to raise the temperature of
1g water at 1oC
Later, more precise, measurements determined
the amount of mechanical energy needed to
raise the temperature of 1g of water from 14.5oC
to 15.5oC
1 cal = 4.186 J
This is known as the mechanical equivalent of
heat
Heat Capacity
• The heat capacity, C, of a particular sample is
defined as the amount of energy needed to raise
the temperature of that sample by 1oC
• If energy Q produces a change of temperature of
DT, then
Q = C DT
Specific Heat
• Specific heat, c, is the heat capacity per
unit mass
• If energy Q transfers to a sample of a
substance of mass m and the temperature
changes by DT, then the specific heat is
Q
c
m DT
Specific Heat
• The specific heat is essentially a measure
of how insensitive a substance is to the
addition of energy
– The greater the substance’s specific heat, the
more energy that must be added to cause a
particular temperature change
• The equation is often written in terms of Q :
Q = m c DT
Some Specific Heat Values
More Specific Heat Values
Sign Conventions
• If the temperature increases:
Q and DT are positive
Energy transfers into the system
• If the temperature decreases:
Q and DT are negative
Energy transfers out of the system
Specific Heat Varies With Temperature
•
•
•
Technically, the specific heat varies with
temperature
Tf
c dT
The corrected equation is Q  m

Ti
However, if the temperature intervals are not
too large, the variation can be ignored and c
can be treated as a constant
There is only about a 1% specific heat
variation between 0o and 100oC
Specific Heat of Water
• Water has the highest specific heat of
common materials
• This is responsible for many weather
phenomena
– Moderate temperatures near large bodies of
water
– Global wind systems
– Land and sea breezes
Calorimetry
• One technique for measuring specific heat
involves heating a material, adding it to a
sample of water, and recording the final
temperature
• This technique is known as calorimetry
– A calorimeter is a device in which this energy
transfer takes place
Calorimetry
• The system of the sample and the water are
isolated
• Conservation of energy requires that the amount
of energy that leaves the sample equals the
amount of energy that enters the water
– Conservation of Energy gives a mathematical
expression of this:
Qcold= -Qhot
Calorimetry
•
•
The negative sign in the equation is critical for
consistency with the established sign convention
Since each Q = mcDT, csample can be found by:
mwcw (T f  Tw )  ms cs (Ts  T f )
cs 
mwcw T f  Tw 
ms Ts  T f 
Technically, the mass of the container should be
included, but if mw >>mcontainer it can be neglected
Phase Changes
• A phase change is happened when a
substance changes from one form to another
– Two common phase changes are
• Solid to liquid (melting)
• Liquid to gas (boiling)
• During a phase change, there is no change
in temperature of the substance
Latent Heat
•
•
•
Different substances react differently to the
energy added or removed during a phase
change due to their different molecular
arrangements
The amount of energy also depends on the
mass of the sample
If an amount of energy Q is required to change
the phase of a sample of mass m,
L = Q /m
Latent Heat
•
•
•
If an amount of energy Q is required to change
the phase of a sample of mass m,
L = Q /m
The quantity L is called the latent heat of the
material
Latent means “hidden”
The value of L depends on the substance as well as
the actual phase change
The energy required to change the phase is
Q =  mL
Latent Heat
• The latent heat of fusion is used when the phase
change is from solid to liquid
• The latent heat of vaporization is used when the
phase change is from liquid to gas
• The positive sign is used when the energy is
transferred into the system
– This will result in melting or boiling
• The negative sign is used when energy is
transferred out of the system
– This will result in freezing or condensation
Sample Latent Heat Values
Graph of Ice to Steam
Warming Ice, Graph Part A
•
Start with one gram of ice at
–30.0ºC
• During phase A, the
temperature of the ice
changes from –30.0ºC to
0ºC. The specific heat of
ice is 2090 J/kg·0C
Q = mi ci ΔT = (1x10-3kg)(2090
J/kg·0C)(30.00C)=62.7J
In this case, 62.7 J of energy
are added
Melting Ice, Graph Part B
•
•
Once at 0ºC, the phase change
(melting) starts
The temperature stays the same
although energy is still being
added. The latent heat of fusion
for water is 3.33x105 J/kg
Q =mi Lf =(1x10-3 kg)(3.33x105
J/kg)=333J
The energy required is 333 J
On the graph, the values move from
62.7 J to 396 J
Warming Water, Graph Part C
•
•
Between 0ºC and 100ºC, the
material is liquid and no
phase changes take place
Energy added increases the
temperature. Specific heat for
water 4186 J/kg·0C.
Q = mwcw ΔT = 419J.
419 J are added
The total is now 815 J
Boiling Water, Graph Part D
•
•
•
At 100ºC, a phase
change occurs
(boiling)
Temperature does not
change. The latent
heat of vaporization for
water is 2.26x106J/kg.
Use Q = mw Lv.
This requires 2260 J
The total is now 3075 J
Heating Steam
•
•
•
•
After all the water is converted to
steam, the steam will heat up
No phase change occurs
The added energy goes to increasing
the temperature. The specific heat for
steam is 2010J/kg∙0C.
Use
Q = mscs ΔT
In this case, 40.2 J are needed
The temperature is going to 120o C
• The total is now 3115 J
If water with a mass mh at temperature Th is
poured into an aluminum cup of mass mAl
containing mass mc of water at Tc, where Th >
Tc, what is the equilibrium temperature of the
system?
If water with a mass mh at temperature Th is
poured into an aluminum cup of mass mAl
containing mass mc of water at Tc, where Th >
Tc, what is the equilibrium temperature of the
system?
Q cold  Q hot





m A lcA l T f  Tc  m ccw T f  Tc   m hcw T f  Th

 m A lcA l  m ccw  T f   m A lcA l  m ccw  Tc  m hcw T f  m hcw Th
 m A lcA l  m ccw  m hcw  T f   m A lcA l  m ccw  Tc  m hcw Th
m A lcA l  m ccw  Tc  m hcw Th

T 
f
m A lcA l  m ccw  m hcw
Molecular View of Phase Changes
•
•
Phase changes can be described in terms of the
rearrangement of molecules (or atoms in an
elemental substance)
Liquid to Gas phase change
Molecules in a liquid are close together
The forces between them are stronger than those in a
gas
Work must be done to separate the molecules
The latent heat of vaporization is the energy per unit
mass needed to accomplish this separation
Molecular View of Phase Changes
Solid to Liquid phase change
•
•
•
•
•
The addition of energy will cause the amplitude of the
vibration of the molecules about their equilibrium
position to increase
At the melting point, the amplitude is great enough to
break apart bonds between the molecules
The molecules can move to new positions
The molecules in the liquid are bound together less
strongly than those of the solid
The latent heat of fusion is the energy per unit mass
required to go from the solid-type to the liquid-type
bonds
Molecular View of Phase Changes
• The latent heat of vaporization is greater
than the latent heat of fusion
In the liquid-to-gas phase change, the liquidtype bonds are broken
The gas molecules are essentially not bonded
to each other
• It takes more energy to completely break
the bonds than to change the type of
bonds
Calorimetry Problem-Solving Strategy
• Units of measure must be consistent
For example, if your value of c is in J/kg.oC,
then your mass must be in kg, the temperatures
in oC and energies in J
• Transfers of energy are given by
•
Q =mc DT only when no phase change
occurs
If there is a phase change, use Q = mL
Calorimetry Problem-Solving Strategy
• Be sure to select the correct sign for all
energy transfers
• Remember to use Qcold = - Qhot
– The DT is always Tf - Ti
State Variables
•
•
•
State variables describe the state of a system
In the macroscopic approach to
thermodynamics, variables are used to describe
the state of the system
Pressure, temperature, volume, internal energy
These are examples of state variables
The macroscopic state of an isolated system can
be specified only if the system is in thermal
equilibrium internally
Work in Thermodynamics
•
•
•
Work can be done on a deformable
system, such as a gas
Consider a cylinder with a moveable
piston
A force is applied to slowly compress
the gas
The compression is slow enough
for all the system to remain
essentially in thermal equilibrium
This is said to occur quasistatically
Work
• The piston is pushed downward by a force
F through a displacement of dr:
dW  F  dr   Fˆj  dyˆj   Fdy   PA dy
A·dy is the change in volume of the gas, dV
Therefore, the work done on the gas is
dW = -P dV
Work
• Interpreting dW = - P dV
– If the gas is compressed, dV is negative and
the work done on the gas is positive
– If the gas expands, dV is positive and the
work done on the gas is negative
– If the volume remains constant, the work done
is zero
• The total work done is:
Vf
W    P dV
Vi
PV Diagrams
The state of the gas at
each step can be plotted
on a graph called a PV
diagram
•
•
This allows us to visualize
the process through which
the gas is progressing
The curve is called the
path
PV diagrams can be
used when the pressure
and volume are known at
each step of the process
PV Diagrams
• The work done on a gas in a quasistatic process that takes the gas from
an initial state to a final state is the
negative of the area under the curve on
the PV diagram, evaluated between the
initial and final states
–This is true whether or not the pressure
stays constant
–The work done does depend on the path
taken
Work Done By Various Paths
• Each of these processes has the same initial
and final states
• The work done differs in each process
• The work done depends on the path
Work From a PV Diagram
•
•
•
The volume of the
gas is first reduced
from Vi to Vf at
constant pressure Pi
Next, the pressure
increases from Pi to
Pf by heating at
constant volume Vf
W = -Pi (Vf – Vi)
Work From a PV Diagram
•
•
•
The pressure of the gas
is increased from Pi to
Pf at a constant volume
The volume is
decreased from Vi to Vf
W = -Pf (Vf – Vi)
Work From a PV Diagram
•
•
•
The pressure and the
volume continually change
The work is some
intermediate value
between Pf (Vf – Vi) and
Pi (Vf – Vi)
To evaluate the actual
amount of work, the
function P(V) must be
known
V
A sample of ideal gas is expanded to twice its
original volume of 1.00 m3 in a quasi-static process
for which P = V 2, with
 = 5.00 atm/m6, as shown
in Figure. How much work is
done on the expanding gas?
A sample of ideal gas is expanded to twice its original volume
of 1.00 m3 in a quasi-static process for which P = V 2, with
 = 5.00 atm/m6, as shown
in Figure. How much work is
done on the expanding gas?
f
W if    PdV
i
.
The work done on the gas is the negative of the area under
2
the curve P  V between V and
i
f

1
3
3
W if    V dV    V f  Vi
3
i
2

V f  2V i  2 1.00 m
3

 2.00 m
3

Vf
A sample of ideal gas is expanded to twice its original volume
of 1.00 m3 in a quasi-static process for which P = V 2, with
 = 5.00 atm/m6, as shown
in Figure. How much work is
done on the expanding gas?
f
W if    PdV
.
i

1 
atm 
5 Pa  
3 3
3
Wif    5.00 6 1.013 10
 2.00m  1.00m
3 
m 
atm 
 1.18MJ
 
 
3
(a) Determine the work
done on a fluid that
expands from i to f as
indicated in Figure.
(b) How much work is
performed on the fluid if it
is compressed from f to i
along the same path?
(a) Determine the work
done on a fluid that
expands from i to f as
indicated in Figure. (b)
How much work is
performed on the fluid if it
is compressed from f to i
along the same path?
(a)
W    PdV


  4.00  10 Pa  3.00  2.00 m
  2.00  10 Pa  4.00  3.00 m
(b)
W   6.00  106 Pa  2.00  1.00 m 3 
6
3
6
3
W i f  12.0 M J

W
fi 
12.0 M J
Heat Transfer
• The energy transfer, Q, into
•
or out of a system also
depends on the process
The piston is pushed
upward, the gas is doing
work on the piston
Heat Transfer
•
•
•
•
This gas has the same initial
volume, temperature and
pressure as the previous
example
The final states are also
identical
No energy is transferred by
heat through the insulating
wall
No work is done by the gas
expanding into the vacuum
Energy Transfer, Summary
• Energy transfers by heat, like the work
done, depend on the initial, final, and
intermediate states of the system
• Both work and heat depend on the path
taken
• Neither can be determined solely by the
end points of a thermodynamic process
The First Law of Thermodynamics
•
•
The First Law of Thermodynamics is a
special case of the Law of Conservation of
Energy
It takes into account changes in internal energy
and energy transfers by heat and work
Although Q and W each are dependent on the
path, Q + W is independent of the path
• The First Law of Thermodynamics states that
DEint = Q + W
All quantities must have the same units of measure
of energy
The First Law of Thermodynamics
•
•
•
One consequence of the first law is that there
must exist some quantity known as internal
energy which is determined by the state of the
system
For infinitesimal changes in a system
dEint = dQ + dW
The first law is an energy conservation
statement specifying that the only type of energy
that changes in a system is internal energy
Isolated Systems
• An isolated system is one that does not
interact with its surroundings
• No energy transfer by heat takes place
• The work done on the system is zero
Q = W = 0, so DEint = 0
• The internal energy of an isolated system
remains constant
Cyclic Processes
•
A cyclic process is one that starts and ends in the
same state
• This process would not be isolated
• On a PV diagram, a cyclic process appears as a
•
•
closed curve
The change in internal energy must be zero since it
is a state variable
If DEint = 0, Q = -W
In a cyclic process, the net work done on the system
per cycle equals the area enclosed by the path
representing the process on a PV diagram
Adiabatic Process
•
An adiabatic process is one
during which no energy enters
or leaves the system by heat
Q=0
This is achieved by:
Thermally insulating the walls
of the system
Having the process proceed
so quickly that no heat can
be exchanged
Adiabatic Process
• Since Q = 0, DEint = W
• If the gas is compressed adiabatically, W
is positive so DEint is positive and the
temperature of the gas increases
• If the gas expands adiabatically, the
temperature of the gas decreases
Adiabatic Processes
• Some important examples of adiabatic
processes related to engineering are:
–The expansion of hot gases in an internal
combustion engine
–The liquefaction of gases in a cooling
system
–The compression stroke in a diesel engine
Adiabatic Free Expansion
•
•
•
•
This is an example of adiabatic
free expansion
The process is adiabatic because
it takes place in an insulated
container
Because the gas expands into a
vacuum, it does not apply a force
on a piston and W = 0
Since Q = 0 and W = 0, DEint = 0
and the initial and final states are
the same
– No change in temperature is
expected
Isobaric Processes
• An isobaric process is one that occurs at a
•
•
constant pressure
The values of the heat and the work are
generally both nonzero
The work done is W = P (Vf – Vi) where P
is the constant pressure
Isovolumetric Processes
•
•
•
•
An isovolumetric process is one in which there
is no change in the volume
Since the volume does not change, W = 0
From the first law, DEint = Q
If energy is added by heat to a system kept at
constant volume, all of the transferred energy
remains in the system as an increase in its
internal energy
Isothermal Process
• An isothermal process is one that occurs
at a constant temperature
• Since there is no change in temperature,
DEint = 0
• Therefore,
Q=-W
• Any energy that enters the system by heat
must leave the system by work
Isothermal Process
•
•
•
At right is a PV
diagram of an
isothermal expansion
The curve is a
hyperbola
The curve is called as
isotherm
Isothermal Expansion
• The curve of the PV diagram indicates
PV = constant
The equation of a hyperbola
• Because it is an ideal gas and the process
is quasi-static, PV = nRT and
V
V nRT
V dV
W    P dV   
dV   nRT 
V
V
V
V
V
 Vi 
W  nRT ln  
 Vf 
 
f
i
f
i
f
i
Isothermal Expansion
• Numerically, the work equals the area
under the PV curve
The shaded area in the diagram
• If the gas expands, Vf > Vi and the work
•
done on the gas is negative
If the gas is compressed, Vf < Vi and the
work done on the gas is positive
Special Processes, Summary
• Adiabatic
No heat exchanged
Q = 0 and DEint = W
• Isobaric
Constant pressure
W = P (Vf – Vi) and DEint = Q + W
• Isothermal
Constant temperature
DEint = 0 and Q = -W
A sample of an ideal gas goes through the process shown
in Figure. From A to B, the process is adiabatic; from B
to C, it is isobaric with 100 kJ of energy entering the
system by heat. From C to D, the process is isothermal;
from D to A, it is isobaric with 150 kJ of energy leaving
the system by heat. Determine the difference in internal
energy Eint,B – Eint,A.
A sample of an ideal gas goes through the process shown in Figure. From
A to B, the process is adiabatic; from B to C, it is isobaric with 100 kJ of
energy entering the system by heat. From C to D, the process is
isothermal; from D to A, it is isobaric with 150 kJ of energy leaving the
system by heat. Determine the difference in internal energy Eint,B – Eint,A.
W BC  PB VC  VB   3.00 atm  0.400  0.090 0 m 3
 94.2 kJ
DEint  Q  W
Eint,C  Eint,B   100  94.2 kJ
Eint,C  Eint,B  5.79 kJ
A sample of an ideal gas goes through the process shown in Figure. From
A to B, the process is adiabatic; from B to C, it is isobaric with 100 kJ of
energy entering the system by heat. From C to D, the process is
isothermal; from D to A, it is isobaric with 150 kJ of energy leaving the
system by heat. Determine the difference in internal energy Eint,B – Eint,A.
W BC  PB VC  VB   3.00 atm  0.400  0.090 0 m 3
 94.2 kJ
DEint  Q  W
Eint,C  Eint,B   100  94.2 kJ
Eint,C  Eint,B  5.79 kJ
T=constant
Eint,D  Eint,C  0
W D A  PD VA  VD   1.00 atm  0.200  1.20 m 3
 101 kJ
A sample of an ideal gas goes through the process shown in Figure. From
A to B, the process is adiabatic; from B to C, it is isobaric with 100 kJ of
energy entering the system by heat. From C to D, the process is
isothermal; from D to A, it is isobaric with 150 kJ of energy leaving the
system by heat. Determine the difference in internal energy Eint,B – Eint,A.
Eint,A  Eint,D  150 kJ  101 kJ  48.7 kJ

 
 

Eint,B  Eint,A    Eint,C  Eint,B  Eint,D  Eint,C  Eint,A  Eint,D 
Eint,B  Eint,A    5.79 kJ 0  48.7 kJ  42.9 kJ
Mechanisms of Heat Transfer
• We want to know the rate at which energy
•
is transferred
There are various mechanisms
responsible for the transfer:
– Conduction
– Convection
– Radiation
Conduction
• The transfer can be viewed on an atomic
scale
– It is an exchange of energy between
microscopic particles by collisions
The microscopic particles can be atoms,
molecules or free electrons
– Less energetic particles gain energy during
collisions with more energetic particles
• Rate of conduction depends upon the
characteristics of the substance
Conduction example
•
•
•
•
The molecules vibrate
about their equilibrium
positions
Particles near the heat
source vibrate with larger
amplitudes
These collide with
adjacent molecules and
transfer some energy
Eventually, the energy
travels entirely through
the pan
Conduction
•
•
•
In general, metals are good conductors
They contain large numbers of electrons that are
relatively free to move through the metal
They can transport energy from one region to
another
Poor conductors include asbestos, paper, and
gases
Conduction can occur only if there is a
difference in temperature between two parts of
the conducting medium
Conduction
•
•
The slab at right
allows energy to
transfer from the
region of higher
temperature to the
region of lower
temperature
The rate of transfer is
given by:
Q
dT
P
 kA
Dt
dx
Conduction
• A is the cross-sectional area
• Δx is the thickness of the slab
or the length of a rod
• P is in Watts when Q is in Joules and t
•
is in seconds
k is the thermal conductivity of the
material
Good conductors have high k values and
good insulators have low k values
Temperature Gradient
•
•
The quantity |dT / dx| is
called the temperature
gradient of the material
It measures the rate at
which temperature varies
with position
For a rod, the
temperature gradient
can be expressed as:
dT Th  Tc

dx
L
Rate of Energy Transfer in a Rod
•
Using the temperature gradient for the rod, the
rate of energy transfer becomes:
 Th  Tc 
P  kA

 L 
Compound Slab
• For a compound slab containing several materials
of various thicknesses (L1, L2, …) and various
thermal conductivities (k1, k2, …) the rate of
energy transfer depends on the materials and the
temperatures at the outer edges:
ATh  Tc 
P
 Li 
 k 
i


i
Some Thermal Conductivities
More Thermal Conductivities
In Figure, the change in internal
energy of a gas that is taken from
A to C is +800 J. The work done on
the gas along path ABC is –500 J.
(a) How much energy must be
added to the system by heat as it
goes from A through B to C?
(b) If the pressure at point A is five
times that of point C, what is the
work done on the system in going
from C to D? (c) What is the energy
exchanged with the surroundings
by heat as the cycle goes from C to
A along the green path? (d) If the
change in internal energy in going
from point D to point A is +500 J,
how much energy must be added
to the system by heat as it goes
from point C to point D?
In Figure, the change in internal energy of
a gas that is taken from A to C is +800 J.
The work done on the gas along path
ABC is –500 J.
(a) How much energy must be added to
the system by heat as it goes from A
through B to C?
(b) If the pressure at point A is five times
that of point C, what is the work done on
the system in going from C to D? (c) What
is the energy exchanged with the
surroundings by heat as the cycle goes
from C to A along the green path? (d) If
the change in internal energy in going
from point D to point A is +500 J, how
much energy must be added to the system
by heat as it goes from point C to point D?
DEint,A BC  DEint,A C
(a)
(conservation of energy)
DEint,A BC  Q A BC  W
A BC
Q ABC  800 J 500 J 1300 J
(First Law)
In Figure, the change in internal energy of
a gas that is taken from A to C is +800 J.
The work done on the gas along path
ABC is –500 J.
(a) How much energy must be added to
the system by heat as it goes from A
through B to C?
(b) If the pressure at point A is five times
that of point C, what is the work done on
the system in going from C to D? (c) What
is the energy exchanged with the
,
surroundings by heat as the cycle
goes
from C to A along the green path? (d) If
the change in internal energy in going
from point D to point A is +500 J, how
much energy must be added to the system
by heat as it goes from point C to point D?
(b)
W CD  PC DVCD DV AB  DVCD
Then,
W CD
, and PA  5PC
1
1
 PA DV AB   W AB  100 J
5
5
(+ means that work is done on the system)
In Figure, the change in internal energy of
a gas that is taken from A to C is +800 J.
The work done on the gas along path
ABC is –500 J.
(a) How much energy must be added to
the system by heat as it goes from A
through B to C?
(b) If the pressure at point A is five times
that of point C, what is the work done on
the system in going from C to D? (c) What
is the energy exchanged with the
surroundings by heat as the cycle goes
from C to A along the green path? (d) If
the change in internal energy in going
from point D to point A is +500 J, how
much energy must be added to the system
by heat as it goes from point C to point D?
(c)
W CD A  W CD
so that
Q CA  DEint,CA  W CD A  800 J 100 J 900 J
(– means that energy must be removed from the system by heat)
In Figure, the change in internal energy of
a gas that is taken from A to C is +800 J.
The work done on the gas along path
ABC is –500 J.
(a) How much energy must be added to
the system by heat as it goes from A
through B to C?
(b) If the pressure at point A is five times
that of point C, what is the work done on
the system in going from C to D? (c) What
is the energy exchanged with the
surroundings by heat as the cycle goes
from C to A along the green path? (d) If
the change in internal energy in going
from point D to point A is +500 J, how
much energy must be added to the system
by heat as it goes from point C to point D?
(d)
DEint,CD  DEint,CD A  DEint,D A  800 J 500 J 1300 J
and
Q CD  DEint,CD  W CD  1300 J 100 J 1400 J
A bar of gold is in thermal contact with a bar of silver
of the same length and area. One end of the
compound bar is maintained at 80.0°C while the
opposite end is at 30.0°C. When the energy transfer
reaches steady state, what is the temperature at the
junction?
A bar of gold is in thermal contact with a bar of silver
of the same length and area. One end of the
compound bar is maintained at 80.0°C while the
opposite end is at 30.0°C. When the energy transfer
reaches steady state, what is the temperature at the
junction?
In the steady state condition,
PA u  PA g
D
T
D
T




so that
kA u A A u 
 kA g A A g 

 Dx  A u
 Dx  A g
A A u  A A g DxA u  DxA g
In this case
DTA u   80.0  T 
DTA g   T  30.0
A bar of gold is in thermal contact with a bar of silver
of the same length and area. One end of the
compound bar is maintained at 80.0°C while the
opposite end is at 30.0°C. When the energy transfer
reaches steady state, what is the temperature at the
junction?
In this case
AA u  AA g
DxA u  DxA g
DTA u   80.0  T 
DTA g   T  30.0
where T is the temperature of the junction.
Therefore,
kA u  80.0  T   kA g  T  30.0
T  51.2C