Download 8. Temperature and Heat - City, University of London

TEMPERATURE
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Heat is the energy that is transferred between objects
because of a temperature difference
Terms such as “transfer of heat” or “heat flow” from
object A to object B simply means that the total energy
of A decreases and that of B increases
An object does not contain heat – it has a certain
energy content, and the energy it exchanges with other
objects due to temperature differences is called heat
Objects are said to be in thermal contact if heat can
flow between them
When a hot object is brought into contact with a cold
object, heat is exchanged
The hot object cools off (its molecules move more
slowly) and the cold object warms up (its molecules
move more rapidly)
After some time in thermal contact, the transfer of heat
ceases – objects are then in thermal equilibrium
Thermodynamics is the study of physical processes
involving the transfer of heat – it deals with the flow of
energy within and between objects
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ZEROTH LAW OF DYNAMICS
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Thermal equilibrium is determined by a single physical
quantity – temperature
Two objects in thermal contact are in equilibrium when
they have the same temperature
If one or the other as a higher temperature, heat flows
from that object to the other until their temperatures are
equal
The zeroth law of thermodynamics is: If object A is in
thermal equilibrium with object B, and object C is also
in thermal equilibrium with object B, then objects A and
C will be in thermal equilibrium if brought into thermal
contact
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THERMAL EXPANSION: LINEAR
EXPANSION
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Most substances expand when heated
In thermometers, the expansion of liquid, such as
mercury or alcohol, results in a column of liquid rising
(or falling) within the glass tube
Water is the exception to the rule
Consider a rod with length L0 at temperature T0
When heated or cooled, its length changes by ∆L in
direct proportion to the temperature change, ∆T
Thus ∆L = (constant)∆T
The constant or proportionality depends, among other
things, on the substance from which the rod is made
In fact experiments show that the change in length, ∆L,
is proportional to both the initial length, L0, and the
temperature change, ∆T
The constant of proportionality is referred to as α which
is the coefficient of linear expansion (K-1)
∆L = αL0∆T
273 Kelvin = 0°C
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THERMAL EXPANSION: LINEAR
EXPANSION - EXAMPLE
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The Eiffel Tower is made from iron. If the tower is 301m
high on a 22°C day, how much does its height
decrease when the temperature cools to 0.0°C?
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THERMAL EXPANSION: AREA
EXPANSION
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Since the length of an object changes with temperature,
it follows that its area changes as well
Consider a square piece of metal of length L, thus the
initial area is L2
If the temperature of the square is increased by ∆T, the
length of each side increases from L to L + ∆L
L + ∆L = L + αL∆T
As a result, the square has an increased area A’
A’ = (L + ∆L)2 = (L + αL∆T)2 = L2 + 2αL2∆T + α2L2∆T2
Since α2∆T2 is very small, it can be ignored
Thus A’ ≈ L2 + 2αL2∆T = A + 2αA∆T
This result also applies to an area of any shape
A circular disk with area πr2 will also increase its area
by 2αA∆T when subject to a temperature change ∆T
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THERMAL EXPANSION: VOLUME
EXPANSION
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If the length and area of an object increase when
subjected to a temperature change ∆T, it follows that
the object’s volume also increases
Consider a cube of length L, and its volume V = L3
Increasing the temperature results in increased volume
V’ = (L + ∆L)3 = (L + αL∆T)3
V’ = L3 + 3αL3∆T + 3α2L3∆T2 + α3L3∆T3
Neglecting the smaller contributions as we did before:
V’ ≈ L3 + 3αL3∆T = V + 3αV∆T
This expression is valid for any volume
Volume expansion is described in the same way as
linear expansion, but with a coefficient of volume
expansion, β (K-1)
∆V = βV∆T ≈ 3αV∆T, where β = 3α
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THERMAL EXPANSION: VOLUME
EXPANSION - EXAMPLE
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A copper flask with a volume of 150cm3 is filled to the
brim with olive oil. If the temperature of the system is
increased from 6°C to 31°C, how much oil spills fro m
the flask?
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SPECIAL PROPERTIES OF WATER
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Water is a substance rich with unusual behaviour
In solid form (ice) it is less dense than the liquid form,
hence icebergs float
The solids of most substances are denser than their
liquids, hence when they freeze, their solids sink!
The density of water changes over a wide range of
temperatures
Its density is a maximum at about 4°C, thus when y ou
heat water from 0°C to 4°C, it shrinks
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HEAT CAPACITY
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Heat (Q) is the energy transferred due to temperature
differences and its units are Joules
It takes 4186J of heat to raise the temperature of 1kg of
water by 1°C
The heat required for a 1°C increase varies from one
substance to another, e.g. it takes only 129J of heat to
increase the temperature of lead by 1°C
The heat required for a given increase in temperature is
given by the heat capacity of a substance
Heat capacity: C = Q/∆T Joules/Kelvin (J/K)
The heat capacity is viewed as the amount of heat
necessary for a given temperature change
An object with a large heat capacity (e.g. water)
requires a large amount of heat for each increment in
temperature
Heat capacity is always positive, just like speed, so Q
and ∆T must have the same sign
Q is positive if ∆T is positive: heat added to the system
Q is negative if ∆T is negative: heat removed from the
system
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SPECIFIC HEAT
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Heat capacity varies not only with the type of
substance, but also with the mass of the substance
Since it takes 4168J to increase the temperature of 1kg
of water by 1°C, it takes twice that much to make the
same temperature change in 2kg, and so on
Thus a new quantity is defined – the specific heat, c –
given by c = Q/m∆T (J/kg.K)
Water has the largest specific heat, which means it can
take in (or give off) large quantities of heat with little
change in temperature
If you take a pie out of
the oven, the filling is
much hotter than the
crust because it has
a high water content
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SPECIFIC HEAT PROBLEM
SOLVING
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Suppose a block of mass mb, specific heat cb, and initial
temperature Tb is dropped into a calorimeter (a
lightweight, insulated flask) containing water
The water has mass mw, specific heat cw and initial
temperature Tw
The aim is to find the final temperature of the block and
water, assuming the calorimeter is light enough for it to
be ignored, and that no heat is transferred from the
calorimeter to its surroundings
Remember that the final temperature of the block and
water will be equal, and that the total energy of the
system is conserved
Thus the amount of energy lost by the block is equal to
that gained by the water (or vice versa)
Mathematically: Qb + Qw = 0 (meaning that the heat
flow from the block is equal and opposite to the heat
flow from the water)
Rewriting Q in terms of specific heats and temperatures
mbcb(T – Tb) + mwcw(T – Tw) = 0
Remember that ∆T = Tfinal – Tinitial
T = [mbcbTb + mwcwTw]/[mbcb + mwcw]
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SPECIFIC HEAT: EXAMPLE
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A 0.5kg block of metal with an initial temperature of
54.5°C is dropped into a container holding 1.1kg of
water at 20.0°C, If the final temperature of the block
water system is 21.4°C, what is the specific heat of the
metal? Assume the container can be ignored and that
no heat is exchanged with the surroundings.
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CONDUCTION
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Conduction is a form of heat exchange – it is the flow
of heat directly through a physical material
From a microscopic point of view, holding one end of a
metal rod with the other end in a fire, the high
temperature of the fire causes the molecules at the far
end of the rod to vibrate with an increased amplitude
Neighbouring molecules start to vibrate as well, and
this effect propagates through the rod, resulting in the
macroscopic phenomenon of conduction
A wooden rod in the same situation would behave
differently – the hot end would catch fire, whereas the
other end would stay cool
Conduction depends on the type of material involved
Conductors conduct heat very well, whereas
insulators do not
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HOW MUCH HEAT FLOWS AS A
RESULT OF CONDUCTION
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Consider a rod length L and cross sectional area A,
with one end at temperature T1 and the other at T2 > T1
Experiments show that the amount of heat, Q, that
flows through this rod:
– Increases in proportion to the rod’s cross sectional
area, A
– Increases in proportion to the temperature
difference, ∆T = T2 – T1
– Increases steadily with time, t
– Decreases with the length of the rod, L
Combining these observations mathematically gives
Q = kA(∆T/L)t
The constant k is referred to as the thermal
conductivity, and varies from material to material
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CONDUCTION: EXAMPLE
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Two 0.525m rods, one
lead, the other copper,
are connected between
metal plates held at
2.0°C and 106°C. The
rods have a square
cross section, 1.5cm on
a side. How much heat
flows through the two
rods in 1.0s? Assume
no heat is exchanged
between the rods and
the surroundings,
except at the ends.
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CONVECTION
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When heating a room with an electric heater, the
heater’s hot coils heat the air in their vicinity
As this air warms up, it expands and thus becomes less
dense
Because of its lower density, the warm air rises, and is
replaced by descending colder, denser air from above
This sets up a circulating air flow that transports heat
from the coils to the air in the room
This type of heat exchange is called convection
Convection occurs when a fluid is unevenly heated
In convection, temperature differences result in a flow
of fluid (hot air rising, cold air sinking)
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EXAMPLE OF CONVECTION
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During the day, the sun warms the land more rapidly
than the water
The rocky land has a lower specific heat than water
The warm land heats the air above it which becomes
less dense and rises
Cooler air from over the water flows in to take its place,
producing a sea breeze
At night, the land cools of more rapidly than water
because of its lower specific heat
Now it is the air above the relatively warm water that
rises and is replaced by cooler air from over the land,
producing a land breeze
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RADIATION
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All objects give off energy as a result of radiation
The energy radiated by an object is in the form of
electromagnetic waves (visible light, infrared, UV)
Unlike conduction and convection, radiation has no
need for a physical material to mediate the energy
transfer, and thus can travel through a vacuum
The radiant energy from the Sun reaches Earth across
150 million kilometres of vacuum
Since radiation includes visible light, it is possible to
see the temperature of an object (i.e. glowing red hot)
The radiated power, P, is the energy radiated per time
by an object
It is proportional to the surface area, A, over which the
radiation occurs
It also depends on the temperature of the object,
specifically the fourth power of temperature T4 (Kelvin)
This behaviour leads to the Stefan-Boltzmann law
P = eσAT4 (Watts)
Stefan-Boltzmann constant: σ = 5.67×10-8 W/(m2.K4)
The emissivity, e, is a dimensionless number between
0 an 1 that indicates how effective the object is in
radiating energy (1 means the object is a perfect
radiator, such as a dark coloured object)
Objects absorb radiation according to the same law by
which they emit radiation
If T is the temperature of an object, and Ts the
surrounding temperature, Pnet is the net radiated power
Pnet = eσA(T4 – Ts4)
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NET RADIATED POWER
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If the object’s temperature is greater than its
surroundings, it radiates more energy than it absorbs
and Pnet is positive
If its temperature is lower than the surroundings, it
absorbs more energy than it radiates and Pnet is
negative
When the object has the same temperature as its
surroundings, it is in equilibrium and Pnet is zero
Example: A person jumps into a lake of icy water. He
has a surface area of 1.15m2 and a surface
temperature of 303K. Find the net radiated power from
this person when he is inside where the temperature is
293K; and when outside where the temperature is
273K. Assume an emissivity of 0.9 for the person’s
skin.
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