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Heat and Temperature
In everyday speech, heat and temperature go hand in hand:
the hotter something is, the greater its temperature. However, there
is a subtle difference in the way we use the two words in everyday
speech, and this subtle difference becomes crucial when studying
physics.
Temperature is a property of a material, and thus depends
on the material, whereas heat is a form of energy existing on its
own. The difference between heat and temperature is analogous to
the difference between money and wealth. For example, $200 is an
amount of money: regardless of who owns it, $200 is $200. With regard
to wealth, though, the significance of $200 varies from person to
person. If you are ten and carrying $200 in your wallet, your friends
might say you are wealthy or ask to borrow some money. However,
if you are thirty-five and carrying $200 in your wallet, your friends
will probably not take that as a sign of great wealth, though they
may still ask to borrow your money.
Temperature
While temperature is related to thermal energy, there
is no absolute correlation between the amount of thermal energy
(heat) of an object and its temperature. Temperature measures the
concentration of thermal energy in an object in much the same way
that density measures the concentration of matter in an object.
As a result, a large object will have a much lower temperature than
a small object with the same amount of thermal energy. As we shall
see shortly, different materials respond to changes in thermal energy
with more or less dramatic changes in temperature.
Degrees Celsius
In the United States, temperature is measured in degrees
Fahrenheit (ºF). However, Fahrenheit is not a metric
unit, so it will not show up on SAT II Physics. Physicists and non-Americans
usually talk about temperature in terms of degrees Celsius,
a.k.a. centigrade (ºC). Water freezes at exactly 0ºC
and boils at 100ºC. This is not a remarkable coincidence—it
is the way the Celsius scale is defined.
SAT II Physics won’t ask you to convert between Fahrenheit
and Celsius, but if you have a hard time thinking in terms of degrees
Celsius, it may help to know how to switch back and forth between
the two. The freezing point of water is 0ºC and 32ºF.
A change in temperature of nine degrees Fahrenheit corresponds to
a change of five degrees Celsius, so that, for instance, 41ºF
is equivalent to 5ºC. In general, we can relate any
temperature of yºF to any temperature
of xºC with the following equation:
Kelvins
In many situations we are only interested in changes of
temperature, so it doesn’t really matter where the freezing point
of water is arbitrarily chosen to be. But in other cases, as we
shall see when we study gases, we will want to do things like “double
the temperature,” which is meaningless if the zero point of the
scale is arbitrary, as with the Celsius scale.
The Kelvin scale (K) is a measure of absolute
temperature, defined so that temperatures expressed in Kelvins are
always positive. Absolute zero, 0 K, which
is equivalent to –273ºC, is the lowest theoretical
temperature a material can have. Other than the placement of the zero
point, the Kelvin and Celsius scales are the same, so water freezes
at 273 K and boils at 373 K.
Definition of Temperature
The temperature of a material is a measure of
the average kinetic energy of the molecules that make up that material.
Absolute zero is defined as the temperature at which the molecules
have zero kinetic energy, which is why it is impossible for anything
to be colder.
Solids are rigid because their molecules do not have enough
kinetic energy to go anywhere—they just vibrate in place. The molecules
in a liquid have enough energy to move around one another—which
is why liquids flow—but not enough to escape each other. In a gas,
the molecules have so much kinetic energy that they disperse and
the gas expands to fill its container.
Heat
Heat is a measure of how much thermal energy is transmitted
from one body to another. We cannot say a body “has” a certain amount
of heat any more than we can say a body “has” a certain amount of
work. While both work and heat can be measured in terms of joules,
they are not measures of energy but rather of energy transfer. A
hot water bottle has a certain amount of thermal energy; when you
cuddle up with a hot water bottle, it transmits a certain amount
of heat to your body.
Calories
Like work, heat can be measured in terms of joules, but
it is frequently measured in terms of calories (cal).
Unlike joules, calories relate heat to changes in temperature, making
them a more convenient unit of measurement for the kinds of thermal
physics problems you will encounter on SAT II Physics. Be forewarned,
however, that a question on thermal physics on SAT II Physics may
be expressed either in terms of calories or joules.
A calorie is defined as the amount of heat needed to raise
the temperature of one gram of water by one degree Celsius. One
calorie is equivalent to 4.19 J.
You’re probably most familiar with the word calorie in
the context of a food’s nutritional content. However, food calories
are not quite the same as what we’re discussing here: they are actually
Calories, with a capital “C,” where 1 Calorie = 1000 calories.
Also, these Calories are not a measure of thermal energy, but rather
a measure of the energy stored in the chemical bonds of food.
Specific Heat
Though heat and temperature are not the same thing, there
is a correlation between the two, captured in a quantity called specific
heat, c. Specific heat measures
how much heat is required to raise the temperature of a certain
mass of a given substance. Specific heat is measured in units of
J/kg · ºC or cal/g · ºC. Every substance
has a different specific heat, but specific heat is a constant for
that substance.
For instance, the specific heat of water, 
, is 
J/kg · ºC or 1 cal/g
· ºC. That means it takes 
joules of heat to raise
one kilogram of water by one degree Celsius. Substances that are
easily heated, like copper, have a low specific heat, while substances
that are difficult to heat, like rubber, have a high specific heat.
, is J/kg · ºC or 1 cal/g
· ºC. That means it takes joules of heat to raise
one kilogram of water by one degree Celsius. Substances that are
easily heated, like copper, have a low specific heat, while substances
that are difficult to heat, like rubber, have a high specific heat.Specific heat allows us to express the relationship between
heat and temperature in a mathematical formula:
where Q is the heat transferred
to a material, m is the mass of the
material, c is the specific heat of
the material, and 
is the change in temperature.
is the change in temperature.Example
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By rearranging the equation above, we can solve for 
:
:
The temperature goes up by 2 Cº, so if the initial temperature
was 12ºC, then the final temperature is 14ºC. Note that when we
talk about an absolute temperature, we write ºC, but when we talk
about a change in temperature, we write Cº.
Thermal Equilibrium
Put a hot mug of cocoa in your hand, and your hand will
get warmer while the mug gets cooler. You may have noticed that
the reverse never happens: you can’t make your hand colder and the
mug hotter by putting your hand against the mug. What you have noticed
is a general truth about the world: heat flows spontaneously from
a hotter object to a colder object, but never from a colder object
to a hotter object. This is one way of stating the Second Law of
Thermodynamics, to which we will return later in this chapter.
Whenever two objects of different temperatures are placed
in contact, heat will flow from the hotter of the two objects to
the colder until they both have the same temperature. When they
reach this state, we say they are in thermal equilibrium.
Because energy is conserved, the heat that flows
out of the hotter object will be equal to the heat that flows into
the colder object. With this in mind, it is possible to calculate
the temperature two objects will reach when they arrive at thermal
equilibrium.
Example
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The heat gained by the gold, 
is equal to the heat lost by the copper, 
. We can set the heat gained by the gold
to be equal to the heat lost by the copper, bearing in mind that
the final temperature of the gold must equal the final temperature
of the copper:
is equal to the heat lost by the copper, . We can set the heat gained by the gold
to be equal to the heat lost by the copper, bearing in mind that
the final temperature of the gold must equal the final temperature
of the copper:
The equality between 
and 
tells us that the temperature change of
the gold is equal to the temperature change of the copper. If the
gold heats up by 30 Cº and the copper cools down by 30 Cº,
then the two substances will reach thermal equilibrium at 50ºC.
and tells us that the temperature change of
the gold is equal to the temperature change of the copper. If the
gold heats up by 30 Cº and the copper cools down by 30 Cº,
then the two substances will reach thermal equilibrium at 50ºC.Phase Changes
As you know, if you heat a block of ice, it won’t simply
get warmer. It will also melt and become liquid. If you heat it
even further, it will boil and become a gas. When a substance changes
between being a solid, liquid, or gas, we say it has undergone a phase
change.
Melting Point and Boiling Point
If a solid is heated through its melting point,
it will melt and turn to liquid. Some substances—for example, dry
ice (solid carbon dioxide)—cannot exist as a liquid at certain pressures
and will sublimate instead, turning directly into gas.
If a liquid is heated through its boiling point, it
will vaporize and turn to gas. If a liquid is cooled through its
melting point, it will freeze. If a gas is cooled through its boiling
point, it will condense into a liquid, or sometimes deposit into
a solid, as in the case of carbon dioxide. These phase changes are
summarized in the figure below.
A substance requires a certain amount of heat to undergo
a phase change. If you were to apply steady heat to a block of ice,
its temperature would rise steadily until it reached 0ºC. Then the
temperature would remain constant as the block of ice slowly melted
into water. Only when all the ice had become water would the temperature
continue to rise.
Latent Heat of Transformation
Just as specific heat tells us how much heat it takes
to increase the temperature of a substance, the latent heat
of transformation, q, tells
us how much heat it takes to change the phase of a substance. For
instance, the latent heat of fusion of water—that is,
the latent heat gained or lost in transforming a solid into a liquid
or a liquid into a solid—is 
J/kg. That means that
you must add 
J to change one kilogram of ice into water,
and remove the same amount of heat to change one kilogram of water
into ice. Throughout this phase change, the temperature will remain
constant at 0ºC.
J/kg. That means that
you must add J to change one kilogram of ice into water,
and remove the same amount of heat to change one kilogram of water
into ice. Throughout this phase change, the temperature will remain
constant at 0ºC.The latent heat of vaporization, which tells
us how much heat is gained or lost in transforming a liquid into
a gas or a gas into a liquid, is a different value from the latent
heat of fusion. For instance, the latent heat of vaporization for
water is 
J/kg, meaning that you must add 
J to change one kilogram of water into
steam, or remove the same amount of heat to change one kilogram
of steam into water. Throughout this phase change, the temperature
will remain constant at 100ºC.
J/kg, meaning that you must add J to change one kilogram of water into
steam, or remove the same amount of heat to change one kilogram
of steam into water. Throughout this phase change, the temperature
will remain constant at 100ºC.To sublimate a solid directly into a gas, you need an
amount of heat equal to the sum of the latent heat of fusion and
the latent heat of vaporization of that substance.
Example
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First, we need to know how much heat it takes to raise
the temperature of the ice to 0ºC:
Next, we need to know how much heat it takes to melt the
ice into water:
Last, we need to know how much heat it takes to warm the
water up to 10ºC.
Now we just add the three figures together to get our
answer:
Note that far more heat was needed to melt the ice into
liquid than was needed to increase the temperature.
Thermal Expansion
You may have noticed in everyday life that substances
can often expand or contract with a change in temperature even if
they don’t change phase. If you play a brass or metal woodwind instrument,
you have probably noticed that this size change creates difficulties
when you’re trying to tune your instrument—the length of the horn,
and thus its pitch, varies with the room temperature. Household
thermometers also work according to this principle: mercury, a liquid
metal, expands when it is heated, and therefore takes up more space and
rise in a thermometer.
Any given substance will have a coefficient of linear
expansion, 
, and a coefficient
of volume expansion, 
. We can use these coefficients
to determine the change in a substance’s length, L,
or volume, V, given a certain change
in temperature.
, and a coefficient
of volume expansion, . We can use these coefficients
to determine the change in a substance’s length, L,
or volume, V, given a certain change
in temperature.
Example
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10–5/Cº, and
the coefficient of linear expansion for brass is 1.9 
10–5/Cº.First, let’s see how much the steel expands:
Next, let’s see how much the brass expands:
The difference in length is 
m. Because the brass expands more than the
steel, the bimetallic strip will bend a little to compensate for
the extra length of the brass.
m. Because the brass expands more than the
steel, the bimetallic strip will bend a little to compensate for
the extra length of the brass.Thermostats work according to this principle: when the
temperature reaches a certain point, a bimetallic strip inside the
thermostat will bend away from an electric contact, interrupting
the signal calling for more heat to be sent into a room or building.
Methods of Heat Transfer
There are three different ways heat can be transferred
from one substance to another or from one place to another. This
material is most likely to come up on SAT II Physics as a question
on what kind of heat transfer is involved in a certain process.
You need only have a qualitative understanding of the three different
kinds of heat transfer.
Conduction
Conduction is the transfer of heat by intermolecular
collisions. For example, when you boil water on a stove, you only
heat the bottom of the pot. The water molecules at the bottom transfer
their kinetic energy to the molecules above them through collisions,
and this process continues until all of the water is at thermal
equilibrium. Conduction is the most common way of transferring heat
between two solids or liquids, or within a single solid or liquid.
Conduction is also a common way of transferring heat through gases.
Convection
While conduction involves molecules passing their kinetic
energy to other molecules, convection involves the
molecules themselves moving from one place to another. For example, a
fan works by displacing hot air with cold air. Convection usually
takes place with gases traveling from one place to another.
Radiation
Molecules can also transform heat into electromagnetic
waves, so that heat is transferred not by molecules but by the waves
themselves. A familiar example is the microwave oven, which sends
microwave radiation into the food, energizing the molecules in the
food without those molecules ever making contact with other, hotter
molecules. Radiation takes place when the source of heat is some
form of electromagnetic wave, such as a microwave or sunlight.
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