Heat
We saw that when two bodies are
not in thermal equilibrium and are in contact with each other, the temperatures
of the objects changes until equilibrium is established. Obviously some
sort of interaction takes place, and as in all interactions, something
is exchanged between the two systems. That something is heat. We
call the exchange of heat between the two systems heat transfer.
Heat transfer has been shown via experiments by Rumford and Joule to be
an energy transfer between the systems.
We define heat transfer as
energy transfer that takes place solely because of a temperature difference.
This does not mean that heat transfer is the only method of transferring
energy. We have seen many examples of other forms of energy transfer throughout
this course. The unit of the quantity of heat transferred with reference
to the temperature change of any particular material is called the calorie.
It is defined to be the amount of heat required to heat one gram of water
from 14.5°C
to 15.5°C.
Since heat is really energy transfer, we can relate the calorie to the
joule, and experimentally we find that
1 cal = 4.186 J
What happens when a heat is
transferred to an object of mass m? We expect that the temperature
will rise. The amount that the temperature changes should also depend on
the mass of the object; it takes more heat to change the temperature of
a massive object than of a non-massive object. This lets us write
where DQ
is the amount of heat added and c is a constant of proportionality.
It varies from material to material, and is called the specific heat
capacity of the material.
Example:
What is the specific heat
capacity of water?
From the definition of a joule,
we see that
Example:
A 1000 kg car is traveling
at 65 m/sec when it hits a tree and comes to a halt. Assuming that the
car is completely made of iron and that all of the energy is converted
to heat, what is the final temperature if the original temperature is 5°C?
The energy released in the
deceleration was
If all of the energy is released as heat
Phase Transitions
We have all seen the situation
where we fill an ice tray with water, put it into the freezer and come
back later to find the tray filled with ice cubes. How can we describe
this physically? Obviously, from (78) we see that as we extract heat from
the water it cools down until it reaches the freezing point of water, 0°C.
However, experience tells us that the water does not instantly change to
a block of ice; rather the change happens slowly. Since the freezer is
continuing to extract heat, what is the work being done by the heat? If
we set up an experiment to monitor the temperature of the ice while it
is freezing, we would find that the temperature does not change until all
of the water has changed to ice. Once the change is completed, the temperature
of the ice will again begin to fall until it reaches equilibrium with the
freezer.
The conversion of water to
ice is an example of a phase change. The water has not changed its
molecular composition. Rather it has transitioned from its liquid phase
into its solid phase. The amount of heat required to compete a phase change
differs from material to material, and must be determined experimentally.
The amount of heat required is given by
where LF is called the heat
of fusion. It is the amount of heat required to convert a unit mass
of the material from its solid to its liquid phase, or vice versa. We expect
that the amount of heat required to undergo a phase change from liquid
to gaseous will be given by a relation similar to (79), but there is no
reason for the constant of proportionality to be the same. Indeed, experiment
shows that it is not, so
where LV is the heat of
vaporization. Finally, under some conditions, it is possible for a
substance to transition directly from solid to gaseous. This process is
called sublimation. Associated with it is the heat of sublimation.
The boiling and freezing
points of a substance depends on the pressure acting on the substance.
This leads to the fact that under certain conditions, it is possible to
cool (or heat) a substance below its freezing (or boiling) point and not
have a phase change occur. The liquid is then said to be supercooled
(or superheated). In both cases if an impurity or disturbance is
introduced in the substance, the phase change will occur locally. This
is the principle used in bubble and cloud chambers.
Example:
A restaurant serves coffee
in copper mugs. A waiter fills a cup of mass 0.1 kg, initially at 20°C,
with 0.2 kg of coffee initially at 70°C.
What is the final temperature after the coffee and the cup attain thermal
equilibrium? Assume that the coffee has the same specific heat capacity
as water.
Let the final temperature
be T. The heat extracted from the coffee is
Qc
= mccc (T -Tc)
Similarly, the heat absorbed by the mug is
Qm
= mmcm (T - Tm)
If we assume that there is no interaction with
the outside air, then this is a closed system, and so by energy conservation
Qm + Qc = 0. Thus
Example:
A physics student wants to
cool 0.25 kg of soda, which is initially at 25°C,
by adding ice initially at -20°C.
How much ice should be added so the final temperature will be 0°C
with all the ice melted? Assume that the heat capacity of the container
is negligible, and the soda has the same specific heat capacity as water.
The heat to be extracted from
the soda is
This must equal the amount of heat provided by
the ice
so
