Introduction:
The amount of heat, Q, required to
raise the temperature of a solid body at constant pressure depends on the
change in temperature, D T, of the body, its mass, m, and a characteristic
of the material forming the body called its specific heat, C. This
relationship is expressed by the equation Q = mCD T
and the dimensions of C
are thus heat per unit mass per unit temperature change. The values of C
do depend on temperature with those of common metals such as aluminum and brass
increasing a few percent as the temperature increases from 20°C to 100°C,
for example, while that for iron or steel increases about 10% over the same
range. Since these are not large changes, average specific heats are often
quoted in handbooks for such fairly broad temperature ranges.
Historically the amount of heat, Q,
was originally expressed in terms of calories. The calorie was defined most
accurately as the amount of heat required to raise the temperature of 1 gram
of water from 14.5°C to
15.5°C at
1 atmosphere pressure. With this definition the specific heat of water between
0°C
and 100°C is
1.00 cal/gm× °C to
within better than 1%. The use of the calorie began before it was established
that heat was a form of energy and that 1 calorie is the equivalent of about
4.18 Joules. Thus in the SI system of units specific heats, that is the
values of C for particular materials, are expressed as J/kg× °C and there is no need for the calorie.
However, since so much work involving heat has used the calorie and
since the specific heat of water is unity when it is employed, it remains a
common unit and will be used in this work. The food Calorie, with a
capital C is 1000 of these calories or 1 kilo-calorie.
The process of measuring quantities of
heat exchanged is called calorimetry. In this experiment your objective will be
to determine the average specific heat of several metals over a certain
temperature range by the calorimeter method of mixtures.
Theory:
We know that when two bodies, initially
at different temperatures, are placed in intimate contact, in time they will
come to equilibrium at some intermediate temperature. Provided no heat is lost
to or gained from the surroundings, the quantity of heat lost by the hotter
body is equal to that gained by the colder body. This is the process which
occurs in the method of mixtures that you will use. The metal sample whose
specific heat is to be measured is heated in boiling water to about 100°C. It is then quickly transferred to an aluminum
calorimeter cup which contains cold water of known temperature. When the metal
sample and calorimeter cup come to equilibrium, the common temperature is
measured with a thermometer. It is assumed that the transfer of heat between
the thermometer and the system is small enough to be neglected. If the net heat
exchange with the surroundings can be kept small, then the heat lost by the
metal sample equals the heat gained by the water and the calorimeter cup.
Let Ms be the mass of
the sample whose specific heat is Cs. Let Ts
be its temperature before it is placed in the calorimeter. Let Mw
and Cw be the mass and specific heat of the water and let Mc
and Cc the mass and specific heat of the calorimeter cup. Denote
the temperature of the water and calorimeter cup before the sample is added by Tw
and the final temperature of the mixture by Tf. Now use these
symbols to express mathematically the situation when a hot object (the sample)
is placed in contact with a cooler one (the water and the calorimeter cup) and
the two are allowed to exchange heat until they reach a common temperature.
From this equation derive an expression for the specific heat of the sample in
terms of the other quantities.
Procedure:
Fill a beaker with enough water so that
the sample when placed in it will be covered with some to spare. Bring the
water to a boil using a bunsen burner. Weigh the aluminum sample and the dry
inner calorimeter cup. Note that the plastic top on the calorimeter is a
thermal insulator whose temperature like that of the outer cup is assumed to be
unaffected by changes in the temperature of the inner calorimeter cup during
the course of experiment. Suspend the sample in the boiling water with string
and a glass rod making sure not to touch the sides or bottom.
While the aluminum sample is reaching
equilibrium, fill the calorimeter cup about 2/3 full of
cool water (about 5°C below room temperature). Cool water can be had
from a water cooler. Ice is also available. Weigh the cup plus water. Care
should be taken that the water is not so cold as to cause condensation on the
outside of the calorimeter. If condensation does occur, dry the outside of the
inner cup before proceeding.
Place the thermometer in the boiling
water near the sample, not touching the bottom. When equilibrium is reached,
record the sample temperature. Remove the thermometer, cool it with tap water,
wipe it dry and then place it in the calorimeter cup. Record the temperature as
soon as it is reasonably steady and then quickly transfer the sample from the
boiling water to the calorimeter. Care must be taken not to carry any hot water
over with the sample nor to splash any cold water out of the calorimeter. Stir
the water in the calorimeter gently with the glass rod and observe the
temperature. When equilibrium is reached, record the temperature. Always read
the thermometer as accurately as you can, interpolating between the marks.
Calculate the average specific heat of
your aluminum sample as determined by your experiment and specify the
temperature range over which your value applies. Since the specific heat of the
aluminum calorimeter cup is also an unknown, approximate its value by assuming
it to be the same as the sample even though the cup and the sample are subject
to different temperature ranges. (Later when you check the effect of this
approximation you will see that it introduces little error.) If the value you
obtain for the specific heat of your sample is not between 0.19 and 0.25 cal/g× °C, repeat the experiment to improve your technique.
You may, for example, want to use a different temperature for the cool water.
The objective is to have the cool water as far below room temperature initially
as the final temperature of the mixture is above room temperature.
Perform the experiment for the other
samples and determine their specific heats. Compare your results with the
accepted values (which your instructor will furnish) and calculate the
percentage error and estimate the uncertainty in your experimental technique.
Questions:
1. What do you see as the major sources of error in
this experiment? Use your calculation for steel as an example and determine the
effect on the measured specific heat if the sample were to cool down 3°C during the transfer. Then calculate the effect if
the net uncertainty in Tf - Tw were ± 0.2°C. These calculations will give you some idea of
the sensitivity of the results to some of the measured variables.