# Specific Heat

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```					                SPECIFIC HEAT AND LATENT HEAT OF FUSION

Introduction:

Heat, Q, is thermal energy transferred as a consequence of temperature differences.
Internal energy, Eint, is the total thermal energy a substance has at some temperature T. The unit
of heat most commonly used is the calorie (cal), it is defined as 4.186 J. The heat Q required to
change the temperature of a mass m of a given material by an amount T = Tf - Ti is
Q = C T = cp m T
where the specific heat cp for a given material is defined as the heat capacity C per unit mass.
Specific heats for some common materials are given in your textbook. More extensive tables can
be found in other references. A useful number is the specific heat for liquid water, 1.00 cal/gC.
(Note that different phases of the same substance can have different specific heats. For example,
the specific heat for ice near 0C is 0.53 cal/gC.)

A change in internal energy is involved when a substance undergoes a change of phase;
the heat Q required is

Q=Lm

where m is the mass of material whose phase is changed and L, called the latent heat, depends on
the nature of the phase change as well as on the properties of the substance. As a substance
changes from a solid to a liquid phase, the heat required to effect this change for a unit mass of
the substance is called its latent heat of fusion LF. The heat Q required to change mass m from
solid to liquid is Q = LF m. As the substance changes from the liquid to the solid phase, it
releases an equal quantity of heat, i.e. Q = - LF m. The latent heat of fusion of water, the heat
required to change ice to liquid water at 0C, is 333 kJ/kg = 79.5 cal/g.
The first law of thermodynamics is a generalization of the law of conservation of energy
to include heat and possible changes in internal energy. It states that the change in internal
energy of a system, Eint = Eint,f - Eint,i, is equal to the heat Q added to the system minus the
work W done by the system:

Eint = Eint,f - Eint,i = Q - W

In this experiment you will determine the specific heats of some samples of various
materials by measuring their change in temperature when a measured amount of thermal energy
is added. You will also measure the latent heat of fusion of water by determining how much
thermal energy must be added to melt a measured mass of ice. No significant changes in
mechanical energy will be involved; only exchanges of thermal energy among the parts of the
system being studied will be considered.

10-1
You will use a calorimeter, a device in which an inner system is isolated as well as
possible from thermal energy transfers to or from the external environment. This inner system
includes known or measured amounts of known substances as well as a sample of a substance x
under study. The initial temperatures of the components of the inner system are known or
measured. The components are allowed to exchange thermal energy and come to a common
equilibrium temperature which is measured. For such a system, energy is conserved (to the
extent that exchanges with the environment can be neglected); the thermal energy gain by some
parts of the system must equal the thermal energy loss by other parts. A single unknown (e.g. cx
or mx) in the equation relating the thermal energy transfers can be determined.

Equipment:
Calorimeter with aluminum inner cup and aluminum stirrer, several metal samples
including aluminum, copper, steel and brass, ice at 0C, laboratory balance, dial thermometer,
tongs, beaker, water near room temperature, cold and hot tap water, facilities for heating
samples.

Inner Al Cup
Phenolic Ring
Lid

Outer
Jacket

Water

Stirrer

Procedure:

A. Specific Heat.
In this part of the experiment the inner calorimeter cup, the water in it, and the stirrer all
gain thermal energy when a heated sample is placed within the cup. The sample loses thermal
energy. First determine and record the room temperature. Note the material used for the
calorimeter's inner cup and stirring device. Make sure the cup and stirrer are dry. Measure their
masses and estimate the errors. Fill the inner cup about half full with water which is about 4 C
colder than room temperature. (This is to minimize net thermal energy exchange with the
surroundings. After adding the hot sample and allowing the system to come to equilibrium, the
final equilibrium temperature should be a few C above room temperature.) Measure the mass
10-2
of the cup plus water and estimate the error. Assemble the calorimeter, including the
thermometer, and stir the water so the whole inner assembly comes to equilibrium at a common
temperature. Record this starting temperature and its estimated error. Take the calorimeter to
the hot water bath. Record the temperature (with error) of the aluminum sample in the hot water
bath and QUICKLY transfer the sample to the inner cup. Cover the calorimeter and observe the
temperature changes as you stir the water. Record the equilibrium temperature and its error.
(The temperature should rise rapidly as parts of the system exchange heat, then decrease slowly
as heat is lost to the surroundings. The best estimate for the equilibrium temperature is the
highest temperature reached.) Remove the sample from the cup and dry it off. Determine the
mass of the (dry) sample and its error. After you are finished with the sample, carefully return it
to the hot water bath.

Repeat the process for two other samples of different materials. Among samples which
may be available are copper, brass, and steel. (Do not use both copper and brass samples.) Note
and record the nature of the samples you use.

Values for the specific heats of the sample substances in cal/gC from various tables
include: aluminum, 0.215; copper, 0.0923; brass, 0.089 to 0.092; and steel, 0.107 to 0.118.

B. Latent Heat of Fusion.

In this case the inner cup should initially contain water about 4 C above room
temperature. (After adding ice and allowing the system to come to equilibrium, the final
equilibrium temperature should be a few C below room temperature.) After determining the
mass of water plus cup and its error, assemble the calorimeter, stir, and determine the initial
temperature and its error. Then drop a sample of ice at 0C (Try not to add any liquid water at
0C.) into the calorimeter, stir, observe the temperature changes, and record the equilibrium
temperature. As before, estimate the errors in all measured quantities. NOTE: Do not add too
much ice or you may end up with some ice still unmelted, the system at 0 C, and no way to
calculate the latent heat. A 10 to 30 gram sample of ice is probably about right. After the
temperature is measured, measure the final mass of water plus cup and its error. The difference
between final mass and the starting mass determines the mass of ice which was added.

Analysis.
In the calculations of this section, you may neglect any uncertainties in given values of
specific heats and latent heats. If you check, you should find that the relative errors in the
masses are much smaller than the relative errors in the temperature changes. If this is true, the
errors in the masses may be neglected in the error calculations.

A. Specific Heat.
In each case, the thermal energy gain of the calorimeter cup c, water w, and stirrer s
should equal the thermal energy loss of sample x as equilibrium is attained:

Qx = Qw + Qc + Qs

(The small contribution of the thermometer and any net heat exchanged with the
environment will be neglected.)

In terms of the masses, specific heats, and temperature changes this becomes:

10-3
-mx cx Tx = mw cw Tc + mc cc Tc + ms cs Tc

where all of the mi and Ti can be determined from your measurements. The specific heat of
liquid water cw is 1.00 cal/gC. The specific heats of the cup and stirrer are cc = cs = cAl.

In the case of the aluminum sample, cx is also cAl which is the only undetermined
quantity in the equation. Solve the equation for cAl. Substitute the values from your
measurements and calculate a value for cAl. Use your error estimates and the rules of error
propagation to calculate the error for your value. Is your measured value for the specific heat of
aluminum consistent with the accepted value given below? Discuss (even if your measurement
is consistent with this value) possible sources of error which have not been considered and which
might contribute to any discrepancy.

For the remainder of the experiment, use the accepted value, cAl = 0.215 cal/gC, for the
specific heat of aluminum. For the other two samples used in your measurements, the only
unknown in the equation is cx. Calculate values and the corresponding errors for cx for your
other two samples using the results of your measurements. Consider the specific heats you
determined for these other two samples. Are they consistent with the values listed in the
Procedure section?

B. Latent Heat of Fusion.
In this case, the thermal energy gained by the ice in melting plus the thermal energy
required to raise the temperature of the water from the melted ice to equilibrium should equal the
thermal energy lost by the calorimeter and the water it initially contained.

Qmelting + Qice water = Qcalorimeter
or
mice LF + mice cw Tice water = -[ mw cw + mc cc + ms cs ]Tc

in terms of masses, specific heats, temperature changes, and LF, the latent heat of fusion for
water.

Here Tice water is the temperature change of the water from the melted ice. Use the
accepted values for cw, cc, and cs. The only quantity not available from your measurements or
from previous knowledge of specific heats is LF. Determine LF and its error using your
measurements. Is your measured value for the latent heat of fusion of water consistent with the
value given in the Introduction? Again, consider and discuss possible sources of error which
may not have been considered in your error calculation.

Questions for the report:

1)     Locate values in handbooks or other references for the specific heats of the materials of
which the samples of part A were made. List the values you find and the sources from
which they come. How do the values you found compare with the values given in these
instructions and in your text. Compare your experimental values with values from your
other sources. Discuss your findings. Can you think of any reasons why the values given
in different references might differ?

10-4
2)     Comment on the differences you would expect to observe in part B. if you added m
grams of water at 0C instead of adding m grams of ice at 0C.

3)     Comment on how the computed value of a specific heat in part A would be affected if
you carried along some boiling water with the metal sample when you transferred it to
the inner calorimeter cup.

4)     Why would it not be practical to heat the metal samples to a much higher temperature in
an oven so a greater change in the temperature of the water in the calorimeter would be
produced?

NOTE: Before you leave the lab, you must have all the measurements and error estimates (for
masses and temperatures) needed to calculate the specific heats for three samples and the latent
heat of fusion of water. If possible, try some calculations before you leave to make sure that you
have all required information and that your measurements seem reasonable. Make sure (as you
always should for all measurements) that appropriate units have been listed for your
measurements. Each lab partner must have a complete data sheet initialed by your TA. The rest
of the work can be done outside of the laboratory.

Data on “Specific Heat & Latent Heat” Experiment
10-5
room temperature = Troom
mass of cup         = mc
mass of stirrer     = ms
error in temperature = T
error in mass        = m

A. Specific Heat for 3 Materials.

sample x      mx       mw         Tin   Tbath    Tfin   Tx=          Tc=
Tfin- Tbath   Tfin-Tin
Al

B. Latent Heat for Ice Fusion
10-6
mass of cup plus water, =mc+ mw
initial temperature ,    = Tin
ice temperature,         = Tice = 0 0C
final temperature,       = Tfin
mass of cup +water +ice,         = mc + mw + mice

mice 
Ticewater  Tfin  Tice =
Tc  Tfin  Tin         =



[mwc w  (mc  ms )c Al ]Tc  micec wTicewater
       LF                                                      =
mice
2
[mwc w  (mc  ms )c Al ]2 [ (Tc )] 2  [micec w (Ticewater )]
(LF ) 2
=
                                            mice 2



10-7

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