Physics 182 - Summer 2013 - Experiment #2
1
Experiment #2, The Specific Heat of Aluminum
1 Purpose
1.
To review the physical concepts and relationships associated with the flow of heat
in and out of materials.
2.
To come to a deeper understanding of the principle of conservation of energy as it
applies to thermal energy.
3.
To determine the specific heat of aluminum in the temperature range between
boiling water and room temperature water.
4.
To look at the impact on an experiment when the thermometer used to make
temperature measurements is not very precise. It has a relatively large reading
uncertainty (error) due the spacing between scales and to reading parallax.
2 Introduction
Heat is the energy associated with the thermally excited motion of atoms. When heat energy is
added to a substance at constant pressure, its temperature usually rises. The exception to this rule
occurs when there is a change of phase (e.g., a solid is changed into a liquid phase or a liquid
state is changed into its vapor phase). A phase change occurs without increase or decrease in the
substance's temperature.
If an object is isolated from the rest of the universe no heat can flow into or leave that object. As
a result its temperature will remain unchanged. If it is brought into thermal contact with another
object that is at a different temperature heat will flow out of the object that is hotter and into that
which is cooler. As a result the thermal energy of the hotter object will decrease and that of the
cooler object will increase. Heat will continue to flow until the temperature of both objects is the
same. At this point they are said to be in thermal equilibrium.
The historical unit of heat energy, the calorie, was defined as the amount of heat energy needed
to raise the temperature of one gram of water by one degree Celsius. The calorie is now defined
in terms of the SI unit of energy, the joule, by:
1 cal = 4.184 J
2.1 Specific Heat
When heat flows into or out of an object, its temperature will change. The connection between
the change in heat energy and the change in temperature is the specific heat. The heat energy ΔQ
needed to raise the temperature T of a substance is related to its mass m according to the
formula:
Q = mcT = mcTf – Ti)
(1)
The specific heat c is defined as the heat energy needed to raise the temperature of one gram of a
substance by one degree Celsius (C). In general, the value of the specific heat of a solid
substance is predominantly a function of temperature, though small variation of the specific heat
occurs due to variation in pressure or volume. The value of c in Eq. (1) is taken as its average
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value, c̄ , over the temperature interval between its initial and final temperatures, Ti and Tf
respectively.
The specific heat of substances varies with the temperature, as for example, the average value of
the specific heat of aluminum is 0.17 cal/(g-C) between room temperature and liquid nitrogen
temperature while it remains essentially constant (0.215 cal/(g-C)) from room temperature to
100 C. In contrast to this, the specific heat of water decreases from 1.00728 cal/(g-C) to
0.99795 cal/(g-C) in the temperature range 0 to 35C and then increases to 1.00697 cal/(g-C) at
100C.
2.2 Determination of the Specific Heat of Aluminum
2.2.1
In the Temperature Range of 100C to 20C
In this experiment, the specific heat of aluminum is measured by immersing an aluminum cube
of known mass mAl and initial temperature Ti,Al (around 100.0C) into water of known mass mw
and initial temperature Ti,w, and measuring the final same equilibrium temperature Tf. At thermal
equilibrium, the cube and the water are at the same temperature. If there is no loss or gain of heat
energy from or to the environment by the water or the aluminum cube (the cube and the water is
an isolated system), the heat lost by the aluminum will be equal to the heat gained by the water.
Conservation of energy, QAluminum + QWater = 0, yields:
-QAluminum = QWater
(2)
The minus sign indicates that heat is flowing out of the aluminum cube and into the water.
Applying the principle of the conservation of energy, -QAluminum = QWater, and using Eq. (1),
one obtains the following equation applicable to this experiment.
m Al c Al ( T i, Al - T f ) = m w c w ( T f - T i,w )
Solving for the specific heat one obtains:
c̄ Al =
m w c w ( T f - T i,w )
m
T w
= w cw
m Al ( T i, Al - T f )
m Al T Al
(3)
(4)
3 Experimental Apparatus and Procedure
3.1 Experimental Apparatus
1.
2.
3.
4.
5.
6.
7.
8.
Celsius Thermometer
Aluminum Cube
Water
Small Double Styrofoam Cup for Water
Beaker
Hot Plate
Digital Thermometer
Double Pan Balance
3.2 Procedure. Data goes into the data section of the Lab Report.
Measurement of the Average Specific Heat of Aluminum in the Temperature Range
100C to 20C. Enter data in the data sheet located at the end of this write-up.
Physics 182 - Summer 2013 - Experiment #2
3
1.
Determine the mass, in grams, of the aluminum cube (mAl). Example: 43.98
grams. Use two numbers after the decimal point. Be sure to place another paper
clip on the right pan of the balance to “balance out” the paper clip attached to the
cube. The mass of the thread is considered to be negligible.
2.
Place water and digital thermometer into a glass beaker. Place the beaker on a hot
plate and heat it. Do not use a setting higher than 4 on the hot plate. Once the
water is at a steady boil, note the temperature on the digital thermometer, and
immerse the cube. The temperature will recede. Two groups may use the same
hotplate. Be sure to keep track of your cube and the boiling time.
3.
While the cube in the beaker is getting to the boiling point, weigh the small
double Styrofoam cup. Fill it with room temperature tap water (60 to70 g) and
weigh it again. Compute the mass (mw) of the water in the cup. (Here, g = grams.)
Example: 60.44 grams. Use two numbers after the decimal point.
4.
When the boiling water returns to approximately the temperature noted in part 2
(before the cube was immersed) the cube should be close to, but not necessarily
at, 100.0C. Let the cube boil around three minutes. It was important to note the
temperature of the boiling water before cube was immersed. (The boiling point
temperature changes with the atmospheric pressure. In addition, there may be a
small error in the setting of the digital thermometer. We will assume a boiling
temperature of 100.0C in order to ignore systematic, and other types of errors.)
5.
Insert a thermometer into the cup with 60 to70 g of tap water, and measure the
initial temperature of the tap water (Ti,w). Allow a few minutes for the
thermometer to come to thermal equilibrium.
6.
When the water in the beaker with the cube has been boiling vigorously for three
minutes, and the temperature returns to approximately 100.0C, record the initial
temperature (Ti,Al) of the cube as 100.0C. Again, we will assume that the
temperature is 100.0C as the digital thermometer may have a calibration error.
7.
Quickly remove the cube and insert it into the cup with the tap water. Be very
careful not to splash water out of the cup. This will result in a measurement
error.
8.
Begin stirring the water once the cube is in. Again, do not splash water out of the
cup. Make sure the cup with the thermometer is not close to the hot plate. When
the water and aluminum cube system attains its highest temperature, measure and
record the final thermal equilibrium temperature (Tf) of the system. Not a bad idea
to take a temperature reading every 30 seconds, as you continue to stir, in order to
truly quantify the highest temperature reached. Make sure the bottom of the
thermometer is close to the bottom of the cup.
4 Calculations and Analysis of the Data and Error for Lab Report
4.1
Computation of the Average Specific Heat of Aluminum in the Temperature Range
of 100C to 20C. Use Excel in the lab and a calculator for your lab report.
1.
Use Eq. 4 to calculate the experimental average specific heat capacity c̄ Al,exp. For
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all calculations, use Excel in the lab and your calculator for your lab report. For
this calculation, assume that the specific heat of water to be 0.998 cal/(g-C) in the
vicinity of room temperature. The accepted value is c̄ Al,acc = 0.215 cal/(g-C) in
this temperature range.
Calculate the percent error in your experimental value of c̄ Al (c̄ Al, experimental).
Percent error is given by:
% error = ( | c̄ Al, experimental - c̄ Al, accepted | / | c̄ Al, accepted | ) (100%)
This should be a positive value due to the use of the absolute value | |. The percent
error is a useful way to measure accuracy.
2.
Use propagation of uncertainty (error) methods to obtain ± c̄ Al,exp, the respective
measurement error in your experimental value of the average specific heat of
aluminum, c̄ Al,exp. Note that the error for c̄ Al,exp, is given by:
from
c̄ Al,exp =
( m - m c )( T f - T i,w )
m w c w T w
= c w c w
m Al T Al
m Al ( T i, Al - T f )
2
2
δcAl, exp = cAl, exp
2
2
δm 2
δm + ( δT 2 ) + ( δT
)
(
) + (
)
Ti, Al - Tf
Tf - Ti, w
m Al
mcw - mc
Or, you can use:
2
δcAl, exp = cAl, exp
2
2
2
δT
δm
δT
)
2(
) + ( δm ) + 2(
) +(
Ti, Al - Tf
Tf - Ti, w
m Al
mcw - mc
Find propagation of error methods in supplementary material entitled "Appendix
on Error Analysis" which is located at www.physicslabs.umb.edu. Errors T and
m should be entered in your data sheet below and in your lab report’s data
δ
δ
as both
section. δm 2 in the above expression is equal to
mc+w and mc have the same uncertainty in the measurement of mass. Same idea
for δT 2 .mandTis the uncertainty (precision) in a single measurement.
5 Questions
1.
List various factors, including errors, which might affect your experimental value
of the specific heat of aluminum at room temperature.
2.
We have ignored the mass and specific heat of the thermometer. Does this make
your measurement of the heat capacity of aluminum higher or lower?
3.
What type of error would result if a digital thermometer was not properly
calibrated? We ignored this error by using 100.0C for boiling water. See the
appendix on error analysis located at www.physicslabs.umb.edu to help you to
answer this question. Same place you download this write-up.
4.
Would the final equilibrium temperature be higher or lower if 200.00 grams of tap
water was used in this experiment? Explain.
Physics 182 - Summer 2013 - Experiment #2
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5.
Change both values of Tf in Equation (4) by adding your error of 0.5oC in
measuring the temperature. Calculate the value of the specific heat of aluminum
with these new value of Tf . Based on the results of this calculation, comment on
how this error in reading the temperature could have affected your experimental
results.
6.
When adding your error in measuring the temperature to Tf, which term in
Equation (4) is most effected by this error; top one (Tf – Ti,w) or bottom one (Ti,Al –
Tf)? We are asking which term has the greatest impact on your calculations of the
specific heat if there is an error in reading the thermometer. You should show
both calculations, as you should change only one Tf for each calculation.
7.
Does c̄ Al,accepted, 0.215 cal/(g-C), fall within the interval determined by the limits
of your experimental error (precision), ±c̄ Al,exp, for c̄ Al,exp , which was determined
in your calculations? We want to know if the following is true:
| c̄ Al, experimental - c̄ Al, accepted |  c̄ Al,exp
The expression | c̄ Al, experimental - c̄ Al, accepted | should be shown numerically. It is
always positive as the absolute value | | is used. It is then compared to c̄ Al,exp, and
commented on. This term, , means less than or equal to.
This question is basically asking if c̄ Al,accepted = 0.215 cal/(g-C) is in the interval
of values given by:
(c̄ Al,exp - c̄ Al,exp , c̄ Al,exp + c̄ Al,exp), or, c̄ Al,exp ± c̄ Al,exp
For your lab report, the final experimental result for the average specific heat of
aluminum is entered as the following, with your values for c̄ Al,exp and ± c̄ Al,exp:
c̄ Al = c̄ Al,exp ± c̄ Al,exp
6 Conclusion
This section should have a clear statement of the results of the experiment and the extent to
which the results are in agreement with the theory being tested. When the experiment results in a
measurement of a constant (e.g., the specific heat of aluminum at a given temperature) compare
it with its established handbook value. Here, 0.215 cal/(g-C). Use percent error for this
comparison.
To make this comparison meaningful, you should include the impact of the experimental error on
your results. This includes errors in plotting and reading linear graphs when determining their
slope and intercept. In addition, please include a statement of what you have learned, a critique
of the experiment, and any suggestions you have which you think could improve the experiment
or the lab handout.
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Specific Heat of Aluminum
Data Sheet
Data goes into the data section of the lab report. Keep it neat and
organized.
3.2
Measurement of the Average Specific Heat of Aluminum in the Temperature Range
100C to 20C.
Mass of Aluminum Cube (mAl)
=______________g
Mass of the Styrofoam cup
=______________g
Mass of the Styrofoam cup plus water
=______________g
Mass of water (mw)
=______________g
Initial temperature of Tap Water (Ti, w)
=______________oC
Initial temperature of Aluminum Cube (Ti, Al )
=
Final temperature of Water (Tf, w)
=______________oC
Final temperature of Aluminum Cube (Tf, Al)
=______________oC
100.0
o
C
Estimated error in reading balance. Errorm =  m = _______ g
Estimated error in reading thermometer. ErrorT =  T = _______ oC
Note that error measurements always go into your lab report’s data section.