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ASEN 2012 Project 1: Calorimetry
This document outlines the experimental process through which a calorimeter may be
used to determine the specific heat of a substance. This unknown specific heat value is found
through heat transfer into an insulated calorimeter with a known specific heat. By measuring
the temperature changes through this process, we can apply the First Law of Thermodynamics
to calculate Specific Heat as shown in the following procedures.
Nomenclature
퐶푠 = specific heat of sample
푚푐 = mass of calorimeter
푚푠 = mass of sample
퐶푐 = specific heat of calorimeter
푇2 = final temperature of sample and calorimeter in equilibrium
푇1 = initial temperature of the sample
푇0 = initial temperature of the calorimeter
I. Introduction
T
he specific heat of sample D is unknown to us and needs to be determined through experimental methods. We
use a heavily insulated calorimeter made with Aluminum 6061 to model an adiabatic system. Because we have no
heat loss to the surroundings, we measure the temperature changes in the calorimeter to observe heat transfer from the
sample. Our measurements require least squares fitting in order to properly model an adiabatic system, as outlined
below. Once the adiabatic assumption holds true, the First Law of Thermodynamics is applicable to the problem, and
we can calculate the unknown specific heat of the sample.
II. Experimental Method
The following section outlines the processes through which the specific heat of a sample were calculated. The
purpose of this experiment is to determine what material the sample is made out of based on the calculated specific heat.
Fig. 1 Calorimeter diagram
[1]
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A. Taking Calorimeter Measurements
There were four thermocouples used while taking temperature measurements throughout the process. These
thermocouples measured the temperature in two instances of the calorimeter, the air of the room, and the boiling
water. These measurements were being taken about ten minutes prior to the insertion of the sample, and kept taking
measurements for about seven minutes after the insertion. These values were then put into a spreadsheet so as to enable
the least-squares fitting of the data. The initial temperature of the calorimeter at time of insertion, the temperature of the
calorimeter at thermal equilibrium, and the initial temperature of the sample are to be determined.
B. Least-Squares Linear Fitting of Data
We based this experiment off the assumption that this would be a adiabatic process; as such, once the data is
formatted for use in MATLAB, we use least-squares linear fitting to account for heat loss and provide a more accurate
adiabatic assumption. We first look for the absolute maximum and minimum values in the data, which represent the
approximate times of equilibrium and sample insertion, respectively. By fitting a line to the data before insertion, we are
able to approximate an initial temperature of the calorimeter before the sample was inserted. By fitting a line to the data
after equilibrium, we can extrapolate backwards to estimate the final temperature at equilibrium with no heat loss to the
surroundings.
C. Calculating Specific Heat of Sample
Once the adiabatic assumption can be held true, the First Law of Thermodynamics can be applied to solve for the
specific heat of the sample. Because heat transfer is assumed to be 0, and this is a closed system with no boundary
transfer energy through work, we can assume that the internal energy of the system doesn’t change while reaching
thermal equilibrium. Hence, we can apply Eq. (2) with our extrapolated values to solve for the specific heat of sample D.
D. Error Propagation of Calculated Value
The error propagation went under the assumption that the error in the masses of the calorimeter and sample would
be negligible. This is because of the lack of knowledge of how the mass measurements were taken, and the overall small
effect any variance in mass would have on the final result. The temperature readings and linear fitting were the greatest
sources of error throughout this process. The error in the equilibrium and insertion temperatures were calculated using
the equation below.
휎푇 =
q
휎
2
퐴
+ 휎
2
퐵
(푡푖) (1)
The values for 휎퐴 and 휎퐵 were taken using the matrices from the least-squares fitting. Because the temperature values
in the calorimeter use the average of two measurements, the error for each set of measurements was found and the errors
were averaged.
The method used to calculate error in 푇0 and 푇2 can’t be applied to 푇1 because there was no least-squares fitting
involved in finding this value. Instead, this value was pulled from the temperature of the boiling water at the time of
insertion. Hence, we concluded that the error in 푇1 is the standard deviation of the boiling water temperature data up
until the time of insertion.
III. Results
A. Specific Heat of Sample
The final result for specific heat of the sample was calculated using Eq. (2) below.
퐶푠 =
푚푐퐶푐 (푇2 − 푇0)
푚푠 (푇1 − 푇2)
(2)
After extrapolating to find the appropriate temperature values the specific heat of sample D was calculated.
퐶푠 = 0.360[퐽/푔
표퐶]
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B. Error Propagation
The error of the maximum temperature 푇2 was found to be 휎푇2 = 0.0201[
표퐶] and the initial temperature 푇0 was
found to be 휎푇0 = 0.0036[
표퐶] These were both calculated using Eq. (1) above. The error in the sample temperature 푇1
was found to be 휎푇1 = 0.0687[
표퐶] as the standard deviation of the boiling water data. While assuming that these errors
were random and independent of each other, they were propagated using the general method
휎퐶푠 = 0.002[퐽/푔
표퐶]
When combining the value calculated for specific heat with the propagated error, we can conclude with 68% confidence
that the specific heat of the sample l was anywhere in the range 퐶푠 ± 휎퐶푠
퐶푠 = 0.360 ± 0.002[퐽/푔
표퐶] = (0.358, 0.362) [퐽/푔
표퐶]
Out of the provided materials for our samples, the material closest to the resulting value for 퐶푠 would be zinc, at a
specific heat of 0.402[퐽/푔
표퐶]. Therefore, this experiment concludes that sample D was made out of zinc.
IV. Discussion
This experiment ended without a definitive answer for the question of which material the sample was made out of.
In fact, the answer concluded was the closest but still a great margin off the experimental results. There are a great
number of reasons as to why this happened, and they are discussed in the sections below.
A. The Calorimeter
One apparent fault with this experiment comes from the nature of the calorimeter and the assumptions imposed. In
order for a calorimeter to provide accurate readings, it must be completely adiabatic. We worked around this constraint
through least-squares fitting of the data to account for heat loss to the surroundings, but this also introduced a great deal
of error into the equation.
As shown in Fig. (1) the least-squares regression lines up very well with the provided data, as it should. However,
this would lead us to conclude an unusually low error in our measurements, as most of the error in our final result
propagated from errors in our regression, rather than our measurements.
While it is impossible to have a truly isolated system for a calorimeter, the heat loss to the surroundings is clearly
outlined in the data as well as the results for this experiment. Further data collection would be required to make
conclusive results, especially with the limitations that a real calorimeter provides.
Fig. 2 Calorimeter temperature with added regression lines
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B. The Sample
Another point of contention would be the initial temperature of the sample at the point of insertion. We estimated
this to be the temperature of the water it was boiled in, though this can’t be entirely accurate. This temperature difference
is similar to how the temperature of the calorimeter should be about room temperature at the point of insertion, yet it
was extrapolated to be almost 5
표퐶 higher than the temperature of the room. That being said, the impact of the sample
temperature on the error of the final result is minimal when compared to the other 2 temperature values we had to
extrapolate, and could probably be ignored unless more precise testing is required.
C. The Final Temperature at Equilibrium
The heaping majority of the error in our final result was the product of error in 푇2. This variable is also the most
likely to have its error underestimated, as the error in the regression is likely relatively small when compared to how
much error would be introduced through our method of solving for this value.
Reaching the true value of the calorimeter temperature at equilibrium is sadly impossible for the reasons discussed
as problems with the calorimeter. On account of this, despite our approximation for 푇2 not being entirely accurate, it’s
about as accurate as we can get in a lab setting.
D. Calculated Value of 퐶푠
The percent error calculated for our value of 퐶푠 came out to only 0.5%. This gives off the impression of a precise
result despite our many undercut assumptions. The percent error when compared to our accepted value for 퐶푍푖푛푐
actually comes out to 10.4%. This means that there was a significant error in our calculations for 퐶푠 that were missed
even when propagating error throughout the process. This missing error was most likely produced from the assumptions
we made surrounding perfectly precise mass measurements, and our assumption of an adiabatic system.
V. Conclusion
To conclude, our experimental data was manipulated under the assumption of an adiabatic system in order to solve
for the specific heat of an unknown substance in sample D. This value was precisely calculated, yet far from accurate to
the accepted values. Sample D was assumed to be Zinc from the list of available materials, though further testing would
be recommended for more accurate results and a more decisive conclusion.
Appendix
A. Derivation of Eq. (2)
Δ퐸 = 푄 − 푊
Δ퐸 = 0
Δ푈 = 0
Δ푈푐 + Δ푈푠 = 0
푚푐퐶푐Δ푇푐 + 푚푠퐶푠Δ푇푠 = 0
퐶푠푚푠 (푇2 − 푇1) = −푚푐퐶푐 (푇2 − 푇0)
퐶푠 =
−푚푐퐶푐 (푇2 − 푇0)
푚푠 (푇2 − 푇1)
퐶푠 =
푚푐퐶푐 (푇2 − 푇0)
푚푠 (푇1 − 푇2)
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B. Error of Eq. (2)
휎푦 =
r
1
푁 − 2
Σ(푇푖 − 퐴 − 퐵푡푖)
2
Δ = 푁Σ푡
2 − (Σ푡)
2
휎퐴 =
휎퐴1 + 휎퐴2
2
−→ 휎퐴푖 = 휎푦
r
Σ푡
2
Δ
휎퐵 =
휎퐵1 + 휎퐵2
2
−→ 휎퐵 푖 = 휎푦
r
푁
Δ
휎푇 푖 =
q
휎
2
퐴
+ 휎
2
퐵
푡
2
푖
푑퐶
푑푇2
=
푚푐퐶푐 (푚푠 (푇1 − 푇2)) − (−푚푠) (푚푐퐶푐 (푇2 − 푇0))
(푚푠 (푇1 − 푇2))2
푑퐶
푑푇0
=
−푚푐퐶푐
푚푠 (푇1 − 푇2)
푑퐶
푑푇1
= −
푚푐퐶푐 (푇2 − 푇0)
(푚푠 (푇1 − 푇2))2
휎퐶푠 =
r
(
푑퐶
푑푇2
휎푇2
)
2 + ( 푑퐶
푑푇0
휎푇0
)
2 + ( 푑퐶
푑푇1
휎푇1
)
2
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C. Code Flow Chart
References
[1] ASEN2012 Team. ASEN 2012 Project 1: Calorimetry https://canvas.colorado.edu/courses/66682/files/folder/Project%201
[2] Bobby Hodgkinson. Thermocouple Calorimeter
https://www.youtube.com/watch?v=3ucpCTUCa7Elist=PLfIpWHYDHoWzrF3dRX6wrAOZb8pkifLgxindex=1t=5s
[3] Bobby Hodgkinson. Calorimeter Demo 9/28/2016
https://www.youtube.com/watch?v=zdR-hhURRn4list=PLfIpWHYDHoWzrF3dRX6wrAOZb8pkifLgxindex=2
[4] Jeff Thayer. calorimeter thermodynamics https://canvas.colorado.edu/courses/66682/files/folder/Project%201?
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