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Lab 15. Heat Capacity
By Jamu Alford With help from Sari Belzycki
Abstract
Using a Styrofoam cup as a calorimeter the specific heat of water was found to be 5. 3 ± 0. 2 JK!^1 g!^1 , having 25% error from the theoretical value of 4. 184 JK!^1 g!^1. This
demonstrated that a systematic error existed in our scientific procedure and that the lab required several modifications before a correct value of the specific heat could be obtained. By allowing lead shot to fall inside a closed pipe, we determined the specific heat of lead to be 833J/kgK. This compared very poorly with the theoretical value of 157J/kgK.
Introduction
One of the most fundamental physical properties is the relation between mechanical energy and heat. Thermodynamics principles were developed during the industrial revolution when the steam engine and later the gasoline engine became the focal point of intense research. At that time, it was found that the specific heat at constant volume ( Cv) of a substance is defined to be its heat capacity per unit mass when all changes are made at a fixed volume. The specific heat at a constant pressure, Cp is defined as the heat capacity per unit mass when all changes are make fro a fixed volume. For a body of mass m to experience a temperature change of ΔT the heat energy required (Q) is given by the first law:
Q = mCv! T +! W Equation ( 1 )
In the equation given above, ΔW is the work done by the system on its environment. For the entirety of the lab, the work performed will be zero. In this case, Equation 1 is simplified to:
Q = mCv! T Equation ( 2 )
This equation is very important and will be used exhaustively during an experiment involving heat.
Experiment
In the first experiment we investigated the specific heat of water. The water was heated by use of an electric current passing through a resistor. Electrical heating allowed us to determine the heat entering the system to a high accuracy.
1006 1ml of water was placed in a Styrofoam cup weighing 2.2grams along with a 2Ω
resistor. The resistor was wired directly to a voltage supply set to deliverer 4.2 6 0.1A of
current at 9.2 6 0.1V. This delivered 38.5 6 0.2W of power or 580J of energy every 15s.
The results of the heating are show in Table1. The water was replaced with 2006 1ml of room temperature water and the experiment repeated. The results are shown in Table2. Next the resistor was exchanged for 6.8Ω resistor, the voltage supply changed to deliver
2.8 6 0.1A of current at 20.0 6 0.1V. This setup delivered 56.0 6 0.2W of power or 840J of
energy every 15 6 1s. Heating of both 100ml and 200ml samples was performed. See Table3, and Table4 for the results.
In the second experiment we considered the conversion of mechanical work to heat. The equation governing this is "! W = mCv! T where – ΔW is the work done on the system by its environment. Since the system was isolated we set Q=0. By rotating a long tube filled with lead shot we approximated a fall of distance Nh where N is the number of times that the tube was rotated by 180 degrees and h is height. The work done on system is then Nmgh. Solving for the specific heat, Cv = Nmgh/mΔT, m is the mass of the shot and, g is of course gravity. The length of the tube was measured and an average distance of the falling show was found to be 90cm. The tube was inverted 30 times and the temperature of the air inside the tube and the shot was measured. This was repeated 2 more times and the final temperature measured each time. The results are given in Table5.
Since we have taken two different messures we can eliminate the Ccup and mcup terms from the problem. Taking the data from the first trial, we graphed Q vs. ΔT and added a line of best fit. See
Graph2. Next, the data was graphed ΔT vs. Q. See Graph3. The lines of best fit for the 100ml cup from Graph2 and Graph3 were:
Q = 452.3ΔT – 220.
ΔT=0.002198Q + 0.
Rearranging the second equation, Q = 456.0ΔT – 247.6. The difference in the first and second equations is the result of the forumla for line of best fit only accounting for error in the y-axis.
Taking the average, we have Q = 454 62 ΔT – 234614
We repeated this procedure with the data taken from the second trial.
Q = 946.1ΔT – 103.
ΔT=0.001047Q + 0.
Rearranging the second equation, Q = 955.1ΔT – 147.5. Taking the average, we have Q
= 950 65 ΔT – 1256 22.
Substituting these values into Equation (4).
sWK gCw gCc (^) up Equation
sWK gCw gCc (^) up Equation
Subtracting Equation (5) from (6) removes the terms related to the cup.
sWK gCw Equation
Thus,
1 (^494 4). 94 1 1 w 100
sWK C JK g g
! = =^!^!
Repeating for the 6.8Ω resistor,
1 (^558 5). 58 1 1 100 w
sWK C JK g g
! = =^!^!
Averaging, Cw = 5. 3 ± 0. 2 JK!^1 g!^1
Using the percent differnce test:
Theoretical-Experimental 4. 1 - 5. 3 Difference = 100 100 25 % Theoretical+Experimental / 2 4. 1 + 5. 3 / 2
x = x =
Comparison to the thoeretical value of 4. 1 JK!^1 g!^1 gives an error of 25%. The
experimental errors calculated from the slope of the curve were only about %1 yet the error of the experiment is 25%. Thus, there were systematic errors in the experiment. Some of the systematic errors are listed below.
Source of errors. There were so many, I don’t even know where to start.
- Loss of energy due to evaporation.
- Energy added by stirring.
- Unequal stirring during between trials.
- Different starting temperatures of water.
- Loss of energy through the cup into the air and table.
In the second experiment the values of shot temperature and air temperature were measured at each trial and are given in Table
Table3 Measured temperatures For each trial.
Trial
Temp. air (oC)
Temp. shot (oC) 0 23 23 1 23 23 2 24 24 3 24 23 4 25 24 5 25 24 6 25 25 7 26 25 8 26 25 9 27 26 10 27 26
The plot of change in shot temperature vs. number of inversions is given in Graph4 A line of best fit through the points in Graph4given the following formula, " T = 0. 0106 K * N! 0. 2273 K where N is the number of inversions. Using the formula Cv = gh /(slopeofTvs.N). The specific heat of lead was found to be 833J/kgoK. The
theoretical value is 157oC. Using the percent differnce test:
Conclusions
The only use of the this experiment was to show how not to do experiments. The styrofoam cup used in experiment 1 was a horrible example of a calorimeter. Adding energy to the cup and measuring the change in temperature of water allowed us to
determine the specific heat to be 5. 3 JK!^1 g!^1 25% error from the theoretical value. This
large error coupled with an measurment error of only %1 demonstrated that were was a huge systematic error in the experiment. In the second experiment involving the falling lead, we determined the specific heat of lead to be 833J/kgK. Using the percent difference test, this was found to have an error 137%. Using the correct formula and specific heat of lead we found that the temperature change from 30 inversions should have been 1.7K.
Sketch1. Surroundings at higher temperature then system.
Sketch2. Surroundings at lower temperature then system.
Sketch3. Surroundings at same temperature as system. Work done on system vs. Temperature
Work done on system vs. Temperature
Work done on system vs. Temperature
Graph1. 100grams of water with 2Ohm resistor.
Tem perature vs. Tim e
y = 0. 0846 x + 294. 69
290
295
300
305
310
315
0 50 100 150 200 250 Tim e (s)
Temperature
(
oK)
Graph2. 100grams of water with 2Ohm resistor Q vs ΔT.
! T^ vs.^ Num^ ber^ of^ Inversions
y = 0. 0106 x - 0. 2273
0
- 5
1
- 5
2
- 5
3
- 5
0 50 100 150 200 250 300 350
Num ber of Inversions
Temp!
( oC)
