Laboratory Experiment pubs.acs.org/jchemeduc
Improved Method for Determining the Heat Capacity of Metals Roger Barth* and Michael J. Moran Department of Chemistry, West Chester University, West Chester, Pennsylvania 19383, United States S Supporting Information *
ABSTRACT: An improved procedure for laboratory determination of the heat capacities of metals is described. The temperature of cold water is continuously recorded with a computer-interfaced temperature probe and the room temperature metal is added. The method is more accurate and faster than previous methods. It allows students to get accurate measurements on multiple samples in a laboratory period, making it suitable for a traditional laboratory course as well as for a discovery lab approach. Results for 267 student trials (excluding 15 outliers) on five metals are reported. The average molar heat capacity for each metal was within 3.3% of the ideal Dulong and Petit value of 3R.
KEYWORDS: First-Year Undergraduate/General, Laboratory Instruction, Physical Chemistry, Hands-On Learning/Manipulatives, Heat Capacity, Metals, Thermodynamics
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INTRODUCTION Heat capacities of metals and the related concept of the law of Dulong and Petit have long been a staple of the general chemistry laboratory. The standard method involves heating a sample in boiling water and then rapidly adding the hot sample to room temperature water to measure the temperature rise.1,2 In this form the experiment is susceptible to heat loss from the hot metal to the environment, leading to large biased errors. The availability of computer-interfaced digital thermometers with 0.01 K resolution makes it possible to get sufficient temperature change from adding room temperature metal to chilled water. Because the item undergoing transfer, the metal, starts at room temperature, heat loss is minimized and the results are potentially much more accurate. In addition, because it is not necessary for metal samples to equilibrate with boiling water, time is saved so more samples can be run in a laboratory period. These two advantages permit the students to investigate the law of Dulong and Petit or, if preferred, to estimate molar masses of metals. The empirical law of Dulong and Petit is a consequence of the equipartition principle, which states that, in the high temperature limit, the molar heat capacity of a substance is equal to 0.5R times the number of quadratic degrees of freedom, where R is the ideal gas constant, 8.3145 J·K−1·mol−1. For a monatomic lattice, there are 6 quadratic degrees of freedom, the vibrational kinetic and potential terms in each spatial dimension. Thus, the ideal value for the molar heat capacity is 3R = 24.954 J·K−1·mol−1. This law was used to help position elements during the development of the periodic table.3 The experiment described was run by students in 14 sections of Experimental General Chemistry I with five different © XXXX American Chemical Society and Division of Chemical Education, Inc.
instructors during the fall semester of 2013. The students worked in pairs. The entire experiment, including calculations and preparation of a report form was allotted 2 h.
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EQUIPMENT AND SUPPLIES Each station was equipped with a Microlab FASTspec interface model 522, a thermistor probe model 103 (Microlab, Inc., Bozeman, MT), and a Lenovo desktop computer with keyboard, mouse, and monitor. The probes were calibrated beforehand by the laboratory staff. A single networked black and white laser printer served all stations. Six top-loading balances with 0.01 g precision were on a side bench. Other items issued were as follows: a 50 mL graduated cylinder, three 10 or 12 oz (∼300−350 mL) polystyrene coffee cups, a 400 mL beaker, a 50 mL beaker, a 1000 mL plastic beaker, ice, three assorted pelleted metal samples that could include aluminum, antimony, copper, lead, magnesium, tin, or zinc. It is best if each pair of students is assigned metals whose molar masses differ by at least 20%.
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PROCEDURE SUMMARY A temperature probe that can measure to 0.01 °C at intervals of 0.5 s was interfaced. The instrumentation was set up to prepare a graph of temperature vs time. Forty milliliters of deionized water was chilled in an ice bath for several minutes. Metal pellets were weighed out (75 g for aluminum or magnesium, 100 g for others) and the mass recorded to 0.01 g. The temperature probe was placed among the pellets and the temperature recorded. The chilled water was poured into a polystyrene coffee cup. The water was stirred for several
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dx.doi.org/10.1021/ed500466m | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Laboratory Experiment
Figure 1. Thermogram for addition of 100.07 g of antimony (initially at 19.07 °C) to 40 mL of water.
the preperiod and postperiod that were relatively free of noise. The lines can be fit mathematically by most interface software, as explained in the Supporting Information. The time for a point that is roughly half way up the temperature rise was recorded and considered to be the time at which the metal and water were brought together (the time of mixing). The y-value (temperature) of the preperiod line evaluated at the halfway time is the initial water temperature at the time of mixing. The value of the postperiod line evaluated at the halfway time is the final temperature of the water and the metal at the time of mixing. This procedure adjusts for the effect of heat leaking into the system from the environment. The exact details of the selection of portions of the preperiod and postperiod, and the halfway time are not critical. If the heat released by the cup and probe can be ignored, the law of conservation of energy requires that the (negative) heat gained by the metal (Qmetal) plus the heat gained by the water (Qwater) add up to zero: Qmetal + Qwater = 0. With the known specific heat of water and the measured ΔT for the water, the heat gained by the water (Qwater) can be determined. The heat gained by the metal (Qmetal, negative because the metal loses heat) is the negative of Qwater. Qmetal and ΔT for the metal can be used to determine the heat capacity, and from it the specific heat capacity and the molar heat capacity of the metal. Our efforts to measure the heat capacity of the apparatus (cup and probe) indicated that their contribution to the total heat
seconds with the temperature probe, then the recording of the temperature was begun while stirring continued. The probe was kept immersed at about the same level in the water throughout the experimental run. After about 2 min, the metal was added all at once, but without splashing. Stirring and measurement continued for about 3 min, then data recording was stopped. A more detailed version of the procedure is included in the Supporting Information.
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HAZARDS The metals are potentially toxic; gloves should be worn. If metal pellets undergo rough handling that could result in the formation of dust, they must be washed and the dust filtered from the wash liquid and disposed of as hazardous waste. This should be done in any case before the first time the metals are issued for student use.
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ANALYSIS A typical data set is shown in Figure 1, with a region, the preperiod, in which the cold water warms slowly, an abrupt and possibly somewhat chaotic temperature increase as the metal is added, followed by another slowly rising temperature region, the postperiod. The lines, boxes, and notation were added for analysis, as explained below. The preperiod and postperiod data were fit to straight lines to approximate the warming rate resulting from heat gained from the surroundings. Lines were fit to the data in portions of B
dx.doi.org/10.1021/ed500466m | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Laboratory Experiment
Table 1. Summary of Molar Heat Capacities Excluding Outliers Metal
Mean/J·K−1·mol−1
SD/J·K−1·mol−1
N
Mean/J·K−1·mol−1
SD/J·K−1·mol−1
N
95% CI/J·K−1·mol−1
Literature/J·K−1·mol−1
Error, %
Al Cu Pb Sb Sn
24.01 24.31 27.07 24.69 27.01
6.04 6.22 11.59 5.40 5.99
91 58 32 43 43
24.62 24.66 24.56 24.97 25.70
3.95 4.25 6.18 1.74 4.30
86 55 30 40 41
±0.8 ±1.1 ±2.0 ±0.5 ±1.3
24.4 24.42 26.3 25.2 25.8
0.9 1.0 7.0 0.9 0.2
addition of metal is accounted for by extrapolating the trend of the water temperature.
capacity was below the threshold of measurability in this system.
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CONCLUSION The availability of high-precision laboratory interfaces makes it feasible to measure specific and molar heat capacities of metals starting at room temperature. This method yields superior results in less time than the standard boiling water method. It has the additional advantage that no heating is required. The quality of the results can, in the hands of careful students, confirm the law of Dulong and Petit. The speed of the experiment allows the testing of multiple metal samples, making it well suited as a discovery activity.6 This can be helpful in revealing and diminishing student misconceptions about heat and energy.
RESULTS Students working in pairs ran thermograms of aluminum and two other metals and calculated the molar heat capacities in a 2 h laboratory period. The students were instructed to save the raw data and to record the identity, mass, and initial temperature of the metal. Some did so. The student data were gathered from the laboratory computers and recalculated by the authors. Those runs that could not be interpreted or were incomplete were discarded, leaving 267 thermograms to be analyzed. The student data were subject to gross errors such as misidentifying the metal, incorrectly recording the sample mass or initial temperature, and using the wrong thermistor calibration file. To get a coherent data set, we applied Chauvinet’s criterion for rejection of outliers to the results for each metal. Chauvinet’s criterion assumes normal distributions and rejects any data point whose Z-score gives it a probability of less than 1/2N, where N is the number of data items. Fifteen runs were rejected as outliers, leaving 252 runs in the data set. New means, standard deviations, and confidence intervals were calculated after application of Chauvinet’s criterion. The results are summarized in Table 1, with the 95% confidence interval and comparisons to the accepted values.
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ASSOCIATED CONTENT
S Supporting Information *
Student-oriented details of procedure and calculations; setup instructions and lab management issues for lab staff; suggestions for lab instructors; possible sources for metals. This material is available via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
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Notes
DISCUSSION Except for lead, each measured heat capacity average is within 1% of the literature value.4 The literature values for all metals fall within the 95% confidence interval of the student results. We expect the specific heat capacity of lead to be more difficult to measure than the other metals because its low specific heat capacity leads to a small temperature change. To get more reliable results for lead, a larger sample could be used to provide a larger temperature rise. In terms of class averages, all of the metals yielded measured molar heat capacities that are near (within 3.3%) the ideal DuLong and Petit value. The relative standard deviations of the results that were not rejected by Chauvinet’s criterion averaged 16%. Students who follow instructions and make careful measurements can achieve greater precision and accuracy than this. The results of our students are more accurate than reported student results obtained by immersing hot metal in room temperature water.5 Ten measurements were reported, eight of which were lower than the accepted values. The average deviation from the literature value was −12%. This is probably due to heat loss when the hot metal was transferred to the water. In our method, the metal is at room temperature when it is transferred, so little heat is exchanged with the environment. The heat gained by the chilled water before and after the
The authors declare the following competing financial interest(s): Roger Barth receives compensation for consulting with Microlab, Inc.
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REFERENCES
(1) Bindel, T. H. Copper/Aluminum Surprise. J. Chem. Educ. 1990, 67, 165−166. (2) Ngeh, L. N.; Orbell, J. H.; Bigger, S. W. A Closer Look at Polystyrene Cup Calorimeters. J. Chem. Educ. 1994, 71, 793−795. (3) Liang, M. Dulong and Petit’s Law: We Should Not Ignore Its Importance. J. Chem. Educ. 2006, 83, 1499−1504. (4) Whitby, M. Periodictable.com. http://www.periodictable.com (accessed Aug 2014). (5) Herrington, D. G. The Heat is On: An Inquiry-Based Investigation for Specific Heat. J. Chem. Educ. 2011, 88, 1558−1561. (6) Bindel, T. H.; Fochi, J. C. Guided Discovery: Law of Specific Heats. J. Chem. Educ. 1997, 74, 955−957.
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