SPECIFIC HEAT CAPACITY OF METALS

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1 SPECIFIC HEAT CAPACITY OF METALS Andreas Kleiven 1, Helge Skarestad 1 a TFY4165 Termisk fysikk, NTNU 014 Abstract The primary intention of this report is to investigate the specific heat capacity of metals. By discussing an experiment in which solid aluminium vaporizes an amount of liquid nitrogen, it is possible to examine temperature dependency of the specific heat capacity. The theoretical background for studying this dependency is Einstein s model for determining the specific heat capacity. Consequently, the report illustrates use of quantum theory and statistical mechanics and how they relate to macroscopic observables. The Einstein temperature for aluminium is estimated to be Θ E = 8 ± 19 K. 1. Introduction The specific heat capacity of metals is a physical quantity which is extremely important for every day life. Systems for cooling electronic components, refrigerators, water heaters and car exhaust systems are some examples of objects which would not work properly without knowledge of how solids act in changing external temperatures. Measurements of heat capacities have been important for the development of quantum theory. According to classical mechanics, heat capacity should not be temperature dependent. However, experiments made before the twentieth century indicated the opposite. Einstein s model, published in 1906 [1], was the first model that showed a correlation between the heat capacity for solids and temperature. Einstein used quantum theory to explain how classical theory differed from experimental measurements. This report deals with a method of finding the specific heat capacity of a metal. An experiment has been conducted in order to illustrate how thermal energy flows through aluminium. The first part of this report explains the theoretical background for the experiment in which the heat capacity of aluminium is found. Here, Einstein s model of harmonic oscillators with quantised energy levels will be explained. The second part presents the method of the experiment which has been carried out as well as the results. The last part contains a discussion on the validity of the results as well as conclusions on the theory and experiment.. Theoretical background.1. Specific heat capacity The specific heat capacity c of a substance is defined as the amount of heat needed to increase the temperature of one unit mass of the substance by 1 K. If the temperature increases from T to T + T, the specific heat capacity at temperature T is given by c = Q m T, (1) where Q is the amount of heat added. It is often more convenient to express the specific heat capacity as molar heat capacity, c pm or c Vm. The indexes p and V indicate that the pressure or the volume is held constant, respectively, during the change in temperature... Kinetic theory Kinetic theory for gases is based on the assumption that the molecules move as free particles. If the atoms in a monoatomic gas are considered identical point masses, each of them will have three translational degrees of freedom, corresponding to their motion in three dimensions. In one mole of the gas, there will be 3N A degrees of freedom. N A expresses Avogadro s constant. The equipartition theorem states that every degree of freedom which appears quadratically in the total energy, contributes 1 k BT to the average energy[]. Therefore we have E = N A E = 3NA 1 k BT = 3 RT, () where k B is the Boltzmann constant and R = N A k B is the gas constant. We get the molar heat capacity by derivation of equation () with respect to time:.3. Heat capacity of solids c Vm = E T = 3 R. (3) Solids that consist of N atoms have 3N modes of vibration and totally 6N degrees of freedom because of both kinetic energy and potential energy for each mode of vibration. The equipartition theorem gives us c Vm = 3R, (4) Preprint submitted to Veileder April, 014

2 also known as the Dulong-Petit law. Dulong-Petit law is approximately correct for temperatures around 300 K, but experiments have shown that c vm is temperature dependent. In other words: classical mechanics is not sufficient to describe the heat capacity. Einstein used quantum mechanics to derive an equation for the heat capacity. He assumed that the atoms in a solid are arranged in a crystal structure. Each atom consists of three independent one-dimensional oscillators and the energy of each oscillator is quantised. The harmonic oscillators have energy E = hν(k + 1/) = E k, (5) where h is Plancks constant and ν is the frequency of the oscillator. The average energy of each oscillator is 1 hνe E0/kBT E = hν + 1 e = 1 hν/kbt hν + hν e hν/kbt 1. (6) Because of three degrees of freedom per oscillator, the total energy per mole is ( ) 1 E = 3N A hν + hν. (7) e hν/kbt 1 The heat capacity from equation (4) is corrected to c Vm = E ( ) hν T = 3R e hν/kbt k B T [e hν/kbt 1]. (8) The behavior of the specific heat capacity is given by the ratio Θ E /T, where Θ E = hν/k B is the Einstein temperature of the solid. This is a characteristic property of metals with different heat capacities. The behaviour of c vm around extreme temperature highs and lows, is determined by equation (8). When T is close to 0, a small temperature increase is not sufficient to excite enough atoms to the first vibrational state, hence c vm is close to 0. For high temperatures, the difference between energy corresponding to vibrational states is small compared to thermal energy, so classical mechanics can be used. When T approaches infinity, c vm converges to 3R. This is the Dulong-Petit law. However, experiments have shown that Einstein s model is not entirely correct. A later addition to heat capacity theory is the Debye model [3], m, specific heat capacity c, and initial temperature T, the heat extracted from the object has to be equal to the heat absorbed by the liquid. Equation (1) and conservation of energy gives us that m 1 c 1 (T 0 T f ) = m c (T f T ), (10) where T f is the shared temperature of the solid and the liquid after the energy exchange. If some of the liquid evaporate at a constant temperature T = T f, the heat change is replaced by the heat of vaporization, Q = L m, (11) where L is the specific heat of vaporization per mass of unit of the liquid and m is the mass of the liquid evaporated because of the heat from the object. Combining equations (8) and (11) leads to [ ] 1 L m = 3nRΘ E e ΘE/T0 1 1, (1) e ΘE/T f 1 where n is the total of moles of atoms in the solid. 3. Method and apparatus This discussion of method and equipment is primarily based on the explanation of the experiment in chapter two of the lab guide which is written for TFY4165 Termisk fysikk[1]. An overview of the apparatus used in the experiment is presented in figure g [ T 3 ] Θ D T c Vm = 9R Θ D 0 x 4 e 4 (e x dx, (9) 1) where Θ D is the Debye temperature and x hν/k B T. The Debye model takes into account that the atoms are not oscillating independently with the same frequency..4. Heat of vaporization If an object with mass m 1, specific heat capacity c 1, and initial temperature T 0, is lowered into a liquid with mass Figure 1: A simple overview of the apparatus which was used during the experiment. Two polystyrene cups are placed on top of an electronic scale. The cups contain liquid nitrogen, and a cube of aluminium is lowered into it. The most important piece of equipment used in the experiment was the electronic scale. The purpose of the scale was to measure how fast liquid nitrogen vaporized in room temperature. On the scale were two polystyrene

3 cups stacked on top of each other. The cups contained liquid nitrogen which is shown with bubbles in figure 1 to indicate that the liquid was boiling during the experiment. The cups were made of polystyrene because they were supposed to act as insulators in order to maintain a controlled heat flow through the liquid-air interface. According to equation (11), the heat required to vaporize an amount of liquid is proportional to the mass of the liquid which is evaporated. By using the scale and a stopwatch, it was possible to record the rate by which mass disappeared from the surface in the cups. Mass [g] Experimental values Linear regression The first part of the experiment was a preliminary study of the rate of evaporation in the cup. Nine observations of weight were made within the first eight minutes. At the ninth observation, a cube of aluminium was cautiously lowered into the liquid by holding a string attached to the cube. As the cube was lowered, the liquid started boiling heavily due to the sudden heat transmission from the metal to the liquid. When the heavy boiling stopped, a new series of observations, similar to those before, were made. These observations provided an overview of the rate of evaporation after adding the aluminium cube. These observations form the basis of calculating how much liquid nitrogen that evaporated as the cube was cooled from room temperature to the temperature in the liquid. The next step of the experiment was to calculate Θ E. By making the assumption that no energy was dissipated during the heat transfer in the cube, the total energy is preserved. Consequently equation (11) and (1) can be rearranged to: [ ] 1 3nRΘ E e ΘE/T0 1 1 L m = 0. (13) e ΘE/T f 1 4. Results In order to obtain an experimental value for Θ E, equation (13) is put to use. The initial temperature of the aluminium cube, T 0 was assumed to be equal to the room temperature before the experiment started. The real temperature was never measured exactly, so the estimate of T 0 is a rough one. This will nonetheless provide a basis for discussing temperature dependency of equation (13). In the following calculations, T 0 is set to be within the range T 0 = ± K. The mass of the aluminium cube with string was measured with the electric scale. The mass of the aluminium cube without the string attached was also measured. From m cube = g, the value of n can be found by using the molecular weight of aluminium [4]: n = 0.3 mol. The latent heat of nitrogen at its boiling point, T f = 77 K [4], is a tabled value: L = J/kg [4]. The amount of liquid nitrogen which evaporated during the cooling of the aluminium cube, m, is found in figure. The two curves in figure show the rate of which Time [s] Figure : The two curves show the measured evaporation rate of liquid nitrogen before and after the aluminium cube was lowered into it. The weight of the aluminium cube with string attached has been subtracted from the lower dataset. nitrogen evaporated from the cup before the aluminium was dropped into it and after the temperature in the cup had stabilized. Consider the area between the two lines. At some time between the last observation on the upper dataset and the first observation on the lower set, m can be read as a discontinuity on the mass-axis due to a heat transfer. One would assume that the two curves had equal slopes. The reason for why the two curves have different slopes is that the temperature of the air around the polystyrene cups weren t room temperature. The outer surface of the cups became colder during the experiment. This was observed as ice formed by water vapour in the surrounding air crystallised on the surface of the cups. Consequently the cup s walls leaked less heat as the experiment went on. This effect was intensified when the heavy boiling of nitrogen occurred. Nitrogen at a temperature just above T f is heavier than room-tempered air. As nitrogen evaporated from the cup, cold gas flowed past the cup walls and cooled the surrounding air. When making the observations for figure, both time intervals and weight measurements were uncertain to a certain degree. The scale changed continuously during the observations, and it was difficult to observe both the stopwatch and the scale at all times. An estimate of deviance in the measured m should account for all the small random errors that occurred during the experiment. It is reasonable to make the assumption that difference between the two lines is evenly distributed and centred around an average value. Due to the fact that the two curves are linear and with different slopes, the mean value of the differences between the curves is the same as the difference in the middle of this interval. Therefore m is found as

4 the mean value of the difference between the curves at time 54 ± 15 s. This is an interval surrounding the exact middle point between the last observation on the top curve and the first observation on the bottom curve. The discontinuity is believed to be contained within this interval. If the distribution of different m is considered to be normally distributed, the standard deviance of many m within many small partitions in a time interval, is a good estimate for the uncertainty in m. The final estimate of mass change is m = 4.7 ± 0.09 g. With all the components of equation (13) known, it is possible to estimate Θ E for aluminium. Due to the fact that the equation can t be solved analytically, figure 3 is included to provide a geometric solution. how Θ E varies with it s parameters. Looking into the equation (13), it becomes clear that the experimental value of Θ E is much more vulnerable to small changes in m, than small changes in T 0. Furthermore, changes in Θ E due to changes in T 0 have positive correlation while changes due to varying m have negative correlation. These observations imply that the largest source of error when finding Θ E is m. Based on the experimentally found Θ E, it is now possible to use Einstein s equation for c Vm to examine the temperature dependency of the specific heat capacity. Figure 4 provides a comparison between heat capacity due to the experimental Θ E and tabulated values[5] Heat difference equation Max. and min. values Zero c Vm [J/(K mol)] With experimental Θ E Tabulated value Dulong Petit law Θ E [K] Figure 3: The geometric solution to equation (13). The experimental Θ E is found where the equation value is zero Temperature [K] Figure 4: Plot of experimental heat capacity as predicted by Einstein s theory and the experimental Θ E. The red dots indicate tabulated values for the heat capacity of aluminium [5]. The green line is Dulong-Petit s law. The blue and green graphs in figure 3 correspond to equation (13) for an array of different values of Θ E. Within the interval of Θ E, the mapping of the heat difference equation crosses zero and this is where the experimental value of Θ E is found. Due to the fact that equation (13) is not suitable for analytical calculations, the following paragraph explains the use of logic in stead of mathematics to discuss how Θ E reacts to changes in parameters. In figure 3, two plots are included to illustrate how Θ E changes with the parameters. The top and bottom, green, curves are plots of how equation (13) would act during a worst case scenario in which both m and T 0 contribute maximally to the equation value. Reading from figure 3 gives that Θ E = 8 ± 19 K. While being an experimental result, this doesn t really say anything about 4 By looking at figure 4, it is obvious that the experimental curve differs from the tabulated values for temperatures greater than and smaller than T 0. Table 1 is included to highlight the quantitative differences between Einstein s model and experimental results. Table 1: Quantitative differences between the experimental and tabulated c Vm. T Vm (K) (J K 1 mol 1 ) (J K 1 mol 1 ) (%) c exp c table Vm These differences were, however, expected. Einstein as-

5 sumed in his model that the oscillators were independent. This assumption gets less valid as temperature gets higher. For high temperatures, more atoms would be excited to higher vibrational states, thus the oscillators could not be considered independent from each other. The conclusion is that higher temperature causes greater chance for interactions between atoms. Also for small temperatures, the experimental curve differs a bit, by the same reason. Atoms with low frequency will be excited at low temperature and will contribute to c V m.this is what equation (9) corrects. 5. Conclusions The main goal of the report has been to investigate specific heat capacity. By measuring how thermal energy flowed from a piece of aluminium when cooled from room temperature to the temperature in liquid nitrogen, an experimental value for the Einstein temperature was estimated to be Θ E = 8 ± 19 K. Using this value in Einstein s model, and comparing the heat capacity to the tabulated value, it showed a slight discrepancy. This was expected due to interactions between atoms, which is not taken into account in Einstein s model. To get a better correlation between experimental data and tabulated values, the Debye model could have been used instead. When it comes to minimising uncertainties connected to the observations made in the experiments, some measures could have been taken. m could have been estimated more accurately if the observations were made by a more precise instrument such as a data logger. Also, the temperature of the aluminium, T 0, could easily have been measured more accurately. This is however not as important as measuring m accurately, as Θ E is more sensitive to m than T 0. The main goal of this report has, however, not been to acquire cutting edge accurate values of specific heat capacity, but to investigate the Einstein model and it s limitations and dependencies. 5

6 6. Reference list [1] NTNU Institutt for fysikk: Laboratorium i emnene TFY4165/FY1005 Termisk fysikk lab/orientering/termisk labhefte 014.pdf, accessed 08 march 014. [] Young H.D. and Freedman R.A. (01) University Physics with modern physics 13th ed. USA: Pearson Education Limited [3] Sweeney S.: The Debye model: phs1ss/ss/ SS%0lecture%08.pdf accessed 15 march 014 [4] Aylward G. and Findlay T. (008) SI Chemical Data 6th ed. Australia: John Wiley & sons [5] Buyco E.H. and Davis F.E. (1970) J. Chem. Eng. Data 15 6