The University of Hong Kong Department of Physics. Physics Laboratory PHYS3550 Experiment No Specific Heat of Solids

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1 The University of Hong Kong Department of Physics 0.1 Aim of the lab This lab allows students to meet several goals: 1. Learn basic principles of thermodynamics Physics Laboratory PHYS3550 Experiment No Specific Heat of Solids 2. Determine the specific heat of cooper, lead and glass 0.2 Introduction When energy is added to a system, its temperature (or internal energy may rise. It was found experimentally, that for a given mass of the substance, the quantity of energy, required to change the temperature by some fixed amount, depends on the material of substance. This experiment is dedicated to the measurement of the specific heat capacities of different substances. 0.3 Theoretical background The heat quantity Q, that is absorbed or evolved when a body is heated or cooled, is proportional to the change of temperature T and to the mass m of the object: Q C T c m T (1 where factor C is the heat capacity of an object, c - specific heat capacity. The specific heat depends on the material. Also, it was found, that specific heat varies with temperature, but if temperature interval is not too great, this dependence can be ignored. From the point of view of thermodynamics, there several types of heat capacity. The two most common are: C P - heat capacity of the system with constant pressure, C V - heat capacity of the system with constant volume. Differences between C P and C V can be seen from principles of thermodynamics. From the first law of thermodynamics: From equation (1, it is easy to see, that heat capacity is defined as: δq du + pdv (2 C δq du + pdv (3 1

2 Figure 1: Schematic description of calorimeter method. If the system keeps the volume as constant, then C V C V ( δq V can be introduced: ( U T V If the system keeps the pressure as constant, then C P can be introduced: ( δq C P ( U T P V ( du(t, V + pdv ( ( U/( T V + ( U/( V T dv + pdv P [ ( U ]( V [ ( U ]( V + p + C V + p + V T T P V T T P It is easy to see, that in most of the cases C P > C V. Students are recommended to check literature by themselves for additional information about heat capacity and thermodynamical potentials. 0.4 Experimental method In this experiment specific heat capacity of different materials is measured. The technique is based on the calorimeter method: a material with unknown heat capacity (a set of shots is pulled down into the water tank (or other liquid with known heat capacity. Schematic description of the experimental method can be found in Fig. 1. By measuring initial and final temperatures of water and a shot one can calculate the heat capacity of a shot. Ideal case. The initial temperature of a shot is T 1, the initial temperature of water with calorimeter is T 2. After the shot is pulled down into the water, temperature of system is changed to T f. The amount of heat, given by the shot to the water tank is calculated in the following way: Q 1 c 1 m sh (T 1 T f (6 where c 1 is the specific heat capacity of the shot (unknown value, m 1 - mass of the shot. Amount of heat Q 1 is spent on the heating of water (eq. (7 and calorimeter (eq. (8. Initial temperature of water tank and calorimeter are the same, as two objects are in thermal equilibrium. We have the following equations: Q 2 c w m w (T f T 2 (7 Q 3 c cal m cal (T f T 2 (8 From the conservation of energy, we obtain the final equation, which will be used to calculate specific P (4 (5 2

3 heat capacity: Q 1 Q 2 + Q 3 c 1 T f T 2 T 1 T f cwm w + c cal m cal m sh (9 To simplify the equation (9, it is convenient to introduce water equivalent of the calorimeter. Water equivalent of the calorimeter (mass of water m K is defined as a certain amount of heat, that is used to raise a temperature of a substance on 1 degree divided by the specific heat capacity of the water: Equation (9 can be simplified to the following form: c cal m cal c w m K m K c calm cal c w (10 c 1 c w m w + m K m sh T f T 2 T 1 T f (11 Real case. In reality, equation (11 is simplified, as it does not take into account heat losses to the surrounding area (experimental hall. To estimate specific heat capacity more precisely, one should take into account heat losses. Equation (9 is modified in the following way: c 1 m sh (T 1 T f c w (m w + m K (T f T 2 + θ (12 Parameter θ stands for the temperature lost to the surrounding area. Dimension parameter θ is K, and it can be estimated experimentally. From (12 it is easy to see, that specific heat capacity will equal to: c 1 c w m w + m K m sh Tf T 2 + θ T 1 T f With proper estimation of parameter θ, one can calculated specific heat capacity of an object using equation ( Experimental Setup and Procedure Schematic view of the experiment is shown in Fig. 2. It consists from the following parts: 1. Dewar vessel (with cover 2. Set of shots: copper, glass, lead 3. Lab balance for weight measurements 4. Thermometer 5. Steam generator 6. Heating apparatus 7. Beaker 8. Stand base 9. Leybold multiclamp 10. Protective cloves (13 3

4 Figure 2: Experimental setup used for the calorimeter method. Experimental procedure: 1. Construct experimental setup, as shown in Fig Fill water into the steam generator, close the device cautiously, and connect it to the top hose connection of the heating apparatus (steam inlet with silicone tubing. 3. Attach silicone tubing to the bottom hose connection of the heating apparatus (steam outlet, and hang the other end in the beaker. See to it that the silicone tubings are securely seated at the connections. 4. Fill the sample chamber of the heating apparatus as completely as possible with lead shot, and seal it with the stopper. 5. Connect the steam generator to the mains, and heat the shot for about minutes in the heating apparatus flowed through by steam. 6. Determine the mass of the empty Dewar vessel, and fill in about 180 g of water. 7. Mount the cover for the Dewar vessel and insert the thermometer. 8. Measure the temperature T 1 of the water. 9. Open the cover of the Dewar vessel and shift it aside: leave the mesh for samples of the cover in the Dewar vessel. 10. Drop the shot with the temperature of T o C into the mesh for samples, close the cover, and thoroughly mix the water with the shot. 4

5 11. Read the mixture temperature as a function of time every t 10 sec until the temperature will reach maximum value. Continue to read values of temperature after it for 5-10 minutes. Create a graph T(t. 12. Determine additional mass m sh of the shot, take m K 23 g. 13. Repeat the experiment with copper and glass shots (If there will be a lack of time, chose only one additional shot for measurements. 0.6 Data Analysis During data analysis for this lab, students are highly motivated to use MatLab. To successfully pass the lab student must complete following tasks: 1. Measure temperature of mixture T (t as a function of time. You start your measurements at initial time t in. Chose time interval as t 10 sec. 2. Mark the time t ext, when T (t ext T f. This value corresponds to the maximum temperature of mixture. 3. Continue to take measurement of temperature of mixture for 3-5 minutes (until you reach t f. Make sure you have good statistics of data points. Check Fig. 3 for reference. 4. Make a plot of temperature of mixture T (t as a function of time. Using experimental data estimate the value of parameter θ (equation ( Calculate value of specific heat capacity, using equation ( Estimate value of θ from data points. Fit the region after t ext with polynomial function of the first order (y Ax + B and estimate the value of θ using equation ( Estimate the uncertainty of specific heat capacity using equation ( Compare your results with literature values. Example of data graph can be found in Fig. 3. Data below t t ext correspond to the increase of temperature of mixture. Estimation of parameter θ. Value of parameter θ can be estimated from data points. It is easy to see, that the heat given by the mixture to the surrounding area is calculated: Q loss c w (m w + m K θ dq loss c w (m w + m K dθ (14 We also know, that heat given by the unit of time can be calculated using Newton s law of cooling: dq loss α A (T (t T in dt (15 where T in is the initial value of temperature, α - heat transfer coefficient, A - heat transfer surface area. From (14 and (15, one can see that: dθ dt α A (T (t T in c w (m w + m K (16 5

6 Figure 3: Experimental data points. From equation (16, one can see, that parameter θ is linear with T. So the dθ can be presented in dt the following way: dθ dt 1 [( dθ ( dθ ] + (17 2 dt t in dt text R. Resnick, D. Halliday, and K. Krane: Physics Volume 1 (John Wiley and Sons, 2002, 5th edition In case, when system does not receive any energy dθ. First fraction in (17 cannot be estimated dt dt during the lab. It is supposed that heat losses to the surrounding area are negligibly small, so 0. Equation (17 can be simplified to: dθ dt in dθ dt 1 ( dθ 2 dt t ext dt 1 ( T θ 1 ( T (t ext t in (18 2 t t ext 2 t t ext Using equation (18, one can estimate the average value of parameter θ, calculate derivate from the experimental data in the region t > t ext. Error analysis. Final result of a measurement of any physical quantity is written in the following way: x x ± x x ± σ tot (19 where x - mean value of a measured quantity (for example, specific heat capacity, σ tot stands for the error bar of a measured quantity. Usually error bar σ tot consists from several parts: systematic uncertainty (σ syst, statistical uncertainty (σ stat. To estimate σ tot, one should add σ syst and σ stat quadratically: σ tot σsyst 2 + σstat 2 (20 In this lab, specific heat capacity is measured indirectly (other quantities are measured, and then heat capacity is estimated. In this case, uncertainty is estimated using error analysis methods. Suppose, we have a function f f(x, y, z, and we know standard deviations σ x, σ y and σ z. It can be shown, that σ tot is expressed in the following way: ( f 2 ( f σ syst x σ x + y σ y 6 2 ( f 2 + z σ z (21

7 Statistical error σ stat is estimated by performing many trials of the same measurement. Students are recommended to study literature for additional information. But due to the lack of time heat capacity in this experiment is measured with only one trial, so σ stat 0. In more general case, variables x, y, z can be dependent. In this case covariances between parameters must be taken into account. Additional information on this case can be found in [3, 4, 5]. During the experiment, students measure (or find in literature the following quantities: m w ± m m sh ± m (Since the same balance is used to weight water in the vessel and shots, uncertainty will be the same T 2 ± T T f ± T (It is an approximation, that uncertainty of the initial temperature of water and final temperature of mixture are the same. Look text for explanation θ ± θ c w ± c w (heat capacity of water is taken from literature, but include also error bar for the data analysis Using formula (21 and expression for c 1 from (13 one can show that uncertainty for the specific heat capacity of a shot is estimated in the following way: c 1 m w + m K 1 [ ( 1 (T m sh T 1 T f T 2 + θ 2 { c 2 w + c 2 1 w + m 2 }+ f m 2 sh (m w + m K 2 ( ( + c 2 w θ 2 (T1 T 2 + θ 2 ] 1/2 ( T 2 (T 1 T f 2 Remark: Initial temperature of water T 2 and highest temperature of a mixture T f are measured with the same thermometer, however extrema near T f in Fig. 3 may not be so sharp. The data near the maximum point can be fit with quadratic (or other function, the value of T f can be extracted from the fit. Questions 1. What type of heat capacity do you measure: C V or C P? Why? What is the deference between these two types of heat capacity? 2. Derive equation (22, assuming that T 2 T f. 3. Determine the value and error bar of the T f point. by performing a fit of the data near the maximum 4. What assumptions are made in the equation (21? How to write equation (21 in general form? (Check [3, 4, 5] 5. What is the difference between measured heat capacity and literature values? Do them match within error bars? 6. How can this experiment be improved? Propose your ideas. 7

8 Bibliography [1] R. Resnick, D. Halliday, and K. Krane (2002. Physics Volume 1, 5th edition. [2] James S. Walker (2007. Physics, Third Edition. [3] James F. (2008. Statistical Methods in Experimental Physics. [4] Leo W. R. (1994. Techniques for Nuclear and Particle Physics Experiments. p. 81 [5] Taylor J. R. (1997. An Introduction to Error Analysis. 8

Copyright 2008, University of Chicago, Department of Physics. Experiment I. RATIO OF SPECIFIC HEATS OF GASES; γ C p

Copyright 2008, University of Chicago, Department of Physics. Experiment I. RATIO OF SPECIFIC HEATS OF GASES; γ C p Experiment I RATIO OF SPECIFIC HEATS OF GASES; γ C p / C v 1. Recommended Reading M. W. Zemansky, Heat and Thermodynamics, Fifth Edition, McGraw Hill, 1968, p. 122-132, 161-2. 2. Introduction You have