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Heat Capacity RatioFrank Perry Rubia1, Joshua Angelo Encarnacion2, Mikoel Miclat3, and Manuel Matthew Chanco V4*1National Institute of Geological Sciences, University of the Philippines-Diliman, Diliman, Quezon City2Department of Geodetic Engineering & Training Center for Applied Geodesy and Photogrammetry, University of the Philippines-Diliman, Diliman, Quezon City 3 Department of Geodetic Engineering & Training Center for Applied Geodesy and Photogrammetry, University of the Philippines-Diliman, Diliman, Quezon City 4Department of Chemical Engineering, University of the Philippines-Diliman, Diliman, Quezon City *Corresponding author: mmchanco5@yahoo.com

AbstractThis experiment was done to understand the concepts of heat capacity and heat capacity ratio. Rchardts method was used to determine the heat capacity ratio of air using simple harmonic motion and damped oscillation concepts. A best-fit line of the T-2 vs. yo-1 plot was determined and the slope and y-intercept of this line was used to calculate the heat capacity ratio and the damping parameter, respectively. The data gave a good fit, with an R2 value of 0.997. The calculated heat capacity ratio only deviated by 0.67% from the theoretical.Keywords: heat capacity ratio, Rchardt method, adiabatic expansion, ideal gas, damped oscillation

IntroductionThe heat capacity of an object is the heat required to raise an objects temperature by one degree Celsius. Determining this is essential in calculating for the change in internal energy of gases if subjected to varying temperatures. For ideal gases, its heat capacity at constant volume and at constant pressure is related by the ratio(1)

where Cp and Cv are the heat capacity of a gas at constant pressure and constant volume, respectively. Theoretically, the heat capacity ratio of an ideal diatomic gas is =1.4. [1]Eduard Rchardt, a German physicist, designed a method to determine the heat capacity ratio of a diatomic gas experimentally. In his experiment, Rchardt put a constant amount of gas in a container by sealing it with a piston on top. The piston was given a small vertical downward displacement which caused to the piston to bounce and oscillate as the set-up creates an air cushion that opposes the vertical displacement. The oscillation has a period that is dependent on the heat capacity ratio of the gas and so, he was able to compute for that ratio.The height of the piston in the set-up was described using Newtons second law yielding to the equation(2)

where m is the mass of the piston, P is the pressure of the gas inside the container, A is the cross-sectional area of the piston, g = 9.81 m/s2 is the acceleration due to gravity and Patm = 101325 Pa is the atmospheric pressure. The gas was given quick changes in volume so it can be assumed that there is no heat transferred in the system. [2] A process which involves almost no transfer of heat is said to be adiabatic and the relationship between its initial and final volumes and pressures is characterized by(3)

Considering this equality and the under damped oscillation that the piston experiences as friction develops between it and the containers walls, it can be incorporated to the previous equation and come up with the relationship between its period and the heat capacity ratio of the gas given by(4)

where the new variables T and b are the period of oscillation and damping parameter, respectively. [2] Using this equation which relates the slope of the T-2 vs. y0-1 plot and heat capacity ratio, the heat capacity ratio can be obtained through the following relationship

(5)

MethodologyThe objective of this experiment is to determine the heat capacity ratio of air using Rchardts method. The materials used were a heat engine apparatus, and a Vernier LabQuest with gas pressure sensor. Figure 1 shows the apparatus used in the experiment.

Figure 1. Gas Enginer Apparatus (left), Vernier LabQuest (right)

At first, the Vernier LabQuest was setup by choosing a duration of 5 seconds with a sampling of 500 samples per second in the Sensors menu. The experiment was done by setting an initial height of the piston, y0, then lightly tapping the piston to compress the air inside. A pressure versus time graph was obtained in the Vernier LabQuest which exhibits damped oscillation. The oscillating part of the graph was isolated by zooming in and the period of oscillation, T, was determined by measuring the time interval for each peak. To make precise measurements, the time interval for several peaks, about four to six, were averaged. Three trials were done and the procedure was repeated for four more values of y0.

Results and DiscussionAfter obtaining the data, a plot of T-2 vs y0-1 was generated. The values of the period were averaged from the three trials. The following figure shows the plot.

Figure 2. T-2vs y0-1 plot. . The data obtained was fitted assuming that the piston movement behaves as damped oscillating

From Figure 2, it can be seen that the experiment gave relatively good results because of the good fit (R2 = 0.9973). An increasing trend can also be observed from the graph as the initial height increases which confirms the direct proportionality between the height and the period of oscillation from equation 4. From equation 5, the experimental heat capacity ratio of air can be calculated, which is equal to 1.409. The damping parameter can also be calculated from the y-intercept using equation 4, which is equal to 5.607 kg/s. If the piston movement is assumed to be a simple harmonic motion, then the data must be fitted to a different equation.(6)

Equation 6 describes a simple harmonic motion where is the angular frequency. From the data, the angular frequency can be obtained with the following relationship. (7)

To calculate the heat capacity ratio assuming simple harmonic motion, a vs y0-1/2 plot must be made from the data and fitted.

Figure 3. vs y0-1/2 plot. The data obtained was fitted assuming that the piston movement behaves as simple harmonic motion

Figure 2 also shows a good fit and same pattern if the piston movement was assumed to behave as simple harmonic motion. However, the calculated heat capacity ratio is equal to 1.680 which deviates from the theoretical value by 20%. This is because by assuming simple harmonic motion, the friction effect (damping) of the wall on the piston was neglected. Hence, the behavior of the piston movement is confirmed as damped oscillating.

ConclusionThe heat capacity ratio was calculated by Rchardts method. The behavior of the piston movement is damped oscillation and a T-2 vs. y0-1 plot was generated. A best fit line was fitted to the data to calculate for the heat capacity ratio. Based on the results, Rchardts method of experimentally determining the heat capacity ratio is reliable because of the very low percent deviation calculated. Also, the direct proportionality between the initial height of the piston and the period of oscillation was confirmed through the experiment.

AcknowledgementsThis experiment was done with the help of lab instructor, Mr. Lean Dasallas.

References1. Hugh D. Young, Roger A. Freedman and Lewis Ford, Sears and Zemanskys University Physics with Modern Physics. 13th Edition, chapter 19. Addison Wesley, Inc. 2008. 1. Lab Manual Authors, "Experiment 4: Heat Capacity Ratio," Physics 73.1 Laboratory Manual 1st Semester A.Y. 2014-2015. pp 3-6, 2014.

Division of Labor

Manuel Matthew Chanco V Author Results and Discussion

Joshua Angelo Encarnacion Introduction

Mikoel Miclat Abstract Methodology Frank Perry Rubia Conclusion

Everyone participated during the experiment.Physics 73.1: Elementary Physics Laboratory IIIUniversity of the Philippines Diliman, Quezon City, Philippines 1