Abstract
Thermal physics is very important when studying the thermal properties of matter. One of the most important characteristics is the heat capacity ratio that compares the heat capacity at a constant pressure (Cp) to that at a constant volume (Cv). By heat capacity, we refer to the quantitative aspect of the heat emitted or gained in a matter in regard to the temperature change experienced. This quantity is expressed in units of Joule per Kelvin. Heat capacity is denoted by the gamma symbol (ɤ). Whenever a gas is heated, the volume increases leading to more force being exerted on the container holding the matter. The energy will be a multiple of the initial volume and the change in temperature(Cv∆T). Many methods have been devised to try to efficiently calculate the heat capacity. The energy exerted will always be proportional to the amount change in the volume. This experiment is aimed at measuring the ɤ of air using the Clement and Desormes method, and Ruchhardt’s method.
Theory
Whenever an ideal gas undergoes an adiabatic process the volume (V), and pressure (P) as well as temperature (T) changes considerably. An ideal gas refers to a hypothetical gas assumed by scientists that are composed of point masses moving in a random, constant but straight line motion (Chirpich 4). Assumptions are made regarding the gas that no forces are acting on the particles. The particles are also said to have no atomic volume. The gas constant is derived from the gas variables of temperature (T), pressure (P), moles (n) and volume (V) that PV=nRT. This means that there are no repulsive or attractive forces involved in the collisions and hence the kinetic energy remains unchanged. For an adiabatic process (a process where there is no heat transfer to the surroundings and between the thermodynamic system, and energy is only transferred as work) the relationship between V and P is PVɤ = constant; where ɤ= Cp /Cv
One of the methods used to measure the ɤ of an ideal gas is the method of Clement and Desormes. This is a simple and straightforward method that was developed by the duo in the 19th century. It involves slightly pressurizing a container of gas above the atmospheric pressure and then allowing it to reach equilibrium with the surroundings. A stopper is then popped from time to time in order to decompress the gas to a single atmosphere. During expansion, the gas cools down, and the immediate temperature (after decompression) is calculated by combining the adiabatic law with that of the ideal gas (Samokhin 372). Letting P1= pressure before decompression, Po= atmospheric pressure, Vc= volume of container, V= expanded gas volume, To= room temperature and T= temperature after decompression, then;
P1Vcɤ =P0Vɤ, P1Vc =nRTo, PoV=nRT
When combined To/T =( P0/ P1)(1-ɤ)/ɤ
The gas warms up again after decompression until it gets to T0 and a P2 pressure. At this period P2>Po and P2<P1. Keeping the n and V constant from the ideal equation, then
PoTo = P2T; Hence
P2/Po = (P1/Po)(ɤ-1)/ɤ . Comparing the two equations I and 2 ɤ = ln (P0/P1))/ln(P2/P1). When a manometer is used to take the measurement, then ɤ=h1/( h1- h2) where h1 and h2 are the column height of the manometer.
The other method is Ruchhardt. This makes the use of a syringe with piston type motion. If the mass (m) and cross-sectional areas (A) are known and given that the piston oscillates at a period, the harmonic motion can be calculated using a differential equation m y+ ɤPA2 vy =0
Y is the piston displacement. Solving for the period and then;
T=2πw=2πmVɤPA2
The γ of the gas enclosed is gotten from γ=4π2 mvA2T2P where P = sum of weight of piston and pressure of enclosed air i.e p=patm+ mgA
Apparatus
Procedure
The connection and operations of the manometer were checked. The connections, settings, control and operation of the oscilloscope and the piezo strip were checked. Finally, the cables and tubing connections were assessed
Clement and Desorme method
The equipment was set up as in shown in the diagram
The paraffin oil was checked on either arm of the manometer to be the same level indicating that the atmospheric pressure was acting on both the arms.
With the Tygon pitching clamp open, the bladder was used to fill the glass flask with air so as to obtain a pressure above atmospheric inside the flask
The flask was then allowed to settle at room temperature and the height difference between the levels in the arms noted, h1
The room temperature reading was recorded as well, T1
The bladder was then disengaged and pinching clamp unscrewed to allow air into the flask until atmospheric pressure was hit.
Once at atmospheric pressure, the clamp was closed. The reading of the height of was recorded too. The process was repeated five times
Figure 1
Ruchhardt method
The equipment was set up as in figure 2 below
The diameter of the syringe was measured using a caliper and the atmospheric pressure recorded
The piston was also weighed using the scale
The oscilloscope was weighed to acquire a single trigger at a time
The syringe was filled with 25 ml amount of air and the volume recorded
The piston was then depressed to a reasonable depth till the pressure rose
It was then placed just touching the piezo strip
The piston was released, and the oscillations of the syringe recorded using the oscilloscope
The printout was taken from the oscilloscope.
Figure 2
Results
Clement and Desorme method
When developed through an excel spreadsheet
Taking the equilibrium to be P0, then from the equation ɤ=h1/( h1- h2) we can be able to calculate the value of gamma. The averages of P1, P2 and equilibrium are
Then, ɤ=h1/( h1- h2)
1.977045
When doing this calculation, we assume that both the mass and cross-sectional area of the arms remains constant. Also, it is assumed that the room temperature and pressure remain constant.
Conclusion
The heat capacity or air or any gas can be easily calculated from the Clement and Desorme method. This method could be faced with uncertainties from taking readings due to parallax error, varying room temperatures and pressure, unleveled working bench, failure to attain the accurate atmospheric pressure when the clamp is unscrewed. However, when multiple readings are taken, then a closer to actual result for ɤ can be obtained. More precise equipment for measuring like digital manometers and calipers should be used for better results.
References
Samokhin, A.A. "On Fluctuations In An Adiabatic Process". Physics Letters A 36.5 (1971): 372. Web.
Chirpich, Thomas P. "Ideal And Non-Ideal Gases. An Experiment with Surprise Value". J. Chem. Educ. 54.6 (1977): 378. Web.
