Determination of coefficient of expansion of air

INTRODUCTION
This experiment is based on investigating the coefficient of expansion of air using a simple laboratory set up; the stopper flask method, where pressure is constant throughout the experiment. The increase in volume of a gas is directly proportional its temperature increase and is expressed as a fractional changed in dimensions per unit temperature change. Air will easily expand when it is heated and contract when it is cooled.
The aim of the experiment was to:
* Determine the coefficient of expansion of air using a stoppered flask method.
The flask was stoppered and a thick tube allowed interactions with the outside. The flask was heated in a beaker (with water) and then transferred immediately to cold water where the cold water was allowed to enter and air within the flask decreased. The initial and final volumes of air and water was calculated (directly or indirectly – whichever appropriate) and the coefficient was calculated from these.
The experiment in its design allowed the calculation of the coefficient of expansion of air to be 3.22 * 10-3 K-1. This was calculated at a temperature of 24oC and pressure of 1 atm, which gives a good approximation compared to the theoretical value of 3.37 * 10-3 at a temperature of 24 oC (297 K).
THEORY
Dooley (1919) indicates that gases are said to be perfectly elastic because they have no elastic limit and expand and contract alike under the action of heat. That is to say, every substance when in the gaseous state and not near its point of liquefaction has the same coefficient of expansion, this coefficient being 1/273 of its volume for each degree Centigrade.

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He further goes on to say that since a gas contracts 1/273 part of its volume when its temperature is lowered 1° C, such a rate of contraction would theoretically reduce its volume to zero at a temperature of – 273° C. Since all gases reach their liquefying point before this low temperature is attained, however, no such contraction exists. At the same time, it may be said that if heat is considered as a motion of the molecules of a substance, that motion is to be considered as having ceased when the temperature has reached – 273° C.
This is the expansion coefficient of an ideal gas.
GAY LUSSACS LAW
Madan (2008: 81) indicates that the coefficient of expansion of a substance at any given temperature, t, is the small fraction of its volume by which one cubic centimeter of the substance will increase when heated from to.
* Gases are affected by changes of temperature in the same general way as liquids and solids, expanding when heated and contracting when cooled.
* For a given change in temperature, they change in volume to a far greater extent than either liquids or solids.
* All gases, at temperatures considerably above their liquefying points, have practically the same coefficient of expansion. This was first observed by Gay Lussac and Charles, and is a very remarkable one, and a great contrast to what has been noticed in the case of solids and liquids, each of which has its own special coefficient of expansion, often differing widely from those of others.
EXPANSION AGAINST CONSTANT PRESSURE
Atkins (2006: p35) indicates that:
By definition:
At constant pressure:
This indicates that the work done is actually the difference between the final and initial volumes multiplied a unit of pressure (which is constant). Once can say therefore that a gas expands (independent of pressure) but dependant on temperature as given by:
METHOD
Method as per hand out, however, a small beaker with water was used to heat the flask and atmospheric pressure was used instead of reading the barometric height (which was not available).
MATERIALS/APPARATUS
· Conical Flask (100 mL)
· Rubber Stopper
· Metal Clip
· Short Glass Tube
· Heater
· Beakers (500 mL) – 2
· Thick Walled Rubber Tube
· Thermometer (0 – 100oC)
· Electric Balance

Weight of flask + fittings

136.4 + 0.1 g

Weight of flask + fitting + water sucked in

168.6 + 0.1 g

Weight of water sucked in

032.2 + 0.1 g

Weight of flask + fittings + full water

279.8 + 0.1 g

Weight of full water

143.4 + 0.1 g

Temperature of boiling water

103.0 + 0.1 oC

Temperature of cold water

024.0 + 0.1 oC

Atmospheric Pressure

1.00 atm

Volume of gas @ 103.0 oC

143.4 + 0.1 cm3

Volume of gas @ 24.0 oC

111.2 + 0.1 cm3

DISCUSSION
The experiment investigated the coefficient of expansion of air. This value was found to be 3.22 * 10-3 experimentally. One would infer, at first glance, that the volume of air initially would have been the volume of the flask (100 mL), as the volume of a gas is the actual volume of the container. But why was the mass of the beaker found (filled with cold water)? Was it to give a better estimation of the volume of the air? By finding the volume using the density of water, it was found to be 143.2 cm3 which is a large difference compared to the 100 mL of the flask. Then one realized that the flask was filled to the top close to the stopper itself, and therefore assuming that the volume of air was 100mL would have been a grave mistake and calculating the volume by density was the best and accurate method to use.
The experiment relies on the fact that the volume of a substance, in this case, air, is dependent on the temperature of the system. The flask (opened) was heated in boiling water, this was indirect heating of the flask, it allowed the inside of the flask to be dry and consequently allowed the air to be dry. In addition, by heating the flask in boiling water, the temperature of the air inside the flask increased as well (according to the zeroth law of thermodynamics), indicating that there will be some form of thermal equilibrium. At this point, the initial volume and temperature of the air will be obtained.
The tube was closed with a clip and placed in the water at a lower temperature. The question that arises at this point is why was the clip closed? A logical assumption is that to disallow further interaction between the atmospheric air (at a lower temperature) and the flask’s air (at a higher temperature), also one can say that because of the temperature gradient, their will want to escape and in so doing create a thermal equilibrium between the two. The water was allowed to enter, to replace the air and thus the volume of air decreased. This method was unique in its design that it used a backward approach. Rather than obtaining the expansion of air from a lower to a higher temperature, it measured the contraction of the air from a higher to a lower temperature. In the end, the initial and final volumes and temperatures of the air being considered were obtained, and thus the coefficient was able to be calculated.
SIGNIFICANCE OF EXPANSION COEFFICIENT
The value ascertained experimentally was 3.22 * 10-3. This can be termed a fractional change as it is very small (0.001th of a value 3.22). It can be inferred that this fractional change affects the volume of the sample when a rise in temperature occurs. It means therefore, that for every change in temperature from to to (t+1)o, the volume of air in one cm3 of air will increase by 3.22 * 10-3 at 1 atm (experimental condition). A small value of α, indicated by Atkins (2006) implies that it responds weakly to changes in temperature i.e. the air responds weakly to changes in temperature which is important in life itself, as air responding strongly to temperature changes would be hazardous to our health, and may even result in cardiac arrests with sudden decreases in temperature (during winter time in north America and Europe among other places) and where there are heat surges.
COMPARING EXPERIMENTAL AND THEORETICAL EXPANSION COEFFICIENT
The theoretical value of the expansion coefficient should be, since. The deviation is (3.37 * 10-3- 3.22 * 10-3) = 1.5 * 10-2. This deviation represented almost 4.66% of the theoretical value! What can account for this deviation? It all leads to experimental errors, since pressure is constant. Obviously, by looking at the formula, the process of obtaining the final and initial volumes and temperatures will have an effect on the expansion coefficient. The volume of water sucked in may not have been at maximum due to hindrances in the tubing attached to the flask, or the water was not allowed to go in as fast as it should. Also, one can consider that the density of water used to calculate the volume of air after the water had been sucked in may have been different and hence affected the calculated the volume). All of these can contribute uncertainties to the coefficient of expansion and can be used to explain the difference observed.
SOURCES OF ERRORS
* The difference between the experimental and established values is therefore attributed to factors such as temperature, volume, and the accuracy at which these values were obtained as described above.
* The density of water probably affected the results when it was used to calculate the final volume of air and initial volumes of air.
* Within the limits of experimental error, the value ascertained was close to the theoretical value with only about 5% deviation.
* The volumes and temperatures had uncertainties of + n, where n represented the volume and temperature. The final result of the coefficient had an uncertainty of 0.41%.
LIMITATIONS
* The method did not allow the calculations of the volumes and temperatures directly but indirectly. A direct method, if possible, would have contributed to a more accurate value of the coefficient of expansion.
* The experiments were not repeated to ascertain different values of the volumes and temperatures. Averaging the values would have allowed a more accurate value of the temperatures and volumes and by extension the coefficient of expansion.
ASSUMPTIONS
* It was assumed that air was ideal in nature and followed the ideal gas equation. Introduction of van der waals coefficient would have proved to be more tedious in calculating the coefficient of expansion of air.
* It was assumed that the volume of dry air in the flask was the volume of the water in cm3. As mentioned previously, the water was filled to the top of the flask (close to the stopper), and assuming 100mL would have been grossly inadequate contributing to more uncertainties and thus a more inaccurate value of the expansion coefficient.
* It was assumed that rate at which the temperature and volume decreased when the flask was placed in the water allowed the expansion coefficient to be ascertained. This was very important, as it implied that the temperature affected the expansion and or contraction of air and water which ultimately enabled the calculation of the coefficient.
CONCLUSION
With reference to the aim, it can be concluded that the experiment in its design allowed the calculation of the coefficient of expansion of air to be 3.22 * 10-3 K-1. This was calculated at a temperature of 24oC and pressure of 1 atm.
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