Gay-Lussac's Law states that when the volume of a container of gas is held constant, while the temperature of the gas is increased, then the pressure of the gas will also increase. In other words, pressure is directly proportional to the absolute temperature for a given mass of gas at constant volume. Although this is, strictly speaking, true only for an ideal gas, most gases that surround us behave much like an ideal gas. Even ordinary air, which is a mixture of gases, can behave like an ideal gas.
In this experiment, a PocketLab that is sealed inside a Mason jar can be used to verify Gay-Lussac's Law as well as extrapolate a value for the absolute zero of temperature. The PocketLab is set to "Two-Graph" mode, recording pressure in mBar and temperature in celsius degrees. Considering the PocketLab specifications for the temperature sensor, it is seen that the allowed range is from -20C to 85C. It would be perfect if we could measure the pressure of the air in the Mason jar for three different temperatures covering the allowed range. The photos in the figure below show three such possibilities.
The photo on the left shows PocketLab sealed in a Mason jar on a table at room temperature, about the middle of the allowed range. The photo in the middle shows the PocketLab Mason jar in a freezer, which will give us a temperature near the low end of the allowed range. The photo on the right shows the PocketLab Mason jar in an oven set to a maximum temperature of 170F (77C), just a little below the high end of the allowed range. It took about an hour for the PocketLab temperature sensor to reach the desired values in the freezer and in the oven, so patience is required.
For safety, protective goggles should be worn. In addition, gloves should be worn when removing the jar from the freezer, as it is cold enough to cost frost bite if handled too long. Gloves should also be worn when removing the jar from the oven, as it will be at a temperature that is not too far from that of boiling water. It is also essential to monitor the temperature on the iPhone to make sure that it doesn't exceed the high end of the allowed range. It should be removed from the oven and the oven turned off a little before reaching the high end of the allowed range. For the author, the stainless steel freezer and oven did not stop PocketLab from communicating data with the iPhone setting on a nearby counter. The author also kept the iPhone charging cord attached during the experiment to avoid running out of charge on the iPhone battery.
The Excel graph shown below summarizes the experimental results. The three data points fall very close to a straight line obtained by doing a linear trend/regression. The line is extended to the left until it reaches the temperature axis. At that temperature, -233C, the pressure would be zero. The value -233C can be obtained from the regression equation by setting y to 0 and solving for x. With the absolute zero of temperature at -273.15C by international agreement, our value of -233C represents an error of about 14.7%. It would likely promote a good classroom conversation to discuss possible causes for this error.
With a pressure sensor built into PocketLab, there must surely be some way to investigate Boyle's Law. This law states that pressure and volume of an ideal gas are inversely proportional to one another provided that the temperature and amount of gas are kept constant within a closed system. What is needed is a closed system that is large enough to hold PocketLab in a way that pressure can be sensed while changing the volume of the enclosed gas (in our case, air).
Educational Innovations, Inc. (teachersource.com) currently sells a Microscale Vacuum Apparatus for $35.95 (as of 8/9/2016) into which PocketLab easily fits. The photo below shows this apparatus with PocketLab inside the polycarbonate bell jar.
The student can determine the volume of the bell jar by a variety of methods. It turns out to be about 220 cc. An estimate of the volume of the air within the attached hose is about 2 cc. Estimating the volume of the solid parts of PocketLab that displace air in the bell jar to be about 12 cc, the volume of the trapped air is then approximately 210 cc when the syringe is at the zero mark.
Before collecting data with PocketLab's pressure sensor, the piston in the syringe is pumped enough times to bring the pressure down to approximately 300 mBar. Then the piston is pulled out and held at the 5 cc (or ml) mark for a few seconds. Then it is pulled further to the 10 cc mark and held for a few seconds. This process is continued by steps of 5 cc through the 30 cc mark. Meanwhile, PocketLab records the pressure of the trapped air, as shown in the video below. We therefore know the pressure readings and the corresponding volume readings (210 cc + 5 cc = 215 cc; 210 cc + 10 cc = 220 cc, etc., through 210 cc + 30 cc = 240 cc).
An Excel chart of the data is shown in the figure below. It clearly shows that increasing pressure results in decreasing volume--some kind of an inverse relationship. But what is the best trend/regression type fit for this data?
When linear, exponential, and power trend/regression types are applied in Excel, all there types provide R-squared value that are very close to one, ordinarily indicative of a good fit. See the three figure below for these charts.
As we can see in all three charts, our data is but a tiny portion of the spectrum of (volume, pressure) pairs possible. This is because of the large space required to hold PocketLab and its pressure sensor. The collected data would likely show more of a curve if this space was small compared to the volume of the syringe. Students can be encouraged to provide intuitive arguments against the linear and exponential trends (questions about what happens if the pressure is zero or the volume is zero--can any finite quantity of gas occupy zero volume?)
This leaves us with the power trend/regression as the only reasonable choice. The exponent is shown as -1.157, close to the value -1 that we would expect if volume and pressure were inversely proportional (i.e., if PV = constant).
In this experiment students will use PocketLab to collect data related to the cooling of a container of hot water as time goes on. Sir Isaac Newton modeled this process under the assumption that the rate at which heat moves from one object to another is proportional to the difference in temperature between the two objects, i.e., the cooling rate = -k*TempDiff. In the case of this experiment, the two objects are water and air. Newton showed that TempDiff = To * exp(-kt), where TempDiff is the temperature difference at time t and To is the temperature difference at time zero.
In order to do this experiment on Newton's Law of Cooling using PocketLab, we need to wrap PocketLab securely in a plastic sandwich bag, so that water cannot leak into the bag and damage PocketLab. The figure below shows how this can be done. PocketLab is wrapped and taped tightly in the plastic bag. It is immersed in water briefly to make sure that it has no leaks. A piece of Velcro is attached as shown on the right of the figure. The purpose of this Velcro is so that PocketLab can then be secured to the bottom of the hot water container without floating.
The figure below shows the complete apparatus setup. PocketLab has been placed and secured to the bottom of a small bottle not much larger than PocketLab. The bottle needs to be as small as possible, for even a small amount of water in the bottle can take several hours to cool back down to near room temperature again. The bottle used by the author required only about 70 ml of water, but data was collected for close to two hours. The room temperature is noted, hot tap water is poured into the bottle until it is full, and data collection is initiated with PocketLab set to record temperature once each second. The purpose of the optional small block of foam is to insulate the bottle some from the table top, resulting in most heat loss into the air.
The image shown below contains a graph of temperature from data produced by the PocketLab app. A red dot is shown once for every two-hundred data points. The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com). The initial rise in temperature is due to the time required for PocketLab to warm up as heat is transferred from the water to PocketLab. Once equilibrium has been reached, the system begins to show cooling as heat is transferred to the surrounding air at room temperature. The cooling appears to be negative exponential. The model fit Temperature = A*exp(-Cx)+B was then applied to the region of the graph shown in dark gray. The correlation coefficient of 1.000 clearly indicates an excellent fit. x corresponds to time in Newton's equation, A to the initial temperature difference, C to the constant of proportionality k, and B to room temperature.