Graham’s Law Actually or gas transfer through an egg

Graham’s Law is one of classic gas laws you learn about in AP or college physics but despite its apparent simplicity it’s very difficult to actually demonstrate the law. Less of a law and more like a tendency under very special circumstances. Many, many things will interfere with a gas moving according to Graham’s Law, like pressure differences, convection currents, and solubility of materials. I originally thought testing out the different sizes of molecules would be as easy as blowing up some balloons but I was sorely mistaken, since the rubber in the balloon interacted with the gases making it impossible to measure Graham’s Law. But after much thought and a little diligence I was able to create an experiment that did reproduce Graham’s Law and demonstrated that Molecules really do move (at least in part) according to their molar masses.

This experiment obviously went through some evolution. Conceptually I had to make the leap of comparing different gases to a control gas that, while not included in the final calculation, stabilized the environment so that comparisons could be made between other gases. What follows is the final version of the experiment.


Materials:
1 to 3 chicken eggs (assuming you don’t have access to a porous cup or enough cups for a classroom demonstration).
Bristle brush, for cleaning the eggs.
Nail or craft knife for poking small holes in the egg.
Vinegar, 5% Acetic Acid from the store is fine.
Glue for tubing, E6000 is a good bet and easy to get at the Hardware store, hot glue may also be okay but provides a poorer seal and can melt the tubing so be careful if you try it.
About 4 ft. (1.2m) of flexible tubing, clear is better.
Another 1 ft. (0.3m) of flexible tubing opening should be just large enough to fit snugly around the other tubing and create a seal.
Dosing syringe from your local pharmacy
Food coloring and water
2 pieces of 3in (about 8cm) diameter pipe, capable of comfortably fitting the egg.
Large manual valve.
Fittings for capping the ends of the pipe and attaching them together with the valve between them (see setup diagram).
Needle nose pliers
Stopwatch or other timer.
Ruler (optional)
Video camera or other device for capturing the manometer level with high fidelity over a long period of time (possibly a laser pointer across the top of the liquid layer).
Helium gas (balloon time tank is fine)
CO2 gas source (bicycle charger, fire extinguisher or baking soda and vinegar reaction)
C2H4F2 gas source (Canned air for electronics, DustOff)


Experimental steps:

  1. Begin by blowing the eggs (see video). You only need one but it’s a good idea to have a spare or two.
  2. Once you have the shell take a small bristle brush and vigorously rub the inside of the egg with the brush to remove as much of the membrane that sticks to the inside of the egg.
    Submerge your egg in vinegar for about 1 hour
  3. Take the egg out and rinse thoroughly, again brush the inside vigorously as the vinegar may have loosened more material from the inner surface.
  4. Before gluing the tubing in make sure the egg is fully dried.
  5. In one side of the egg widen the diameter of the hole, if needed, to accommodate the width of your tubing, though usually all the cleaning will have made one hole large enough already.
  6. Glue in a 1.5 to 2 ft piece of your thinner tubing. You may need to brace the tubing with a toothpick and be careful not to close over the tubing with glue it needs to be open to the inside of the egg.
  7. Also glue shut the opposite hole from blowing the egg.
  8. While the egg is drying assemble your manometer. Start by connecting two pieces of tubing, one thick and one thin. The thick one will be where your egg is connected later.
  9. Add glue to create a permanent connection between the two different diameter tubes. This will form the bottom of the “U” in your manometer.
  10. After the glue is dry attach the U tube shape to a vertical surface, a piece of foam core works fine, wood or just taping it to the wall is also sufficient, but the straighter the tubing the better.
  11. Mix up some food coloring and water, the food coloring just makes the water level easier to see. Suck up some of this into a dosing syringe so that it can be suirted into the manometer tubing.
  12. Fill the tubing about 5 to 6 in. (10 to 15 cm.) deep. You will likely get bubbles. You can gently blow and suck on one side of the manometer to slosh the fluid around and this usually gets rid of most of them. Another trick is to insert a thin piece of wire into the tube to break the surface tension and the water will flow along the wire toward the bottom.
  13. Next connect the tube from the egg to the thicker side of the manometer and test the seal by cupping the egg in your hand. The heat from your hand should cause the gas inside to expand tilting the water in the manometer.
  14. Assuming the egg tests good the next phase is getting the egg into the piping that will late connect to the valve. Begin by carving a small hole for the tube connected to the egg to escape through.
  15. Next insert the egg into the tube and use pliers to carefully pull the tubing through the hole you just made. Glue around the edge to make a seal.
  16. Next cap off the pipe with the fittings and arrange the pipe pieces into a closed column with the valve in the middle and the egg with it’s manometer connection going out the side (see the end of the video for an example).
  17. Once assembled hook the manometer up again to the egg.
  18. Fill the top pipe with helium gas. This should immediately cause the manometer to show an overpressure (the side the egg is connected to goes down and the thinner side goes up)
  19. Cover the opening and attach to the valve. Below the valve attach the other pipe, this time just leaving air (mostly nitrogen) inside the bottom.
  20. Wait until the manometer even out again before the next step. This is important since if you go too fast the timing will be wrong.
  21. Flip the manometer and open the valve. Immediately start the timer. An assistant can be good at this point.
  22. Watch the manometer closely, as the Helium leaves the egg you should see a small dip in pressure inside the egg. Continue timing until the manometer returns to equilibrium.
  23. Depending on how your egg/manometer was built this can take about an hour and small changes may not be easily visible. A video recorder or other apparatus to monitor the fluid level in the manometer is strongly suggested.
  24. Record the time that the manometer stopped moving. Again reviewing footage may kane this more accurate.
  25. Repeat steps 19 through 25 but this time add either CO2 or C2H4F2 into the bottom chamber first. If everything is working properly you should get longer equilibrium times for them.
  26. After recording all the times (this may take several hours) compare them via the mathematics explained below.

Data:
Here’s the basic data that was used for my gas comparison.

Molar mass:
N2=28
CO2=44
C2H4F2=66
*Note these values were gathered by adding up the atomic weights from the periodic table of all the atoms in each molecule.

Time of diffusion from He to fully stabilized:
N2=2572s ±120
CO2=3135s ±120
C2H4F2=4109s ±120

Math:
Graham’s law is a relationship between the molar mass, or mass per molecule of a gas and the rate at which it moves through a barrier. The reason this works (or should) is because when the temperature is kept constant all the molecules in a sample have the same average kinetic energy; however since they do not have the same mass they must move at different speeds to have that value of kinetic energy. Conceptually you can imagine a large molecule like a bowling ball and the small ones like ping-pong balls. If all the ping bong balls are bouncing wildly about they can move the bowling ball a little but even when the ping pong balls transfer their momentum to the bowling ball the bowling ball never gets moving very fast. For gas molecules the same holds true and is encapsulated in Graham’s Law:

Grahams Law basic
(eq1)

This is all well and good but the problem is that unless I can know the rate of one gas as it moves through a tiny hole or across a room with no other atoms it’s very difficult to get a rate (volume of gas per amount of time) to compare with other gases. This is why the final experiment uses the control gas as a background medium for the gas transfer comparison. Mathematically it cancels out in the end but it needs to be used to provide a basic measurement against which other gases can be compared. So the idea is set up like this mathematically:

Test gas grahams law
(eq2)

then we would make the same set up for the second gas

Test gas 2 grahams law
(eq3)

Next we would divide eq1 by eq2

devided test gases
(eq4)

using the reciprocal rule

test gas ricripricol
(eq5)

canceling the test gas out we get Graham’s Law again but each step was able to be measured against a test gas.

Grahams Law basic

Note that the right hand side is a rate, that is some volume of gas (1 eggs worth) per some units of time (seconds) so we should expand Graham’s Law to show the times and volumes.

egg rates
(eq6)

Using the reciprocal rule again

egg rates ricripricol.png
(eq7)

canceling the egg volumes

time ratio.png
(eq8)

we can now see that because of the special set-up we have the ratio of the times for each gas to move through the egg should be equivalent to the ratio of the molar masses. Remember this only works because the control gas cancels out, and we can only justify this cancellation if we wait long enough that the test gas has completely penetrated the egg and no longer has some mixture of gases. If the experiment is stopped too soon this method will not work.

Still we now have the tool we need to make a prediction of the relative times of diffusion of the three gases of interest, Nitrogen (N2), Carbon Dioxide (CO2) and Difluoroethane (C2H4F2). Helium (He) being the control gas and thus not part of the measurement directly. The predicted rates for how much slower CO2 and C2H4F2 should be than N2 can be written as follows:

CO2 predict.png
(eq9)

C2H4F2 predict.png
(eq10)

Next we take the average value. I repeated the experiment 3 times for each gas before I ran out of helium as the control gas and had about 2 min of variation in the times.

CO2 entering the egg vs N2 entering the egg
CO2 time ratio.png
(eq11)

C2H4F2 entering the egg vs N2 entering the egg

C2H4F2 Time ratio.png
(eq12)

Error analysis: Overall the agreement with the predicted values was fairly good despite only attaining a 4%-5% accuracy in the measurement of the times. The actual percent error from the predicted value is:

CO2 entering the egg vs N2 entering the egg
CO2 Error.png
(eq13)

C2H4F2 entering the egg vs N2 entering the egg
C2H4F2 Error.png
(eq14)

Again by no means perfect but it is a considerably better demonstration than letting a balloon deflate, where carbon dioxide and hydrocarbons pass through the rubber faster than helium does. It could even contend with the NH3 and HCl diffusion down a tube demonstration since this is far less dangerous, if somewhat slow.

Discussion:
I’m finally happy with the result! ^_^

But more seriously I did in fact start the erstwhile balloon experiment way back in August before I even launched my channel and if you’ve seen that video you saw that it did not go according to plan. Sure, I could have simply left it at that with the knowledge that Graham’s Law doesn’t work under rubbery and organic circumstances but I wanted to KNOW. So I kept at it watching videos, reading up on other demonstrations. Eventually i came up with this, and not before seriously considering doing NH3 and HCl diffusion but I hardly have any glassware and I really wanted a test an ameture (like myself) could perform without much risk.

I wasn’t sure, even until the very end if this one would work, but still only 4.7% and 2.4% errors is pretty darn good after all that did go wrong. Here’s hoping you try it and please comment away (here or on the video) if you have any questions about the execution.

Linkography:
How to blow eggs
https://www.youtube.com/watch?v=OE19g_OCZms

Porous cup demonstrations
https://www.youtube.com/watch?v=w5ktwcDyOWo
https://www.youtube.com/watch?v=nuADfRR9Yto

Graham’s Law
https://www.youtube.com/watch?v=TLRZAFU_9Kg
https://www.youtube.com/watch?v=g6QuuoTs2Oo

Dalton’s Law
https://www.youtube.com/watch?v=90825On7Nfs
https://www.youtube.com/watch?v=gvXMg3J5j_Y
https://www.youtube.com/watch?v=JbqtqCunYzA

How do chicks breath inside a shell – porosity of an egg shell.
https://www.youtube.com/watch?v=QVc8Stq3LFI
http://www.scielo.br/scielo.php?pid=S1516-635X2000000200007&script=sci_arttext

More detailed explanations and references
http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Reaction_Mechanisms/Rate_of_Diffusion_through_a_Solution
https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
http://web.cecs.pdx.edu/~fishw/FT_L3_ADE_Derive.pdf

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