29th October 2005 14:41
Partial Pressure Maths Explained
Partial Pressure Explained
By Ron Micjan
One of the harder things for new Rebreather and mixed gas divers to understand is the math involved in figuring Partial Pressures of the gas they are breathing. The first thing to get clear is what exactly are we talking about when we discuss the PP of a particular gas. Then we will get into how to calculate things like best mix, equivalent air depth (EAD), equivalent narcotic depth (END), and the all important thing for a Rebreather diver to understand, how to quick calculate what your mix should be when you do a diluent flush.
Composition of Air
In the above diagram we can see the approximate composition of air, there are about 1% of other trace elements in this mix and oxygen is generally referred to as having 20.9% of oxygen and nitrogen as 78% but for our purposes we will refer to air as 21/79.
Pressure is defined as the movement of the gas particles bouncing around and impacting everything they are exposed to. Gas, unlike water, or solids, is easily compressible and the pressure of the air around us is about 14.7 pounds per square inch at sea level. This pressure is created by gravity, pulling the planets surrounding atmosphere toward the core of the earth. Gas, like anything, has mass, and mass attracts mass according to Newton’s law of gravitation. What does air weigh? Glad you asked. If you were able to contain one square inch of gas in a vertical column stretching from the ground, all the way up to space, that column of gas would weigh…14.7 pounds. So it’s the mass of the air above us, being pulled down by the earth’s gravity, that causes the pressure we have surrounding us. We call this amount of pressure, 1 Atmosphere, or ATM.
1 ATM = 14.7 Pounds per square inch or PSI = 1 BAR (metric equivalent)
We must remember to include the pressure of our surrounding atmosphere when we do our calculations. We are going to compare our pressures to a perfect vacuum, or zero pressure or 0 psi. We also call this type of pressure measuring “absolute pressure”. So we are surrounded by 1 ATA, which means Atmosphere Absolute. If our atmosphere was twice as thick, we would have 29.4psi of atmospheric pressure, 2x14.7psi=29.4psi.
So, what does water weigh? Lets use that same one square inch tube we used for the air measurement. We will find out that our water column only has to be 33 feet high to get to that same 14.7 pounds, so to double the ambient pressure around us, we would have to dive 33 feet below the surface and then the pressure would be 29.4psi, or 2 ATA. So we have all the atmosphere above us, 1 ATA, and an additional 33 feet of water, another ATA, added together gives us 2 ATA of pressure. If we went down another 33 feet under water, we would be at 3 ATA and so on. So depth can be expressed in feet, meters, BAR or ATA. See the below table for some continuance of this.
Also notice Dalton’s law in effect. As we descend, if we were to take a bag of gas with us, say one cubic foot of air, in an elastic bag, like a balloon, we would see the volume of that bag shrink with the additional pressure of the water around us. At 165 feet, that bag would be one sixth of its size at the surface. It still contains the exact same air we filled it with on the surface, but now the molecules are much closer together, we say it has a higher density.
Some Fun Stuff to tease your OC buddies with!
Lets toss in another advantage of diving a rebreather and compare how we utilize an aluminum 80 compared to a single 13cf oxygen cylinder we might have on our rebreather. The divers RMV (respiratory minute volume) is based on a .6 cfm converted to liters, so .6x28.3= 16.98 or 17 lpm. The same diver (me actually) uses about 1 liter of oxygen per minute as a diving metabolic rate or DMR. Yeah, I know my RMV sucks since I went to the rebreather, it used to be around .4cfm. Hey, but so what, I'm still diving with 4+ hours of gas on me.
So, as you can see, our diver diving open circuit must take down 360 cubic feet of gas to do the same one hour dive at 132 feet as our rebreather diver does on 2.1 cubic feet of oxygen. Now we must also add that we will actually use more gas than this because of mask clearing and equalizing as well as the gas we bleed off during ascent, but the facts still show the RB diver uses much less gas than an OC diver.
Lets calculate how much oxygen is in an aluminum 80. 80 CF of gas, of which 21% is oxygen, so 80x.21=16.8 cubic feet of oxygen, converted to liters is 16.8x28.3= 475 liters of oxygen, this amount of oxygen would allow the RB diver to stay under about 475 minutes or almost 8 hours!!! Amazing! So at 132 feet, the open circuit diver is only using 1/85th or 1.1% of the oxygen he inhales and the rest of it gets wasted as bubbles on the surface.
Ok, back to the lesson…
Lets go back to our Air mix.
The oxygen component of air is 21%, or in decimal, .21 or 21/100 of the whole amount. If the whole amount is 1, as in 1ATA, then the Partial Pressure of the oxygen in air, at the surface, is .21, easy, right? Now if we go under water to the 33 foot level, we just added another atmosphere of pressure didn’t we? So the total pressure we now have is 2 ATA, but the FRACTION (remember this word) of the oxygen is still .21. To get the Partial pressure of oxygen at 2 ATA we multiply the FRACTION of the gas by the pressure absolute or ATA to get the partial pressure of that gas. At depth if we measure the number of molecules of oxygen we would count 3 times the number in the same volume, at 3 ATA of depth, than at 1 ATA of depth (the surface). Make sense?
2 ATA(our depth measured in ATA) x .21 (the fraction of oxygen)= .42 Partial Pressure of Oxygen, or .42 PPO2.
This works for nitrogen too, if nitrogen is 79% of the whole of air, or .79 in decimal, then it stands to reason that at 2 ATA our partial pressure of nitrogen is…
2 ATA x .79 = 1.58 PPN2 or partial pressure of Nitrogen.
Lets see a graphical representation of that formula.
Some folks call this the Pigs on Top formula, some call it the T formula, call it whatever you want.
T Formula or Pigs on Top Formula
This is something you might want to memorize, it will come in handy as long as you are a diver.
To use this formula, you cover the data you want to find and do the resulting math. For example, cover the PPg, what’s left? Fg and Depth, so Fg (or Fraction of Gas x Depth (in ATA) = the Partial pressure of that particular gas, at that depth.
Cover the Depth and you are left with PPg “over” Fg, so if you need the depth where a fraction of oxygen reaches, say, 1.6 PPO2, the Fg (fraction of gas) or oxygen content of your mix is 32%, or .32, then you divide (remember PPg/Fg, same as PPg divided by Fg) 1.6 (our PPg) divided by .32 (the O2 of our mix) equals 5. OK 5 what? Remember our units, 5 ATA of pressure. How do we get depth out of that? Simple, if that is the pressure we are at, then subtract one ATA from it (remember one of our ATA’s is the weight of the earth’s atmosphere) so we are left with 4 and we know that every ATA of pressure is equal to 33 feet of water, so 4ATA x 33feet = 132 feet. So if we wanted the safe depth of a 32% nitrox mixture this is how we would arrive at that.
Lets do some more examples
If we had a nitrox mix of 32% oxygen, and we wanted to know what the PPO2 would be at 100 feet, how would we do it? Well we want the PP of a gas, in this case oxygen, so we put our thumb over the PPg part of the formula and we are left with Fg and Depth, so we multiply our Fg by our Depth in ATA.
Ok, the fraction of oxygen is 32%,or expressed as a decimal, .32. So Fg=.32 multiply that by our depth, but we cannot use 100 feet, because that is not in ATA.
So how do we get ATA from feet of water? Simple, just think about what is over your head, 100 feet of water, plus the earths atmosphere. How many 33’s are in 100? 100/33=3.03 (lets call it 3) So that is 3 ATA of water we have, now add that to the pressure of the atmosphere 1 ATA so 3+1=4 ATA of pressure at 100 feet.
Now take our fraction of gas, .32 and multiply that by 4, which is our depth, but measured in ATA. .32 x 4 ATA = 1.28 ATA of oxygen in that 32% mix at a depth of 100 feet. Since the human body can tolerate between .16 (hypoxia)and 1.6 (hyperoxia) partial pressure of oxygen, this 1.28 is right in the ballpark.
Really stop and think about what is going on here, if you can get this concept stuck hard in your head, you will have no trouble with the rest of this. Always remember to keep your units consistent and carry them through the equation, there is nothing more frustrating than to end up with a pure number that you have no idea what it means.
More about units, where are we going to see these units
PP: This will be the readout on our PPO2 meter at depth or when we calibrate sensors.
This will also be a way to figure how much narcosis we can tolerate, if we start getting narc’ed at around 100 feet of depth, on air, then we need to keep in mind we want our gas mixes to not allow a PPN2 higher than 3.2. more on how to do this later.
You may not need this, but you can also figure what the PP of your helium is in a particular mix. We are more concerned with PP’s of Oxygen and Nitrogen than helium but it is a good way to check your math at the end of mix calculations.
Depth in ATA: This is our depth unit, we will need to convert this to feet to make it useful to us, if we are plugging in a depth in feet, we divide that number by 33 and then add one to it. Feet into ATA (D in feet/33)+1 = D in ATA. If we are pulling depth out of the formula in ATA and converting to feet, we subtract one and multiply by 33. (Depth in ATA – 1) / 33 = Depth in feet. Instead of remembering the formulas remember the concepts. What’s over your head? if you are given depth in ATA, subtract the weigh of the air, and only count the water.
Fg: This is our fraction of gas, if we are pulling this out of the formula, we might be mixing some gas up for a dive. If we are plugging this into the formula, then we might be figuring out how deep we can take a particular mix, either limiting by Oxygen limits, or Narcosis limits.
So here are some more ways we can use this formula.
We want to do a dive to 175 feet to a wreck and want a bottom mix, or diluent to be a maximum of 1.6 PPO2 and a max narcotic depth to be equivalent of air at 100 feet. So this will be a trimix blend. First figure how much oxygen will be in the mix. So this would be a fraction of gas. We cover the Fg of the formula. Leaves us with PPg over Depth. The PPg is going to be 1.6ATA, the max PP of oxygen we feel safe with. The depth is going to be 175 feet, but we cannot use this measurement, so convert to ATA, divide by 33 and add 1 to the quotient. 5.3+1=6.3 ATA of depth.
Now 1.6 divided by 6.3, remember our algebra, the top number is divided by the bottom, the answer is .254 and now which way do we round? Think about what we are doing here, figuring an oxygen content of a mix to be used at 175 feet! Do we want to round up, hell no! Yeah, round down to .25 or 25% oxygen content in our mix.
Ok, now we know how much O2 to put in our mix, how about the other two gasses. Well the most important component of our mix is the oxygen, so we figured that first. The next most important component is the nitrogen and remember we wanted the narcosis to be like air at 100 feet. So what is the PP of nitrogen, on air, at 100 feet? What is the unit we want out of our calculations? OK PPN2, that’s the top of the formula, cover it, what’s left? Fg and depth. The fraction of nitrogen in air is .79 or 79% (mathematically the same). So .79 X depth. Our depth is 100 feet (remember we want the PPN2 on air at 100 feet) Depth is (100/33)+1 = 4.03ATA Lets do the math (I always wanted to say that). Fg of .79 X 4.03 Depth in ATA=3.18 PPN2. So we are happy with a PPN2 of 3.18, or 3.2. Narcosis is so variable it wont matter which way we round this. So now we have a safe PPN2 of 3.2 and we want to know what Fg, or fraction of N2 in our mix will yield a PPN2 of 3.2 at 175 feet. Cover the Fg, plug in the PPN2 of 3.2 in the top of the formula and divide by the Depth in ATA, we already figured it to be 6.3ATA so 3.2 divided by 6.3 = .507 or .51 = 51% of nitrogen, so we have 25% oxygen, and 51% nitrogen, what do we use to fill the rest of the mix? Helium, of course. We will end up with 25% O2, 51% N2 and 24% HE. Or TMX 25/24. There are several ways to state trimix blends, one comes from the Compressed Gas Industry, one used by divers and ANDI has there own. It doesn’t matter what you use, just so everyone involved is using and understands the same units.
Check our sensors at depth:
Many of us manual CCR divers only have two oxygen sensors and readouts to refer to. So what if they disagree during a dive? OK, do you know the Muffin Man? How about the oxygen content of your diluent? Because knowing the Muffin Man is not going to help you out at 150 feet, which is where our sensor took a hike. 150 feet is about 5.5 ATA in depth and the oxygen content of our diluent is normoxic or the same as air, .21. We flush the loop with diluent, exhaling out our nose and sucking in through our loop three times, or if we have a purge button, we could loosen lips and flush all at once. Our two sensors are now reading 1.1 and 2.1, which one is correct? Well you might want to think the higher reading is correct if you were just guessing, but I have personally seen an oxygen sensor that failed high for about 2 hours before it quit working altogether, so don’t count on that. Its best to do the math. Back to our T formula. We need to know what the PP of an air diluent is at 150 feet or 5.5 ATA. .21 X 5.5, cant do this in your head, better learn. One way is to carry a chart (see below) of what PP’s your diluent is at various depths, great if you use the same dil for every dive, or vary between just a couple mixes. The other way is to round, like round .21 to just plain 2 and multiply that by your five and a half and get 11, plug back in the decimal point and viola! 1.1, end your dive and pay attention to the one good sensor. You might also, if its possible, to make your depth an even ATA number, like 33, 66, 99, 132, 165 etc, to make your calculation easier. 2.1X5 is easier than 2.1x5.5
Below is a table for even ATA’s on air or 21 % oxygen mixtures. It is normal to have some variation in oxygen sensors, do not expect them to agree exactly.
Below is a table for 10/50 trimix.
Equivalent Air Depth EAD
This refers to comparing the nitrogen content in the mix you are breathing and matching it to a like nitrogen exposure if one was diving air. This is helpful if one is calculating decompression for a nitrox mix and the only thing you have to use is air tables. In my humble opinion, one should stick a crow bar in your wallet and buy the correct tables, or a nitrox computer rather than waste time learning to use this formula. ‘nuff said.
Equivalent Narcotic Depth END
Now here is a comparison I can reason with. If a diver knows that he/she will likely become narced at any depth below 130 FSW, then one would plan your mixes to have an equivalent narcotic depth of less than that, say 100 fsw or less. Like we did in the best mix calculations above. The PN2 of air at 100 feet is about 3.2. If you don’t know how I got that number, you have not been paying attention, go back to the T formula and catch up with the rest of us later. So, to make a mix with a PN2 of 3.2 or less, we will want to pull from our T formula a Fg, or fraction of gas, in this case nitrogen. If our depth is to be 140 FSW (5.2 ATA), again you should know by now how we got the 5.2, we covered the Fg, and 3.2 PN2/5.2 ATA Depth = .61, or 61% of nitrogen. So what is the rest of the mix? Well lets try oxygen, 100%- 61% = 39% of gas we need to make the rest of our mix. But 39% oxygen at 140 FSW is a bit hot, like 2.0 PO2, that’s likely to make you do the funky chicken, bad at any depth. So what would work? Well at 5.2 ATA of depth, we might want the mix to be a 1.5 PO2, so same formula, different gases, 1.5 PO2 of oxygen/5.2 ATA of depth =.29 or 29 % oxygen, so our mix is now 29% O2 and 61 % N2 which is 90% of the full monty, OK, 10 % helium will make the best mix for this dive. There is no reason to use more HE than you need, pick an END that you are comfortable with and plan your mixes accordingly.
Remember that this information is not placed here to provide you with a way out from taking a class from a certified and qualified instructor, this is here to help you on your way. Just because I have taught you everything you know, doesn’t mean that I have taught you everything you need to know. On line and book learning is no substitute for a knowledgeable instructor working with just you or a small class. Find a good instructor and grow with him or her.
When we dive on a closed circuit rebreather, we are said to be diving a constant PO2. What exactly does that mean?
Ok, as an open circuit diver we calculate our PO2 at the deepest part of the dive and mix our gas for that depth so as to not go over 1.6 PO2, the reason we get as close to that as possible, is so that we do not take on as much nitrogen and have to spend too much time on the deco line. The only depth that mix is really good for is at the deepest part of the dive. As we ascend, that mix becomes less and less PO2 so that as technical divers we bring along extra mixes with higher PO2’s to increase the gradient of oxygen and speed up decompression. Lets look at another table.
The diver on open circuit’s PO2 is steadily increasing as he descends and maxes out at 132 feet with a 1.6. The closed circuit diver is using a setpoint of 1.2 and as he descends his PO2 remains at 1.2 throughout the whole dive, while his Fg, or fraction of gas decreases from a high of 60% to a low of 24% at 132 fsw. He does not have to run his mix as high as 1.6 at any time, this reduces his OTU’s or Oxygen Tolerance Units, a measure of the pulmonary toxicity of oxygen. The CC diver has his FG increasing during ascent and this helps with the gradient of oxygen and washes out the inert gas faster. Look at the same diagram with air as an OC dive gas.
Now the OC diver is really not getting washed out of nitrogen, at 155 feet the mix is almost exactly the same, but our CC diver is at his worst fraction, it only gets better as he ascends. The OC diver will take a much longer time to decompress. Its even worse at the safety stop, our CC diver is breathing the equivalent of 80% O2 at 15 feet while our OC diver is breathing .31, quite the opposite of what you would want to do.
I ran a couple profiles on GAP deco software, 2 dives to 132 fsw, one diver on CC with a 1.2 setpoint, the other on air Open circuit, the bottom time was 20 minutes, the air OC diver had a total runtime including deco of 57 minutes and stops beginning at 70 feet, showing more nitrogen loading than the CCR dive who had only a 31 minute total run time, stops starting at 50 feet. Constant PPO2 is better for bottom time and deco, no doubt about it.
Lets do another thinking exercise
Lets say you are diving your CCR at a resort and your constant PO2 dive computer takes a dive without you. The only thing extra laying around is a normal nitrox computer, can you use it? What should you set the mix for? The first answer is yes, the second takes a bit of thinking. Decompression is based on inert gas, in this case nitrogen, so lets look at what the computer is calculating and what we are breathing.
Ok lets look at our nitrogen uptake, think about this. Our air computer is thinking we are breathing 79% nitrogen on open circuit. As long as we are not breathing a higher concentration of nitrogen than that, we will fall below the model and be diving more conservative than the computer thinks we are. Remember also as we ascend our Fg O2 gets higher on CC, while the air computer still thinks our Fg O2 is getting lower, so if the computer says we can ascend we are for sure, good to go.
But where does this model BREAK down? Look at the table above. AT 33 fsw the computer thinks we are breathing 1.6 partial pressure of nitrogen, we are actually only breathing .8. Terrific! How about at 132 feet, computer thinks we are sucking on 3.95 PN2 while we are at only 3.8, still pretty good. But what about if we drop down to 188 keeping that 1.2 setpoint, whoops, we just blew the model and are actually taking on more nitrogen than the model thinks we are, in fact we are only breathing the equivalent of 17% oxygen whereas the model thinks we are breathing 21%. How to fix this? Glad you asked. Its quite simple really, we just increase the Setpoint of our rebreather to 1.5 for the time we are at that depth. Do the math. Remember the T formula, to find the PO2 that will match .21 at 7 ATA, .21x7= 1.47PO2. its really quite simple, we just have to think about what we are breathing, its not rocket science, just a bit of math. Approach the mixture from all the angles and compare it. How much nitrogen? How much Oxygen? Holding a setpoint, or Open circuit? If you plan to stay shallow, you can make up a chart like this for a 32 percent setting on your nitrox dive computer and adjust your setpoint accordingly. Remember, on decompression its not the oxygen, it’s the nitrogen.
Thanks to Leon Scamahorn of ISC for the above enlightening, he showed me this during my Megalodon training and I just had to pass it along.
Problems for the student
An OC dive to 185 feet to a wreck with no penetration planned. Max PPO2 to be 1.5 ATA at depth/deco stops. Max Narcotic depth to be equal to 90 fsw. Gas changes at 70 fsw and 30 fsw.
Best Bottom Mix
Two deco gases
Closed circuit rebreather dive to 183 feet, two sensors do not agree, your diluent is normoxic trimix and you do a flush with that, what should your PPO2 read?
If your great grandmother gets narced anytime she dives below 70 fsw, figure her out a batch of trimix that will allow her to dive happily to 245 FSW, assuming you like your great g’ma, so no hot mixes.
Open circuit dive to 165 feet, you are dropped off by a skiff on a lovely wreck, do your bottom time and there is a swim in on a shelf for about 15 minutes at 65 feet before you reach another wall. What mix would you carry for your first deco mix to take advantage and off gas most efficiently during that 15 minute swim?
OK, I guess I have confused everyone enough for now. If this helped you, please let me know, if it confused you irretrievably, let me know that too. If I have missed anything, or you find errors in my math, call my math teacher and bitch at him.
3 Dec 2004
Last edited by Louby Lou; 3rd December 2008 at 08:43.