# Calculating Bubble Diameter at Surface: A Diver's Dilemma

• talaroue
In summary, the diver's bubble has an initial pressure of 3.38 atm and a final pressure of 393044 Pa. The bubble's diameter just as it reaches the surface of the lake, where the water temperature is 25°C, is 2.19 cm.
talaroue

## Homework Statement

A diver 38.8 m deep in 14°C fresh water exhales a 2.19 cm diameter bubble. What is the bubble's diameter just as it reaches the surface of the lake, where the water temperature is 25°C?

PV=nRT

## The Attempt at a Solution

I don't know where to start on this problem at all I can find the intial pressure of the bubble but I don't see how that matters?

Helium balloons expand as they rise in the atmosphere, because the air gets thinner (and the air pressure reduces as a result). The same thing happens here. The pressure falls with rising height. So you can calculate the pressure at the two depths and see how it changes. However, the temperature also changes. As you know, from the ideal gas law, both of these things affect the volume.

I didn't think about that way. Let me work through it and then post what I got, thanks

What would be the initial pressure be? I know the final would be sea level of 101300 Pa.

I found Pi should be 3.38 atm x 101300 Pa= 393044 Pa because since he is 38.8 meters below sea level the pressure is 3.38 atm

Yeah, I guess the pressure at the surface would be be atmospheric pressure. Do you know how the pressure varies with depth in a fluid?

i approximated it that ever 10 meters below sea level= 1 atm is there a way to solve it for another way?

talaroue said:
because since he is 38.8 meters below sea level the pressure is 3.38 atm

How do you figure that?

talaroue said:
i approximated it that ever 10 meters below sea level= 1 atm is there a way to solve it for another way?

Interesting. Where did you find this approximation? It turns out to be more or less correct (for water). However, it might be helpful for you to know where it came from. Have you seen this before?

http://hyperphysics.phy-astr.gsu.edu/hbase/pflu.html#fp

Now you know the principle behind it, so you can figure it out precisely (it's not exactly 1 atm per 10 m) and you can also do this for any fluid.

I am still wrong here is my work...

By the way, 38.8 m / 10 m = 3.88, not 3.38 as you have written.

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PiVi/Ti=PfVf/Tf solved for Vf

Vf=PiViTf/TiPf then plugged in 1/6d^3 for the volumes and then found out that...

df^3=PiTfdi^3/TiPf...(380240*298*.0219^3/(285*101300))^1/3 and then got .125 m

The volume of a sphere is given by (4/3)πr3

Even converting to diameter, that turns into (π/6)d3, which is not the same as what you have written there.

I haven't checked your other work closely yet.

Oh right, but it cancels either way

that is the work i did before i switched the pressure so the anwser i really got is not .0344 m( either way that's wrong too) but .1235 m

The initial temperature is 287 K, not 285 K.

I should have an answer soon, and then we can compare.

ok thank you

I'm not going to reveal my answer yet ;)

Here's what I have:

$$V_f = \frac{P_i}{P_f}\frac{T_f}{T_i}V_i$$

$$\frac{\pi}{6}d_f^3 = 3.88\frac{298}{287} \frac{\pi}{6}d_i^3$$

$$d_f = \sqrt[3]{3.88\frac{298}{287}d_i^3}$$​

Is it any different from your work so far?

EDIT: I see you have exactly the same thing as I do. So, do you get the right answer with the corrected numbers?

nope it is the same except i changed my pressures to Pa instead of leaving it in atms like you.

talaroue said:
nope it is the same except i changed my pressures to Pa instead of leaving it in atms like you.

Yeah that's a good point. If I don't use the approximation, I get that the ratio of pressures should be closer to 3.76 rather than 3.88.

Also, your initial pressure seems a bit off.

And doesn't 1 atm = 101 325 Pa?

What is the answer *supposed* to be?

I don't know it is online. It just told me my anwser is wrong.

I am thinking that there is something wrong with his C++ language in the program.

Edit: I think I was just being stupid. $\rho gh$ is the pressure CHANGE (Pfinal - Pinitial). Therefore, the pressure at the original depth is:

Pinitial + $\rho gh$

I knew that.

Another way to think about it using the approximation is that if it increases by 1 atm every 10 m, then the pressure at 38.8 m is given by

1 atm + (38.8 m)*(1 atm/10 m) = 1 atm + 3.88 atm

emphasis on the PLUS sign, LOL. Duh!

so instead of Pi=density*g*h it should be Pi=Pf+ghdensity?

talaroue said:
so instead of Pi=density*g*h it should be Pi=Pf+ghdensity?

Yeah. I guess our definitions of "initial" and "final" were reversed because I was going downward from the surface (in the direction of increasing pressure), and you were going upward (like the bubble).

For absolute clarity:

(pressure at depth h) = (pressure at surface) + density*g*h

OOOOOOOOh makes since now

I got .0373 m but that is still wrong!

I get the same answer.

The initial diameter is given in cm. What units is your program expecting the final answer to be in?

It doesn't matter as long as I put the corresponding units next too it. I figured out the problem even though he had posted he wanted the diameter of the bubble he really only wanted the radius. So our work was right just he was wrong. Thank you for all your help!

You're welcome!

Boy, if I were you, I would be annoyed with the teacher right now!

I know I am because this is like the third or fourth time it as happened! I have posted question after question on here thinking i am doing something wrong and then to find out it was an error on his part. It gets frustrating, but everyone makes mistakes.

## What is the purpose of calculating bubble diameter at the surface for a diver?

The purpose of calculating bubble diameter at the surface for a diver is to determine the potential risk of decompression sickness, also known as "the bends." By understanding the size and number of bubbles in a diver's body, we can estimate the likelihood of nitrogen bubbles forming in the bloodstream and causing symptoms.

## What factors affect the calculation of bubble diameter at the surface?

Factors that can affect the calculation of bubble diameter at the surface include the depth and duration of the dive, the diver's breathing rate, the type of gas mixture used, and the diver's physical condition. These factors can impact the amount of nitrogen absorbed by the body and the rate at which it is released during ascent.

## How is bubble diameter at the surface calculated?

The most common method for calculating bubble diameter at the surface is through the use of mathematical models, such as the Haldane or Bühlmann decompression algorithms. These models take into account the factors mentioned above and use equations to estimate the size and number of bubbles that may form during a dive.

## What are the potential risks of not accurately calculating bubble diameter at the surface?

If bubble diameter at the surface is not accurately calculated, there is a risk that a diver may experience decompression sickness. This can range from mild symptoms, such as joint pain and skin rashes, to more serious and potentially life-threatening conditions, such as neurological damage or arterial gas embolism.

## Are there any precautions that a diver can take to reduce the risk of decompression sickness?

Yes, there are several precautions that a diver can take to reduce the risk of decompression sickness. These include following safe diving practices, such as ascending slowly and making decompression stops, monitoring their air consumption and depth, and properly maintaining and using their equipment. Additionally, divers can also consider using nitrox or other gas mixtures with lower levels of nitrogen to reduce the amount of gas absorbed by their body during a dive.

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