Physics Applied to Diving 1.- Boyle's Law
Boyle's law shows the relationship between the pressure of a gas and
its volume in the Ideal Gas Model. This law gives reason to a type of diving accidents call
"barotraumas".
The law enunciates that:
"For a sample of gas at constant temperature, the product of the pressure
and the volume is constant" [1]
The mathematical way to
show this law is:
V*P=k
Where V is the volume of
the gas, P is the pressure of the gas and k is a constant value.
Pressure and depth
When you're diving,
there's a column of water over you exercising a certain amount of pressure
called hydrostatic pressure. In consequence, the deeper you are the bigger the
column and the more pressure you will experience. If you want to know the total pressure at
a certain depth, the only thing you have to do is to add the hydrostatic pressure
and the atmospheric pressure.
P
= Ph + Pa
Where P is the total
pressure, Ph is the hydrostatic pressure and Pa is the
atmospheric pressure. Now to transform a
certain hydrostatic pressure to it’s value in depth (and viceversa) we commonly
say that every 10 meter of column height is equal to 1 atmosphere (atm) of hydrostatic
pressure regarding of whether it is salt or fresh water.
Example 1a.- Depth to pressure
What is the pressure a
diver experience when being at a depth of 20 meters at sea level.
Depth =D = 20 m
Because we have to "transform" meters, we have to multiply it by a fraction with meters in the denominator so:
1 atm = 10 m
(1 atm / 10 m) = (10 m /
10 m)
(1 atm / 10 m) = 1
Next we take the value
of our depth (20 m) and multiply it by our fraction.
20m * (1 atm / 10 m) = 2
atm
So a depth of 20 meters
is equal to a hydrostatic pressure of 2 atm. Now, if we want the total
pressure we have to add the atmospheric pressure. In this case we are at sea
level so the atmospheric pressure is 1 atm.
P = Ph + Pa
P = 2 atm + 1 atm = 3
atm
P = 3 atm
Example 1b .- Hydrostatic
pressure to depth
If a diver experience a hydrostatic
pressure of 3.7 atm, at what depth is s/he?
Hydrostatic pressure = Ph = 3.7
Because we have to transform
atmospheres, we have to multiply it by a fraction with atmospheres in the denominator so:
1 atm = 10 m
(1 atm / 1 atm) = (10 m
/ 1 atm)
1 = (10 m / 1 atm)
Next we take the value
of our pressure (3.7 atm) and multiply it by our fraction.
3.7 atm * (10 m / 1 atm)
= 37 m
So the hydrostatic pressure
of 3.7 atm is equal to a depth of 37 m
Volume and pressure
In order to calculate the volume of a gas at a certain pressure only using this law, you need a reference state. Imagine you have a state "1" where the pressure is P1 and the volume is V1, then you change
the pressure to a known value we call P2. This new state where the pressure is P2 will have a different volume (V2) than V1. To calculate
this, we must do the procedure shown next.
V1*P1
= k
V2*P2
= k
Knowing that k is a
constant, meaning it doesn't change value, it has the same value in both cases which gives us:
V1*P1
= k = V2*P2
V1*P1
= V2*P2
The equation shown
before is useful to calculate any volume or pressure whenever we have a
reference state. In this case we want to calculate the V2,
so we do the next clearence.
V1*P1 = V2*P2
(V1*P1)/P1 = (V2*P2)/P1
V1 = (V2*P2)/P1
Example 2a .- What's the final depth?
You are outside the
water holding a 3 L balloon. Next you jump into the water and start descending
until the balloon is a third it's initial size (1 L). What is your final depth?
What we know:
P1 = 1 atm
V1 = 3 L
V2 = 1 L
P2 =?
We must find the pressure
to find the depth after so:
V1*P1 = V2*P2
P2 = (V1*P1)/V2
P2
= (3 L * 1 atm)/1 L = 3 atm
Know to
find the depth:
P2 = Ph2 + Pa
P2 - Pa = Ph2 + Pa
- Pa
Ph2 = P2 - Pa
Ph2 = 3 atm - 1 atm = 2 atm
And we
know that every 10 meters is equal to 1 atm so:
Depth = 2 atm (10 m/ 1 atm) = 20 m
Example 2b .- What is the final volume?
You have a balloon at a
depth of 45 with a volume of 1 L. Then you go up to depth of 10 meters, what is
the size of the balloon now?
What we know
Initial depth = D1
= 45 m
V1 = 1 L
Final Depth = D2 = 10 m
First, we must know the initial
total pressure so:
Ph1 = D1(1 atm / 10 m)
Ph1 = 45 m (1
atm / 10 m) = 4.5 atm
P1 = Ph1
+ Pa
P1 = 4.5 atm
+ 1 atm = 5.5 atm
Then the final total
pressure
Ph2 = D2(1 atm / 10 m)
Ph2 = 10 m (1
atm / 10 m) = 1 atm
P2 = Ph2
+ Pa
P2 = 1 atm +
1 atm = 2 atm
So, to finally get the
volume (V2) we are looking for:
V1*P1
= V2*P2
V2 = (V1*P1)/P2
V2 = (1L *
5.5 atm) / 2 atm
V2 = 2.75 L
[1] J. Giordano Nicholas, College Physics: Reasoning and Relationships, Brooks/Cole, 2009, page 471
Next see Gay-Lussac's Law
Next see Gay-Lussac's Law
March 25, 2020
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