Physics Applied to Diving 2.- Gay-Lussac's Law
Gay-Lussac's law shows the relationship between the pressure of a gas
and its temperature in the Ideal Gas Model. This law gives reason to why you shouldn't leave gas tanks
under the sun beam and why the tanks fill warm after you fill them.
The law enunciates that:
"For a sample of gas held in a container with constant volume, changes in
pressure are proportional to changes in temperature." [1]
The mathematical way to
show this law is:
P=T*k
Where P is the pressure
of the gas, T is the ABSOLUTE temperature of the gas and k is a constant value.
This relationship can be understood quite simple from a molecular point of view
by describing pressure and temperature for a gaseous system.
Pressure of a gas inside
a container
The pressure that a certain gas exercises on the walls of a
container is proportional to frequency in which the gas molecules bounce with
the walls.
Temperature and kinetic
energy
When we are speaking of
temperature in a gaseous system, we are mainly referring to the kinetic energy
of the molecules. This is because the molecules are mainly "free" with
no bonds between them.
Analyzing the phenomena
of pressure and temperature we can determine that the relationship between these
two parameters comes from the molecules movement inside the system. What's
happening is that the molecules move at a certain speed showing a certain
temperature and, in consequence, bouncing with the walls showing a certain
pressure. So, if we increase the temperature, the molecules will move faster,
and they will bounce more often with the walls of the container, increasing the
pressure.
Calculations
In order to calculate the pressure of a gas at a certain temperature only using this law, you need a reference state. Imagine you have a state "1"
where the pressure is P1 and the
temperature is T1, then you
change the temperature to a known value we call T2.
This new state where the pressure is T2
will have a different unknown pressure value that we call P2. To calculate the pressure P2 , we must do the procedure shown
next.
P1/T1
= k
P2/T2
= k
Knowing that k is a
constant, meaning it doesn't change value, it has the same value in both cases
which gives us:
P1/T1 = k = P2/T2
P1/T1 = P2/T2
The equation shown above
is useful to calculate any temperature or pressure whenever we have a reference
state. Next we show an example.
The equation shown above
is useful to calculate P1, T1, P2 or T2 with the proper clearance whenever we have 3 of
the 4 variables. In this case because we want to calculate T2 we do the next clearance.
P1/T1 = P2/T2
(P1*T2)/T1 = (P2*T2)/T2
P2 = (P1*T2)/T1
Example 1.- Warm tank
Imagine you are at the
beach in Chile and you decide to go SCUBA diving, so you go to a store and pick
up a tank. This tank feels quite warm to the touch and the store manager tells
you that's normal. Next, you check the pressure inside the tank and it's 3000
psi. You get a little bit worried because the water in the ocean is quite cold
and you know the pressure in the tank will drop. In consequence you decide to
take temperatures of the tank (36.7 °C) and the ocean (10.4 °C). What will be the
pressure of the tank when it gets to water temperature?
What we know:
Tank initial temperature
= T1 = 36.7 °C
Ocean temperature = T2
= 10.4 °C
Tank initial pressure =
P1 = 3000 psi
The first thing we must
do is to transform the temperature values to absolute values. I like to use
Kelvin units for this where the only thig you must do is to add 273.15 to your
Celsius data, so:
T1 = 36.7
+ 273.15 = 309.85 K
T2 = 10.4 +
273.15 = 285.55 K
Then to find the
Pressure we are looking for we must do some clearance, so:
P1/T1 = P2/T2
(T2*P1)/T1 = (T2*P2)/T2
(T2*P1)/T1 = P2
P2 = (285.55 K * 3000 psi) /
309.85 K = 2764.72 psi
In conclusion, when you
get into the water the pressure in your tank will drop all the way to 2764.72 psi, losing more than 200 psi.
[1] J. Giordano Nicholas, College Physics:
Reasoning and Relationships, Brooks/Cole, 2009, page 471
Next see Charles's Law
Next see Charles's Law
March 26, 2020
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