Charles' Law
Gas Volumes and Heat
There is another way to make a balloon smaller other than pushing it
underwater. You can put it in the freezer. When you heat up a gas, the
molecules that the gas is made up of move faster. In our balloon example,
this increase in molecular motion causes the molecules to hit the sides of
the balloon more often, and with more force, making the balloon expand.
Cooling the gas would have the inverse effect, making the balloon smaller.
While not too important when dealing with balloons, this concept has other
applications. For example a full scuba tank, if left in the sun, will heat
up. This causes the molecules in the air in the tank to move faster.
Unlike the balloon which would expand, the tank is a rigid container that
will not expand. This increase in motion then raises the pressure inside
the tank. In fact, a full scuba tank will gain about 5-6 psi for every
degree of temperature increase.
This is one reason that full tanks should not be left in a hot trunk of a
car. A tank filled to 3000psi could easily reach 3500 psi if the
temperature of it increased substantially. There have also been several
cases where full scuba tanks, involved in boat fires, have exploded.
This is due to the weakening the metal and the increased pressure from the
heat.
If go back to Boyle's Law and read a
little more carefully we will see that Boyle's Law states: If the
temperature remains constant, the volume of a given mass of
gas is inversely proportional to the absolute pressure. Boyle did not
concern himself much with changing temperature. This was,however, the main
goal of a French scientist Jacques Charles.
Charles showed that raising the temperature of a gas would tend to increase
the volume of thegas, if its pressure remained constant. A few other laws,
like Amonton’s Law, "The pressure of a fixed amount of a gas maintained at a
constant volume is directly proportional to the gas pressure." or
P µ T, P/T = constant or P1/P2 = T1/T2 all came together to help make what
has come to be known as the General Gas Law.
T1 is the temperature at the first location, T2 is the
temperature at the second. Remember when we were dealing with the pressure,
we used the absolute pressure for our calculations. We must start at the
zero mark. As stated in the Charles' law we must also use absolute
temperature in the calculations as well. Absolute zero
(** Raising the temperature of a gas causes the molecules in the gas to move
faster. Lowering the temperature causes the molecules to slow down.
Absolute zero is the theoretical temperature where all molecular motion
stops.) is 460 degrees below Fahrenheit zero. This means we must add 460 to
our temperatures before applying them to the formula. This is known as the
Rankine scale. For example, 40 degrees Fahrenheit would be 460+ 40, or 500
degrees Rankine.
Using the formula, we can explore the second half of Charles' law. This
part of the law states that if the volume was kept constant, raising the
temperature would increase the pressure of the gas. A good example of this
can be found by an example with a scuba tank. Let us look at a scubatank
that shows it is filled to 3000 psi. We will assume the tank reads this
pressure while sitting in an air-conditioned room which is at 70 degrees
Fahrenheit. How much pressure would there be in the tank if it were left
in a trunk of a car where the temperature climbed to 140 F?
Let's put our numbers into the formula. The first thing we can do is take
the V's out of the formula. Since the volume of the scuba tank will remain
the same, we can cancel the V's and change our formula to:
P1 / T1 = P2 / T2
Our starting pressure, P1, at first appears to be 3000 psi. We must
remember to use absolute numbers though, so to obtain absolute pressure, we
add in atmospheric pressure of 14.7. P1 =3000 + 14.7. T1 would be our
starting temperature in degrees Rankine. T1 = 460 + 70. And T2 would be
460 + 140. Let's now look at our formula with its numbers in place:
adding up our numbers we get:
3014.7/530 = P2/600
We then multiply both sides of the equation by 600 to get the P2 on one side
by itself.
(600 x 3014.7) / 530 = P2 or P2 = 3412.8
It is important to note that 3412.8 is the absolute pressure at the second
location. If we were asked "What is the gauge pressure at the second
location?" we would subtract 14.7 from 3412.8 for an answer of 3398.1.
This is a substantial increase in pressure. It turns out that a full 80
cubic foot scuba tank will have a pressure change of approximately 5-6 psi
for every degree oftemperature change.
Since Charles' law also deals with changing the volume of a gas with a
change in temperature, we can use the General Gas Law formula to determine
the answer to the following question. If a balloon is inflated to one
cubic foot at the surface, with air that is 85 degrees, how large would the
balloon be if taken into 50 degree sea water to a depth of 40 feet?
Solving for our Ps, Vs, and Ts, we get:
P1 = 14.7 V1 = 1 T1 = 85 + 460
P2 = [40 x .445] + 14.7 T2 = 460 + 50
Using these numbers, we can solve for V2.
(14.7 x 1) / 545 = ({[40 x .445] + 14.7} x V2) / 510
solving further we get:
14.7 / 545 = (32.5 x V2) / 510
Since we want to get the V2 by itself on one side of the equation, we will
multiply both sides by 510 over 32.5. This will leave the V2 alone on the
second side:
(510 x 14.7) / (32.5 x 545) = V2
using a calculator we get:
7497 / 17712.5 = V2
or: V2 = 0.4232604093155
Thus the volume of our balloon at its second location would be about .42
cubic feet.
Unlimited Sunshine & Blue Water,
Scubabuf
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