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 3000 psi 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 the gas, if its pressure remained constant. He also showed that if the volume was kept constant, raising the temperature would increase the pressure of the gas.

It was shown that the amount of change in pressure, or volume of a quantity of gas is directly proportional to the absolute temperature change.

This added the variable T to Boyle's law, and the formula is called the General Gas Law:

        
       P1 x V1    P2 x V2
       _______ = ________
          T1        T2

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 scuba tank 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:

             3000 + 14.7        P2      
            _____________ = _________                  
              460 + 70      460 + 140  

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 of temperature 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.



Dalton's Law To continue with your gas theory knowledge.

Java Pressure / Volume Calculator

Uses Boyle's Law to calculate volume changes with depth.