Boyle's Law and the Manometer
Boyle's Law
The most important thing to remember about Boyle's Law is that
it only holds
when the temperature and amount of gas are constant. A state of constant
temperature is often referred to as isothermal conditions. When these two
conditions are met, Boyle's law states that the volume V of a gas varies
inversely with its pressure P. The equation below expresses Boyle's law
mathematically:
C is a constant unique to the temperature and mass of gas involved. plots pressure versus volume for a gas that obeys Boyles law.
Figure %: Pressure versus Volume
You will get the most mileage out of another incarnation of Boyle's law:
The subscripts 1 and 2 refer to two different sets of conditions. It is easiest
to think of the above equation as a "before and after" equation.
Initially the gas has volume and pressure
V_{1} and
P_{1}. After some event,
the gas has volume and pressure
V_{2} and
P_{2}. Often you will be given three
of these variables and asked to find the fourth. You should realize that this
is a simple case of algebra. Separate the knowns and unknowns on two different
sides of the "=" sign, plug in the known values, and solve for the unknown.
The Manometer
Boyle used a manometer to discover his gas law. His manometer had an odd
"J" shape:
Figure %: A Manometer
As you can see from , there are two ends to Boyle's
manometer. One end is open to the atmosphere. The other end is sealed, but
contains gas at atmospheric pressure. Since the pressure on both ends of the
tube is the same, the level of mercury is also the same.
Next Boyle added mercury to the open end of his manometer.
The volume of the gas at the closed end of the manometer decreased, but since
gas can't get in or out of the closed end, the amount of gas does not change.
Likewise we can assume that the experiment occurs under isothermal conditions.
Boyle's law should hold, meaning that the initial volume times pressure should
equal the volume times pressure after the additional mercury was added. Let's
use the equation below on the gas at the sealed end:
The pressure of the gas before mercury is added is equal to the atmospheric
pressure, 760 mm Hg (let's assume that the experiment is run at
^{o}C
so that 1 torr = 1 mm Hg). So
P_{1} = 760 mm Hg. The volume
V_{1} is measured
to be 100 mL.
After Boyle added mercury, the volume of the gas, V_{2}, drops to 50 mL. To
find the value of P_{2}, rearrange the equation above and plug in values:
P_{2}  =  P_{1}V_{1}/V_{2} 

 =  (100 mL)(760 mm Hg)/(50 mL) 

 =  1520 mm Hg 

If you look back at , you'll notice that the difference
P_{2}  P_{1} = 760 mm Hg, and that this exactly equals the difference in mercury levels
on the two sides,
h. In fact, Boyle's manometer illustrates a truism common
to
all manometers:
h corresponds to the difference in pressure
between the two ends of the manometer.
Boyle's manometer is only one of the many kinds of manometers you'll face.
Don't be disheartened; all manometers are practically the same. Realize that
each end of a manometer can only be:
 sealed and contain a vacuum (P = 0)
 open to the atmosphere (P = P_{atm})
 open to a sample of gas with pressure P
This is the key to solving manometer problems. Once you figure out the pressure
at both ends of the manometer, you can use the difference to determine the
height
h of the liquid column, and vice versa.