E-13
GAS LAWS
NGSS: HS: CC Patterns. SEP Using Mathematical and Computational Thinking.
In E-11, Boyle's Law, you investigated the relationship between the volume and pressure of a fixed
amount of air at a constant temperature. In E-12, Charles' Law, you investigated the relationship
between the temperature and volume of a fixed amount of air at a constant pressure. From these
experiments, one can conclude that to specify the state of a gas, one must know the values of four
variables: amount, volume, temperature, and pressure. All gases respond to changes in these variables in
a predictable, quantitative manner. To solve gas laws problems it is necessary to review individual
relationships first, and combine these relationships into a single equation. In addition, it will become
apparent that a special method is required to deal with gases collected over water.
PROPERTIES OF GASES
Temperature: the measure of the average kinetic energy of particles

measured in Kelvin when using gas law calculations

TK = ToC + 273
Volume: the amount of space the gas occupies; it is controlled by the size of the container

measured in L, mL or any cubic units in the SI system

1 L = 1 dm3 = 1000 cm3 = 1000 mL
Pressure: the amount of force exerted on a given area

measured in Pascals (Pa) in the SI system

some relationships between the various measures of gas pressure are as follows:
101300 Pa = 101.3 kPa =1 atm = 760. mmHg = 760. torr = 14.7 psi = 1 bar
Amount: the amount of gas is measured in moles, which will learn more about later this semester

assume that the amount of gas is unchanged in these gas law problems
CONVERSIONS BETWEEN UNITS:
1.
Water changes to steam at 100.00 oC. What temperature is this in Kelvin?
2.
A gas liquefies at -178 oC. What is this temperature in Kelvin?
For Questions 3-6, remember from E-1 that a conversion factor is made up of 2 equal quantities. When
multiplying by the conversion factor, the numerator should contain the desired units, and the denominator
contains the units matching those in the original measurement.
3.
A gallon jug of milk holds 3.78 L. How many mL is this?
4.
A balloon contains 347 cm3 of air. How many liters is this?
5.
An announcer on a radio station in Montreal reports the atmospheric pressure to be 99.6 kPa. What
is the pressure in atmospheres?
6.
Express a pressure of 729 mm Hg in atmospheres.
1
KINETIC THEORY OF MATTER:
The kinetic theory of matter states that all particles (atoms and molecules) are in constant motion at
all times. There are 3 main types of motion – vibrational, rotational, and translational. Solids are
substances that are fixed in volume and shape – their “fixed” shape is directly related to the fact that
they only exhibit vibrational motion (they vibrate, but do not change position). Liquids are more
fluid, and while their volume is fixed, their shape is not. Liquids also have rotational motion
(particles move around one another, but do not travel a significant distance). Since gases have no
fixed shape or volume, they demonstrate all three types of motion – translational motion means that
the particles can move to another location freely. For example, gases that make up our atmosphere
move freely from one location to another, which explains air pollution or ozone depletion in areas
that are not large polluters.
Since temperature is directly related to kinetic energy, the greater the temperature of a substance
the faster the particles move. When solids are heated enough, they begin to rotate more and change
into liquids (melting). Liquids can be heated enough so that the particles begin to move more
freely, and eventually they change to a gas (boiling).
IDEAL GASES:
An ideal gas is characterized by 4 main conditions:
1. Gases consist of small particles that have mass (although very small).
2. The volume of the particles in negligible compared to their containers.
3. Their collisions are elastic – they can collide, but do not lose any energy in the process.
4. Gases exert pressure due to their collisions with the walls of their containers.
5. The speed at which the particles move is directly related to their temperature (kinetic energy)
BOYLE'S LAW – the inverse relationship between pressure and volume
At a constant temperature, as the pressure on a fixed amount of air increases, the volume decreases.
P is inversely proportional to volume, or, mathematically, V = constant/ P. If the same gas sample
is compared at two different conditions, the algebraic equation: P1V1 = P2V2
PROBLEM SOLVING USING THE NON-ALGEBRA METHOD
Step 1: In order to predict an outcome, it is necessary to know what existed before and what is
desired after. When this is applied to solving gas law problems, it means identifying what is known
and unknown. Since all of our problems will include volume, pressure, and temperature, it will be
very helpful if the following format is used when setting-up the problem:
BEFORE
V =
P =
T =
AFTER
V =
P =
T =
** Once you have recorded the known information, make sure the units match. If not, convert so
that units will be the same.
Step 2: Identify the relationship involved. Is it V-P; T-V; or T-P?
Step 3: Set up the problem so that the result will conform to the relationship.
Step 4: Solve and check your answer to see if it agrees with the relationship.
2
SAMPLE PROBLEM: At room temperature, 2.3 L of helium (He) gas in a flexible container at a
pressure of 2.1 atm has its volume changed to 3500 mL. What is the new pressure of the gas?
Step 1:
BEFORE
V = 2.3 L
P = 2.1 atm
T = room temp.
AFTER
V = 3500 mL = 3.5 L
P = ?
T = room temp.
Step 2: Since temperature is the same before and after, the relationship only involves the variables
of V, volume, and P, pressure, which is Boyle's Law. From the laboratory investigation, you know
that as the pressure increased on the syringe, the volume of trapped air decreased. In this problem,
however, the volume is increasing, so, the pressure must be decreasing. We want to set up the
problem so that the pressure after will be less than the starting pressure.
Step 3: Which volume fraction is less than 1, so that when multiplied by the pressure of 2.1 atm it
will result in a smaller pressure 2.3 L or 3.5 L ?
Step 4:
2.1 atm
= 1.4 atm (remember: use significant figures)
The result, 1.4 atm, is indeed less than the 2.1 atm we started with, so the solution does agree with
what was observed in the E-7 lab - if the volume of a gas is increasing then the pressure must be
decreasing!
ALGEBRA METHOD (for the same sample problem):
Step 1 is the same, but Step 2 is to substitute the values into the formula P 1V1 = P2V2, and then
solve.
P1V1 =
P2V2
(2.1 atm) ∙ (2.3 L) = (P2) ∙ (3.5 L)
4.8 atm∙L
= (P2) ∙ (3.5 L)
1.4 atm
= P2
SOLVE THE FOLLOWING BOYLE'S LAW PROBLEMS.
1.
A sample of gas occupies a volume of 2.00 L at 27 oC and 1.00 atm pressure. What volume
will this sample occupy at the same temperature, but at a pressure of 2.75 atm?
2.
A balloon is inflated to a volume of 12.6 L on a day when the atmospheric pressure is 674
mm Hg. The next day, as a storm front passes, the atmospheric pressure drops to 651 mm
Hg. Assuming the temperature remains constant, what is the new volume of the balloon?
3.
A sample of carbon dioxide (CO2) gas occupies a volume of 5.75 L at 0.890 atm. If the
temperature remains constant, calculate the volume when the pressure is increased to 1.25
atm.
4.
The pressure of a 5.00-L sample of xenon (Xe) gas is 745 mm Hg. If the temperature
remains constant, calculate the new pressure when the volume becomes 8.75 L.
3
CHARLES' LAW: the direct relationship between temperature and volume
At a constant pressure, as the temperature of a fixed amount of air increases, its volume increases. T is
directly proportional to volume, or, mathematically, T = constant x V. Algebraically speaking,
T - V problems are solved in the same manner as Boyle's Law problems, except all temperatures must be
converted to the absolute temperature scale, Kelvin (K)
SAMPLE PROBLEM: On a cold day, a person takes a breath of 450. mL of air at 756 mm Hg and -10. oC.
What is the volume of this air in the lungs at 37oC and 756 mm Hg?
Step 1:
BEFORE
V = 450. mL
P = 756 mm Hg
T = -10. oC  263 K
AFTER
V = ?
P = 756 mm Hg
T = 37 oC  300. K
Step 2: Since the pressure is the same before and after, the relationship only involves the variables of V,
volume, and T, temperature, which is Charles' Law. From the laboratory investigation, you know that as the
trapped volume of air was heated up, the volume increased. That's what's happening in this problem, so the
volume at the end should be larger than the volume at the beginning. However, we must first change the
temperatures to K. -10. oC becomes 263 K, and 27 oC becomes 300. K.
Step 3: Which temperature fraction is greater than 1, so that when multiplied by the volume of 450 mL it
will result in a larger volume 263 K or 300. K?
Step 4:
450 mL
= 513 mL (use significant figures!)
The result, 513 mL, is indeed greater than the 450 mL we started with, so the solution does agree with what
was observed in the E-8 lab, as the temperature increases the volume increases.
ALGEBRA METHOD (for the same sample problem):
(263 K) ∙ V2 = (300. K) ∙ (450. mL)
(263 K) ∙ V2 = 135,000 K∙mL
V2 = 513 mL
SOLVE THE FOLLOWING CHARLES' LAW PROBLEMS.
1.
A balloon is filled with helium (He) gas. Its volume is 5.0 L at 27 oC. What will its volume be at
2.
-73 oC assuming no change in pressure?
3.
785 L of oxygen (O2) can be obtained from a tank at 1.00 atm and 21oC. What would the volume of
oxygen be at 28 oC?
4.
A sample of gas occupies a volume of 345 mL at a pressure of 758 torr and a temperature of 22 oC.
If the volume increases to 480. mL, what is the new temperature, pressure remaining constant?
5.
Argon (Ar) gas is contained in a steel cylinder with a volume of 9.76 L and a temperature of 24 oC.
To reduce the volume to half its original size, to what temperature will the tank have to be cooled?
4
GAY-LUSSAC’S LAW: the direct relationship between pressure and temperature.
At a constant volume, as the temperature of a fixed amount of air increases, its pressure increases. T is
directly proportional to pressure, or, mathematically, T = constant x P. These problems are solved in exactly
the same manner as the other two sets. The algebraic formula is
SAMPLE PROBLEM: An automobile tire is inflated to 30. psi at 16 oC. After driving, the tire and the air
inside are heated to 85 oC. What is the pressure after driving?
Step 1:
BEFORE
V = constant
P = 30. psi
T = 16 oC  289 K
AFTER
V = constant
P = ?
T = 85 oC  358 K
Step 2: Since the volume is the same before and after, the relationship only involves the variables of T,
temperature, and P, pressure. We want to set the problem up so that the pressure after will be greater than
the pressure before. 16 oC becomes 289 K; 85 oC becomes 358 K.
Step 3: Which temperature fraction is greater than 1, so that when multiplied by 30. psi it will result in a
greater pressure 289 K or 358 K ?
Step 4:
30. psi
= 37 psi (use significant figures!)
The result, 37 psi, is greater than the 30. psi we started with, so the solution does agree with what is known if the temperature of a fixed amount of air is increasing the pressure will also increase.
ALGEBRA METHOD (for the same sample problem):
(289 K) ∙ P2 = (358 K)∙(30. psi)
(289 K) ∙ P2 = 11,000 K∙ psi
P2 = 38 psi
* Due to rounding, the answer is slightly
different, but since the last significant
figure is “the estimated digit”, this answer
is also correct.
SOLVE THE FOLLOWING GAY-LUSSAC’S PROBLEMS.
1.
At 127 oC the pressure in an autoclave used to sterilize surgical instruments is 2.0 atm. What will
the pressure be at 27 oC?
2.
Normal tire inflation is 32 psi. If a tire was filled when the temperature was 44 oC, what will the
pressure be when the temperature falls to -40. oC?
3.
If the pressure inside a hot air balloon is 850. torr at a temperature of 22 oC, what will the new
temperature be if the pressure changes to 105 kPa?
4.
A steel cylinder contains a gas sample at a pressure of 690. mm Hg and a temperature of 0. oC. To
what temperature must the tank be warmed to change the pressure to 104,000 Pa?
5
THE COMBINED GAS LAWS
All of the above relationships can be combined into one simple equation. Often these problems deal with
STP - standard temperature and standard pressure. Standard temperature is 0 oC; standard pressure is 760.
mm Hg. Remember: all temperatures must be converted to K! The combined gas law formula is
SAMPLE PROBLEM: A quantity of hydrogen (H2) gas has a volume of 5.73 L at STP. What would the
pressure be if the volume is changed to 10.3 L at 30. oC?
Step 1:
BEFORE
V = 5.73 L
P = 760 mm Hg
T = 0 oC
AFTER
V = 10.3 L
P = ?
T = 30. oC
Step 2: Each variable must be independently compared to the unknown. Volume is increasing, so pressure
should be getting smaller, so the volume fraction has to be less than 1. The temperature is rising, so pressure
increases; so that fraction has to be greater than 1.
Step 3: Both fractions must be greater than 1!
Step 4:
760. mm Hg
=
469 mm Hg
The result, mm Hg, 469 is much smaller than 760 mm Hg, so this tells us that the volume change had a
much greater effect on the pressure, which should make sense mathematically. The volume changed much
more than the temperature did!
ALGEBRA METHOD (for the same sample problem):
1,320,000 mmHg∙L∙K = P2 ∙ 2,810 K∙L
470. mmHg
= P2
* Again this is slightly
different due to rounding.
SOLVE THE FOLLOWING COMBINED GAS LAWS PROBLEMS.
1.
A sample of gas occupies a volume of 4.00 L at 27 oC and 780. mm Hg. What will its volume be
when changed to STP?
2.
A sample of nitrogen (N2) gas collected at 18 oC and 12.6 psi has a volume of 2.67 L. What is the
volume at STP?
3.
A balloon contains hydrogen (H2) gas with a volume of 1.05 L at 20. oC and 755 mm Hg. What will
the new volume be if the pressure becomes 625 torr and the temperature drops to -10. oC?
4.
A weather balloon was filled at ground level with helium (He) gas to a volume of 45 L at a pressure
of 0.97 atm at 21 oC. After rising to an altitude of three hundred meters, the volume had changed to
41 L at a temperature of 10. oC. What is the pressure of the helium inside the balloon?
6
DALTON'S LAW OF PARTIAL PRESSURE
The total pressure of a gas mixture is the sum of the partial pressures of the components of the
mixture. For a mixture of two gases A and B,
Ptotal =
PA + PB
An illustration of this is standard pressure, 760 mm Hg. Our atmosphere is comprised primarily of
nitrogen (N2) gas - 80 % - and oxygen (O2) gas - 20%. If we multiply standard pressure by the
percent composition of each of the 2 gases in the atmosphere, we find that nitrogen's partial
pressure is 608 mm Hg and oxygen's partial pressure is 152 mm Hg.
The implication of this to gas laws involves the way in which most gases are collected - over water!
This means the total measured pressure is made up of the partial pressure of the gas and the partial
pressure of the water vapor. This is corrected by using a water vapor table, which provides a value
for the mm Hg due to the presence of water vapor at a specific temperature.
SAMPLE PROBLEM: A student prepares a sample of hydrogen (H2) gas by electrolyzing water at
25 oC. She collects 152 mL of the gas at a total pressure of 758 torr. What is the volume of the
collected gas at STP?
Step 1:
BEFORE
V
= 152 mL
Ptotal = 758 torr
T
= 25 oC
AFTER
V = ?
P = 760. torr
T = 0 oC
Step 2: Find the corrected pressure. From the table provided, at 25 oC, 23.8 mm Hg are due to the
water vapor pressure. Subtract 23.8 mm Hg from 758 torr (remember they're equal!). The pressure
of just the gas = 734 torr. Use this pressure in all further calculations!
Step 3: Change the temperatures to K. 25 oC becomes 298 K; 0 oC becomes 273 K.
Step 4:
152 mL
=
134 mL
SOLVE THE FOLLOWING DALTON'S LAW PROBLEMS.
1.
A sample of oxygen gas is collected over water at 25 oC. The wet gas occupies a volume of 12.83 L
at a total pressure of 745 mm Hg. If all of the water is removed, what volume will the dry oxygen
occupy at a pressure of 762 mm Hg and a temperature of 50. oC?
2.
A student collects 355 mL of oxygen saturated with water vapor at 27 oC. The mixture exerts a total
pressure of 775 torr. What will the volume of dry gas be at standard temperature and pressure?
3.
Hydrogen gas is collected over water at 24 oC and a total pressure of 104.5 kPa and a volume of 4.05
L. What is the dry volume of gas at STP?
4.
A sample of methane gas is bubbled through water at 20. oC and 99.8 kPa, and it has a volume of
1.00 L. What is the volume of the dry methane gas at 28 oC and 775 mmHg?
7
Extra E-13 Gas Laws Problems
BOYLE’S
1. A sample of carbon dioxide gas occupies a volume of 8.75 L at 0.940 atm. If the temperature
remains constant, calculate the volume when the pressure is changed to 1.10 atm?
2. At STP a balloon has a volume of 1.75 L. When the balloon is moved to a higher elevation, the gas
expands to 1900 mL. What is the pressure of the air at the higher elevation?
3. What will the new volume be if a 4.25 mL sample of gas collected at a pressure of 0.75 atm is
subjected to a new pressure of 1.25 atm?
CHARLES’S
4.
What is the final volume when a 1.00 L gas sample, originally at STP, is heated to 25 oC at constant
pressure?
5.
If 255 mL of H2 is produced, measured at 22oC and 740. mmHg, what will the volume be at STP?
6.
A 50. mL sample of gas is originally at a temperature of 50 oC. To what temperature must it be
heated to triple its volume, pressure remaining constant?
GAY-LUSSAC’S LAW
7.
An aerosol spray can contains gas under pressure of 2.00 atm at 27 oC. The can itself can only
withstand a pressure of 3.05 atm. To what temperature can the can be heated before it explodes?
8.
A bottle of pop is sealed with a pressure of 2.0 atm at 25oC. When the bottle (and dissolved gas) is
heated to 100 oC the bottle explodes. What is the pressure of the gas in the bottle when it explodes?
9.
An “empty” pop bottle is tossed into the backseat of your car (22oC & 760.0 mmHg). Overnight the
temperature of the car decreases to 0 oC. What is the pressure in the bottle at the lower temperature?
COMBINED GAS LAWS
10.
A 2.90 mL air bubble forms in a deep lake where the temperature is 8 oC at a total pressure of 1850.
torr. The bubble rises to a depth where the temperature and pressure are 15 oC and 900. torr,
respectively. Assuming the amount of air in the bubble has not changed, calculate its new volume.
11.
A gas with a pressure of 605 torr at 27 oC occupies a volume of 450. mL. What will the volume be at
STP?
12.
A quantity of gas has a volume of 1.23 L at STP. What would the pressure be if the volume is
changed to 862 mL at 30. oC?
DALTON’S LAW
13.
Suppose a 2.00 L sample of hydrogen gas is collected over water at 25 oC and a total pressure of 756
mm Hg. What is the volume of dry hydrogen gas at STP?
14.
If a 5.00 L sample of gas is collected over water at 24 oC and a pressure of 805 torr. What will the
volume be at STP?
15.
If a 4.00 L sample of gas, stored at STP, is bubbled through water at 22 oC and a pressure of 955
mmHg, what is the new volume?
8