17. The Gas Laws
The earth's atmosphere is a complex mixture of several "gases", either atomic or molecular in nature. Air consists primarily of N2 (78%) and O2 (21%), with small amounts of several other substances, including Ar (0.9%).
The gaseous form of substances that are solids or liquids under normal conditions are often called vapors.
Gases differ from solids and liquids in several significant ways;
Among the most easily measured properties of a gas are its pressure, temperature, and volume.
- A gas expands spontaneously to fill its container. The volume of a gas equals the volume of the container in which it is held.
- A gas is highly compressible. When pressure is applied to a gas, its volume readily decreases.
- Gases form homogeneous mixtures with each other regardless of the identities or relative proportions of the component gases.
- Compared to solids and liquids, the molecules of gases are relatively far apart. In air, the molecules take up only about 0.1% of the total volume - compared to the individual molecules of a liquid that occupy about 70% of the total space.
Gases exert a pressure, or a force, on any surface with which they are in contact. Pressure (P) is the force (F) that acts on a given area (A).
P = F / A
Even though the kinetic energy of individual gas molecules in the atmosphere override the gravitational acceleration of gravity, the atmosphere as a whole presses down on Earth's surface, creating an atmospheric pressure.
The force exerted by the column of air represented by the diagram on the right can be calculated as:
F = ma = (104 kg) (9.8 m/s2) = 1 X 105 N
This force can be converted to pressure by:
P = F / A = 1 X 105 N / 1 m2 =
The SI unit of pressure, N/m2, is given the name pascal, Pa.
1 X 105 Pa = 100 kPa
Standard atmospheric pressure, the typical pressure at sea level, is the pressure sufficient to support a column of mercury 760 mm high. In SI units, this pressure equals 1.01325 X 105 Pa.
These conversion factors are used in gas calculations. They are all equal:
In laboratories, a device called a manometer is often used to measure the pressure of enclosed gases. Although it operates on a principle similar to that of a mercury barometer, one end of the manometer tube is usually open to the atmosphere instead of being sealed.
In the drawing above, manometer 1 indicates a gas pressure in the container higher than atmospheric pressure. Manometer 2 indicates a gas pressure in the container lower than atmospheric pressure. Notice the difference in the two pressure calculations.
- 1 atmosphere, atm =
- 760 mm Hg =
- 760 torr =
- 1.01325 X 105 Pa =
- 101.325 kPa =
- 1013 millibars, mb
- 14.7 lb/in2, psi
Temperature is a measure of the "heat content" or "particle motion" of matter. Temperature changes will cause the volume of matter to change. This volume change is most obvious in the gas phase of matter.
The volume of a gas means nothing unless the
conditions under which it was measured are known.
Tips for working with gas laws:
- All gas calculations must use Kelvin temperatures.
- The conditions 0 oC and 1 atm are referred to as standard temperature and pressure - STP.
- The volume occupied by one mole of a gas at STP, 22.4 liters, is referred to as molar volume.
- Read the problem to see what conditions change.
- Decide which gas law to use and write its equation.
- Reread the problem to see what question is asked.
- If needed, manipulate the gas law equation.
- Plug numbers and units into the equation.
- Pickup your calculator and punch buttons.
- Write the answer to the problem, don't forget significant figures, and circle it.
Boyle's Gas Law:
The Pressure-Volume Relationship was established by Robert Boyle
Boyle's Law states: the volume of a fixed quantity of gas maintained at constant temperature is inversely proportional to the pressure.
When two measurements are inversely proportional, one gets smaller as the other gets bigger.
Boyle's Law is expressed by the equation:
P1V1 = P2V2
Boyle's Law Problems
Charles' Gas Law:
The Temperature-Volume Relationship was established by Jacques Charles
Charles' Law states: the volume of a fixed amount of gas maintained at constant pressure is directly proportional to its absolute temperature.
When two measurements are directly proportional, as one changes in size the other undergoes the same size change.
Charles' Law is expressed by the equation:
Charles' Law Problems
The Combined Gas Law:
Boyle's Law and Charles' Law can be used in combination when both pressure and temperature change. This relationship produces the Combined Gas Law Equation.
Combined Gas Law Problems
Dalton's Law of Partial Pressures, established by John Dalton, states: the total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone.
The pressure exerted by a particular component of a mixture of gases is called the partial pressure of that gas.
Dalton's Law of Partial Pressures is expressed by the equation:
Ptotal = P1 + P2 + P3 . . .
Dalton's Law is helpful when collecting a gas "over water". This diagram shows the collection of a gas by water displacement.
A collecting tube is filled with water and inverted in an open pan of water. Gas is then allowed to rise into the tube, displacing the water. By raising or lowering the collecting tube until the water levels inside and outside the tube are the same, the pressure inside the tube is exactly that of the atmospheric pressure.
A gas collected "over water" is a mixture of the gas and water vapor. Dalton's law of partial pressures describes this situation as:
Ptotal = Pgas + PH2O
Charts like this one are readily available that give water vapor pressure at any common temperature.
Dalton's Law Problems
The Quantity-Volume Relationship is named for Amedeo Avogadro
Avogadro's Hypothesis states: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
At 0 oC and 1 atm, 22.4 L of any gas contains 6.02 X 1023 gas molecules.
Avogadro's Law states: The volume of a gas maintained at constant temperature and pressure is directly proportional to the number of moles of the gas.
Avogadro's Law is expressed by the equation:
The Ideal-Gas Equation
An ideal gas is a hypothetical gas whose molecules have no volume and no attraction to other molecules. While real gas molecules do have volume and are attracted to other molecules, at common temperatures the difference is so small that it can be ignored.
The ideal-gas equation is: PV = nRT
An ideal-gas equation modification
- P is standard pressure in kPa
- V is molar volume
- n is number of moles
- T is standard temperature in K
- R is called the gas constant.
The value and units of R depend on the units of P, V, n, and T. Pressure units are the ones that most often are different.
NOTE: here are two commonly used values for R:
- Pressure units in atm, R = 0.0821 L-atm/mol-K
- Pressure units in Pa, R = 8.314 J/mol-K
Sample problems using the ideal gas equation:
- The number of moles, n, can be expressed as:
mass (m) / moleculear mass (M)
- The equation then becomes:
- How many moles of gas are found in a 500 dm3 container if the conditions inside the container are 25 oC and 200 kPa?
- What volume will 50 grams of chlorine gas occupy at STP?
- What is the molecular weight of a gas if 150 grams of the gas occupy 250 dm3 at 500 mm Hg and 30 oC?
(This problem requires the ideal gas equation modification.)
Gas density can also be calculated using the ideal-gas equation.
Density is equal to mass divided by volume, d = m/v.
The ideal-gas equation can be arranged to give density in g/L:
This equations shows the density of a gas depends on its pressure, molar mass, and temperature. The higher the molar mass and pressure, the greater the gas density; the higher the temperature, the less dense the gas.
Even though gases form homogeneous mixtures regardless of their identities, a less dense gas will lie above a more dense one if they are not physically mixed. The differences between the densities of hot and cold gases is responsible for CO2 being able to keep oxygen from reaching combustible materials (thus acting as a fire extinguisher) and for many weather phenomena, such as the formation of large thunderhead clouds during thunderstorms.
Ideal Gas Equation Problems
Fuel in a "butane" lighter