Gas Laws

Gases
Pressure
Pressure
The most readily measured properties of a gas are:
•   Temperature
•   Volume
•   Pressure
Pressure (P) is the force (F) which acts on a given area (A)

The gas in an inflated balloon exerts a pressure on the inside surface of the balloon

Atmospheric Pressure and the Barometer
Due to gravity, the atmosphere exerts a downward force and therefore a pressure upon the earth's surface
•   Force = (mass*acceleration) or F=ma
•   The earth's gravity exerts an acceleration of 9.8 m/s2
•   A column of air 1 m2 in cross section, extending through the atmosphere, has a mass of roughly 10,000 kg

(one Newton equals 1 kg m/s2)

The force exerted by this column of air is approximately 1 x 105 Newtons
The pressure, P, exerted by the column is the force, F, divided by its cross sectional area, A:

The SI unit of pressure is Nm-2, called a pascal (1Pa = 1 N/m2)
•   The atmospheric pressure at sea level is about 100 kPa
Atmospheric pressure can be measured by using a barometer
•   A glass tube with a length somewhat longer than 760 mm is closed at one end and filled with mercury
•   The filled tube is inverted over a dish of mercury such that no air enters the tube
•   Some of the mercury flows out of the tube, but a column of mercury remains in the tube. The space at the top of the tube is essentially a vacuum
•   The dish is open to the atmosphere, and the fluctuating pressure of the atmosphere will change the height of the mercury in the tube

The mercury is pushed up the tube until the pressure due to the mass of the mercury in the column balances the atmospheric pressure
Standard atmospheric pressure
•   Corresponds to typical atmospheric pressure at sea level
•   The pressure needed to support a column of mercury 760 mm in height
•   Equals 1.01325 x 105 Pa
Relationship to other common units of pressure:

(Note that 1 torr = 1 mm Hg)
Pressures of Enclosed Gases and Manometers

A manometer is used to measure the pressure of an enclosed gas. Their operation is similar to the barometer, and they usually contain mercury
•   A closed tube manometer is used to measure pressures below atmospheric
•   An open tube manometer is used to measure pressures slightly above or below atmospheric
In a closed tube manometer the pressure is just the difference between the two levels (in mm of mercury)
In an open tube manometer the difference in mercury levels indicates the pressure difference in reference to atmospheric pressure
Other liquids can be employed in a manometer besides mercury.
•   The difference in height of the liquid levels is inversely proportional to the density of the liquid
•   i.e. the greater the density of the liquid, the smaller the difference in height of the liquid
•   The high density of mercury (13.6 g/ml) allows relatively small manometers to be built
  

1996 Michael Blaber
Gases
The Gas Laws

The Gas Laws
Four variables are usually sufficient to define the state (i.e. condition) of a gas:
•   Temperature, T
•   Pressure, P
•   Volume, V
•   Quantity of matter, usually the number of moles, n
The equations that express the relationships among P, T, V and n are known as the gas laws
The Pressure-Volume Relationship: Boyle's Law
Robert Boyle (1627-1691)
Studied the relationship between the pressure exerted on a gas and the resulting volume of the gas. He utilized a simple 'J' shaped tube and used mercury to apply pressure to a gas:

•   He found that the volume of a gas decreased as the pressure was increased
•   Doubling the pressure caused the gas to decrease to one-half its original volume
Boyle's Law:
The volume of a fixed quantity of gas maintained at constant temperature is inversely proportional to the pressure

•   The value of the constant depends on the temperature and the amount of gas in the sample
•   A plot of V vs. 1/P will give a straight line with slope = constant

The Temperature-Volume Relationship: Charles's Law
The relationship between gas volume and temperature was discovered in 1787 by Jacques Charles (1746-1823)
•   The volume of a fixed quantity of gas at constant pressure increases linearly with temperature
•   The line could be extrapolated to predict that gasses would have zero volume at a temperature of -273.15°C (however, all gases liquefy or solidify before this low temperature is reached
•   In 1848 William Thomson (Lord Kelvin) proposed an absolute temperature scale for which 0°K equals -273.15°C
•   In terms of the Kelvin scale, Charles's Law can be restated as:
The volume of a fixed amount of gas maintained at constant pressure is directly proportional to its absolute temperature
•   Doubling the absolute temperature causes the gas volume to double

•   The value of constant depends on the pressure and amount of gas
The Quantity-Volume Relationship: Avogadro's Law
The volume of a gas is affected not only by pressure and temperature, but by the amount of gas as well.
Joseph Louis Gay-Lussac (1778-1823)
Discovered the Law of Combining Volumes:
•   At a given temperature and pressure, the volumes of gasses that react with one another are in the ratios of small whole numbers
•   For example, two volumes of hydrogen react with one volume of oxygen to form two volumes of water vapor
Amadeo Avogadro interpreted Gay-Lussac's data
•   Avogadro's hypothesis:
Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules
•   1 mole of any gas (i.e. 6.02 x 1023 gas molecules) at 1 atmosphere pressure and 0°C occupies approximately 22.4 liters volume
•   Avogadro's Law:
The volume of a gas maintained at constant temperature and pressure is directly proportional to the number of moles of the gas

•   Doubling the number of moles of gas will cause the volume to double if T and P remain constant

•   1996 Michael Blaber



Gases
The Ideal-Gas Equation

The Ideal Gas Equation
The three historically important gas laws derived relationships between two physical properties of a gas, while keeping other properties constant:

These different relationships can be combined into a single relationship to make a more general gas law:

If the proportionality constant is called "R", then we have:

Rearranging to a more familiar form:
This equation is known as the ideal-gas equation
•   An "ideal gas" is one whose physical behavior is accurately described by the ideal-gas equation
•   The constant R is called the gas constant
•   The value and units of R depend on the units used in determining P, V, n and T
•   Temperature, T, must always be expressed on an absolute-temperature scale (K)
•   The quantity of gas, n, is normally expressed in moles
•   The units chosen for pressure and volume are typically atmospheres (atm) and liters (l), however, other units may be chosen
•   PV can have the units of energy:




•   Therefore, R can include energy units such as Joules or calories
•  
Values for the gas constant R
Units
Value
L atm/mol K
0.08206
cal/mol K
1.987
J/mol K
8.314
m3 Pa/mol K
8.314
L torr/mol K
62.36
Example:
If we had 1.0 mol of gas at 1.0 atm of pressure at 0°C (273.15 K), what would be the volume?
PV = nRT
V = nRT/P
V = (1.0 mol)(0.0821 L atm/mol K)(273 K)/(1.0 atm)
V = 22.41 L
•   0 °C and 1 atm pressure are referred to as the standard temperature and pressure (STP)
The molar volume of an ideal gas (any ideal gas) is 22.4 liters at STP
Example: Nitrate salts (NO3-) when heated can produce nitrites (NO2-) plus oxygen (O2). A sample of potassium nitrate is heated and the O2 gas produced is collected in a 750 ml flask. The pressure of the gas in the flask is 2.8 atmospheres and the temperature is recorded to be 53.6 °C.
How many moles of O2 gas were produced?
PV = nRT
n = PV/RT
n = (2.8 atm * 0.75 L) / (0.0821 L atm/mol K * (53.6 + 273)K
n = (2.1 atm L) / (26.81 L atm/mol)
n = 0.078 mol O2 were produced
Relationship Between the Ideal-Gas Equation and the Gas Laws
Boyle's law, Charles's law and Avogadro's law represent special cases of the ideal gas law
•   If the quantity of gas and the temperature are held constant then:
PV = nRT
PV = constant
P = constant * (1/V)
P 1/V (Boyle's law)
•   If the quantity of gas and the pressure are held constant then:
PV = nRT
V = (nR/P) * T
V = constant * T
V T (Charles's law)
•   If the temperature and pressure are held constant then:
PV = nRT
V = n * (RT/P)
V = constant * n
V n (Avogadro's law)
•   A very common situation is that P, V and T are changing for a fixed quantity of gas
PV = nRT
(PV)/T = nR = constant
•   Under this situation, (PV/T) is a constant, thus we can compare the system before and after the changes in P, V and/or T:

Example:
A 1 liter sample of air at room temperature (25 °C) and pressure (1 atm) is compressed to a volume of 3.3 mls at a pressure of 1000 atm. What is the temperature of the air sample?

  

1996 Michael Blaber

Gases
Gas Mixtures and Partial Pressures

Gas Mixtures and Partial Pressures
How do we deal with gases composed of a mixture of two or more different substances?
John Dalton (1766-1844) - (gave us Dalton's atomic theory)
The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone
The partial pressure of a gas:
•   The pressure exerted by a particular component of a mixture of gases
Dalton's Law of Partial Pressures:
•   Pt is the total pressure of a sample which contains a mixture of gases
•   P1, P2, P3, etc. are the partial pressures of the gases in the mixture
Pt = P1 + P2 + P3 + ...
If each of the gases behaves independently of the others then we can apply the ideal gas law to each gas component in the sample:
•   For the first component, n1 = the number of moles of component #1 in the sample
•   The pressure due to component #1 would be:

•   For the second component, n2 = the number of moles of component #2 in the sample
•   The pressure due to component #2 would be:

And so on for all components. Therefore, the total pressure Pt will be equal to:

•   All components will share the same temperature, T, and volume V, therefore, the total pressure Pt will be:

•   Since the sum of the number of moles of each component gas equals the total number of moles of gas molecules in the sample:

At constant temperature and volume, the total pressure of a gas sample is determined by the total number of moles of gas present, whether this represents a single substance, or a mixture
Example
A gaseous mixture made from 10 g of oxygen and 5 g of methane is placed in a 10 L vessel at 25°C. What is the partial pressure of each gas, and what is the total pressure in the vessel?
(10 g O2)(1 mol/32 g) = 0.313 mol O2
(10 g CH4)(1 mol/16 g) = 0.616 mol CH4
V=10 L
T=(273+25K)=298K


Pt = PO2 + PCH4 = 0.702 atm + 1.403 atm = 2.105 atm
Partial Pressures and Mole Fractions
The ratio of the partial pressure of one component of a gas to the total pressure is:

thus...

•   The value (n1/nt) is termed the mole fraction of the component gas
•   The mole fraction (X) of a component gas is a dimensionless number, which expresses the ratio of the number of moles of one component to the total number of moles of gas in the sample
The ratio of the partial pressure to the total pressure is equal to the mole fraction of the component gas
•   The above equation can be rearranged to give:

The partial pressure of a gas is equal to its mole fraction times the total pressure
Example
a) A synthetic atmosphere is created by blending 2 mol percent CO2, 20 mol percent O2 and 78 mol percent N2. If the total pressure is 750 torr, calculate the partial pressure of the oxygen component.
Mole fraction of oxygen is (20/100) = 0.2
Therefore, partial pressure of oxygen = (0.2)(750 torr) = 150 torr
b) If 25 liters of this atmosphere, at 37°C, have to be produced, how many moles of O2 are needed?
PO2 = 150 torr (1 atm/760 torr) = 0.197 atm
V = 25 L
T = (273+37K)=310K
R=0.0821 L atm/mol K
PV = nRT
n = (PV)/(RT) = (0.197 atm * 25 L)/(0.0821 L atm/mol K * 310K)
n = 0.194 mol
  

1996 Michael Blaber

Gases
Volumes of Gases in Chemical Reactions

Volumes of Gases in Chemical Reactions
•   Gasses are often reactants or products in chemical reactions
•   Balanced chemical equations deal with the number of moles of reactants consumed or products formed
•   For a gas, the number of moles is related to pressure (P), volume (V) and temperature (T)
Example
The synthesis of nitric acid involves the reaction of nitrogen dioxide gas with water:
3NO2(g) + H2O(l) -> 2HNO3(aq) + NO(g)
How many moles of nitric acid can be prepared using 450 L of NO2 at a pressure of 5.0 atm and a temperature of 295 K?
(5.0 atm)(450 L) = n(0.0821 L atm/mol K)(295 K)
= 92.9 mol NO2
92.9 mol NO2 (2HNO3/3NO2) = 61.9 mol 2HNO3
Collecting Gases Over Water
•   Certain experiments involve the determination of the number of moles of a gas produced in a chemical reaction
•   Sometimes the gas can be collected over water
Potassium chlorate when heated gives off oxygen:
2KClO3(s) -> 2KCl(s) + 3O2(g)
•   The oxygen can be collected in a bottle that is initially filled with water

•   The volume of gas collected is measured by first adjusting the beaker so that the water level in the beaker is the same as in the pan.
•   When the levels are the same, the pressure inside the beaker is the same as on the water in the pan (i.e. 1 atm of pressure)
•   The total pressure inside the beaker is equal to the sum of the pressure of gas collected and the pressure of water vapor in equilibrium with liquid water
Pt = PO2 + PH2O
•   The pressure exerted by water vapor at various temperatures is usually available in tables:
•  
Temperature (°C)
Pressure (torr)
0 4.58
25 23.76
35 42.2
65 187.5
100 760.0
Example
A sample of KClO3 is partially decomposed, producing O2 gas that is collected over water. The volume of gas collected is 0.25 L at 25 °C and 765 torr total pressure.
a) How many moles of O2 are collected?
Pt = 765 torr = PO2 + PH2O = PO2 + 23.76 torr
PO2 = 765 - 23.76 = 741.2 torr
PO2 = 741.2 torr (1 atm/760 torr) = 0.975 atm
PV = nRT
(0.975 atm)(0.25 L) = n(0.0821 L atm/mol K)(273 + 25K)
n = 9.96 x 10-3 mol O2
b) How many grams of KClO3 were decomposed?
9.96 x 10-3 mol O2 (2KC lO3/3 O2) = 6.64 x 10-3 mol KClO3
6.64 x 10-3 mol KClO3 (122.6 g/mol) = 0.814 g KClO3
c) If the O2 were dry, what volume would it occupy at the same T and P?
PO2 = (Pt)(XO2) = 765 torr (1.0) = 765 torr (1 atm/760 torr) = 1.007 atm
(1.007 atm)(V) = (9.96 x 10-3 mol)(0.0821 L atm/mol K)(273 + 25 K)
V = 0.242 L
Alternatively...
If the number of moles, n, and the temperature, T, are held constant then we can use Boyle's Law:
P1V1 = P2V2
V2 = (P1V1)/ P2
V2 = (741.2 torr * 0.25 L)/(765 torr)
V2 = 0.242 L

1996 Michael Blaber

Gases
Kinetic-Molecular Theory

Kinetic-Molecular Theory
The ideal gas equation
PV = nRT
describes how gases behave.
•   A gas expands when heated at constant pressure
•   The pressure increases when a gas is compressed at constant temperature
But, why do gases behave this way?
What happens to gas particles when conditions such as pressure and temperature change?
The Kinetic-Molecular Theory ("the theory of moving molecules"; Rudolf Clausius, 1857)
Gases consist of large numbers of molecules (or atoms, in the case of the noble gases) that are in continuous, random motion
The volume of all the molecules of the gas is negligible compared to the total volume in which the gas is contained
Attractive and repulsive forces between gas molecules is negligible
The average kinetic energy of the molecules does not change with time (as long as the temperature of the gas remains constant). Energy can be transferred between molecules during collisions (but the collisions are perfectly elastic)
The average kinetic energy of the molecules is proportional to absolute temperature. At any given temperature, the molecules of all gases have the same average kinetic energy. In other words, if I have two gas samples, both at the same temperature, then the average kinetic energy for the collection of gas molecules in one sample is equal to the average kinetic energy for the collection of gas molecules in the other sample.

Pressure
•   The pressure of a gas is causes by collisions of the molecules with the walls of the container.
•   The magnitude of the pressure is related to how hard and how often the molecules strike the wall
•   The "hardness" of the impact of the molecules with the wall will be related to the velocity of the molecules times the mass of the molecules


Absolute Temperature
•   The absolute temperature is a measure of the average kinetic energy of its molecules
•   If two different gases are at the same temperature, their molecules have the same average kinetic energy
•   If the temperature of a gas is doubled, the average kinetic energy of its molecules is doubled


Molecular Speed
•   Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds
•   Some are moving fast, others relatively slowly

•   At higher temperatures at greater fraction of the molecules are moving at higher speeds
What is the speed (velocity) of a molecule possessing average kinetic energy?
•   The average kinetic energy, e, is related to the root mean square (rms) speed u

Example:
Suppose we have four molecules in our gas sample. Their speeds are 3.0, 4.5, 5.2 and 8.3 m/s.
•   The average speed is:

•   The root mean square speed is:


•   Because the mass of the molecules does not increase, the rms speed of the molecules must increase with increasing temperature

Application of the "Kinetic Molecular Theory" to the Gas Laws
Effect of a volume increase at a constant temperature
•   Constant temperature means that the average kinetic energy of the gas molecules remains constant
•   This means that the rms speed of the molecules, u, remains unchanged
•   If the rms speed remains unchanged, but the volume increases, this means that there will be fewer collisions with the container walls over a a given time
•   Therefore, the pressure will decrease (Boyle's law)
Effect of a temperature increase at constant volume
•   An increase in temperature means an increase in the average kinetic energy of the gas molecules, thus an increase in u
•   There will be more collisions per unit time, furthermore, the momentum of each collision increases (molecules strike the wall harder)
•   Therefore, there will be an increase in pressure
•   If we allow the volume to change to maintain constant pressure, the volume will increase with increasing temperature (Charles's law)
  

1996 Michael Blaber

Gases
Molecular Effusion and Diffusion

Molecular Effusion and Diffusion
Kinetic-molecular theory stated that
The average kinetic energy of molecules is proportional to absolute temperature
•   Thus, at a given temperature, to different gases (e.g. He vs. Xe) will have the same average kinetic energy
•   The lighter gas has a much lower mass, but the same kinetic energy, therefore its rms velocity (u) must be higher than that of the heavier gas
where M is the molar mass
Example
Calculate the rms speed, u, of an N2 molecule at room temperature (25°C)
T = (25+273)°K = 298°K
M = 28 g/mol = 0.028 kg/mol
R = 8.314 J/mol °K = 8.314 kg m2/s2 mol °K

Note: this is equal to 1,150 miles/hour!
Effusion
The rate of escape of a gas through a tiny pore or pinhole in its container.
•   Latex is a porous material (tiny pores), from which balloons are made
•   Helium balloons seem to deflate faster than those we fill with air (blow up by mouth)
The effusion rate, r, has been found to be inversely proportional to the square root of its molar mass:

and a lighter gas will effuse more rapidly than a heavy gas:

Basis of effusion
•   The only way for a gas to effuse, is for a molecule to collide with the pore or pinhole (and escape)
•   The number of such collisions will increase as the speed of the molecules increases

Diffusion: the spread of one substance through space, or though a second substance (such as the atmosphere)
Diffusion and Mean Free Path
•   Similarly to effusion, diffusion is faster for light molecules than for heavy ones
•   The relative rates of diffusion of two molecules is given by the equation

•   The speed of molecules is quite high, however...
the rates of diffusion are slower than molecular speeds due to molecular collisions
•   Due to the density of molecules comprising the atmosphere, collisions occur about 1010 (i.e. 10 billion) times per second
•   Due to these collisions, the direction of a molecule of gas in the atmosphere is constantly changing
The average distance traveled by a molecule between collisions is the mean free path
•   The higher the density of gas, the smaller the mean free path (more likelyhood of a collision)
•   At sea level the mean free path is about 60 nm
•   At 100 km altitude the atmosphere is less dense, and the mean free path is about 0.1 m (about 1 million times longer than at sea level)

1996 Michael Blaber

Gases
Deviation from Ideal Behavior

Deviations from Ideal Behavior
All real gasses fail to obey the ideal gas law to varying degrees
The ideal gas law can be written as:

For a sample of 1.0 mol of gas, n = 1.0 and therefore:

Plotting PV/RT for various gasses as a function of pressure, P:

•   The deviation from ideal behavior is large at high pressure
•   The deviation varies from gas to gas
•   At lower pressures (<10 atm) the deviation from ideal behavior is typically small, and the ideal gas law can be used to predict behavior with little error
Deviation from ideal behavior is also a function of temperature:

•   As temperature increases the deviation from ideal behavior decreases
•   As temperature decreases the deviation increases, with a maximum deviation near the temperature at which the gas becomes a liquid
Two of the characteristics of ideal gases included:
•   The gas molecules themselves occupy no appreciable volume
•   The gas molecules have no attraction or repulsion for each other
Real molecules, however, do have a finite volume and do attract one another
•   At high pressures, and low volumes, the intermolecular distances can become quite short, and attractive forces between molecules becomes significant
•   Neighboring molecules exert an attractive force, which will minimize the interaction of molecules with the container walls. And the apparent pressure will be less than ideal (PV/RT will thus be less than ideal).
•   As pressures increase, and volume decreases, the volume of the gas molecules becomes significant in relationship to the container volume
•   In an extreme example, the volume can decrease below the molecular volume, thus PV/RT will be higher than ideal (V is higher)
•   At high temperatures, the kinetic energy of the molecules can overcome the attractive influence and the gasses behave more ideal
•   At higher pressures, and lower volumes, the volume of the molecules influences PV/RT and its value, again, is higher than ideal
The van der Waals Equation
•   The ideal gas equation is not much use at high pressures
•   One of the most useful equations to predict the behavior of real gases was developed by Johannes van der Waals (1837-1923)
•   He modified the ideal gas law to account for:
•   The finite volume of gas molecules
•   The attractive forces between gas molecules

van der Waals equation:

•   The van der Waals constants a and b are different for different gasses
•   They generally increase with an increase in mass of the molecule and with an increase in the complexity of the gas molecule (i.e. volume and number of atoms)
•  
Substance
a (L2 atm/mol2)
b(L/mol)
He 0.0341 0.0237
H2 0.244 0.0266
O2 1.36 0.0318
H2O 5.46 0.0305
CCl4 20.4 0.1383

Example
Use the van der Waals equation to calculate the pressure exerted by 100.0 mol of oxygen gas in 22.41 L at 0.0°C
V = 22.41 L
T = (0.0 + 273) = 273°K
a (O2) = 1.36 L2 atm/mol2
b (O2) = 0.0318 L /mol


P = 117atm - 27.1atm
P = 90atm
•   The pressure will be 90 atm, whereas if it was an ideal gas, the pressure would be 100 atm
•   The 90 atm represents the pressure correction due to the molecular volume. In other words the volume is somewhat less than 22.41 L due to the molecular volume. Therefore the molecules must collide a bit more frequently with the walls of the container, thus the pressure must be slightly higher. The -27.1 atm represents the effects of the molecular attraction. The pressure is reduced due to this attraction.

1996 Michael Blaber