Gas pressure is a measure of the frequency and energy of the molecules of a gas striking a surface. An increase in either the frequency or energy of collisions results in an increased gas pressure.
The energy of collision depends on the velocity and mass of the molecule. The collisions are continually occurring between the molecules, it is only against a surface that we measure pressure.
Pressure (P) is a measure of force (F) per unit area (A).
P = F/A
A device used to measure atmospheric pressure
The gases of the Atmosphere also exert a pressure on all surfaces. Atmospheric pressure is the result of compression of the atmospheric gases from gravity. The pressure of the atmosphere will vary with elevation and with the composition of the atmosphere, which changes with storm systems.
The Standard Atmospheric Pressure is defined at sea level.
Standard Atmospheric Pressure
|
Unit |
Standard Pressure |
|
atmosphere |
1 atm (exact) |
|
millimeters of mercury |
760 mm Hg (exact) |
|
torr (1 torr = 1 mm Hg) |
760 torr (exact) |
|
Pascal (Pa) |
101,325 Pa |
|
kilopascal |
101 kPa |
|
Bar |
1.01 bar |
Pressure = force/area
Pascal = Newton/m2
1 torr = 1 mm Hg
1 bar = 105 Pa
A device for measuring pressure. (not necessarily atmospheric pressure)
A unit of pressure based on the height of a column of mercury in a manometer or barometer, equal to the torr.
Density times column height = d * h = (g/cm3)
(cm) = (g/cm2) = mass/ area
= (force/area)(constant) = pressure (constant)
With this line above we can see that d*h is directly proportional to pressure, so if we want the height of a second liquid based on the height (pressure) of the first liquid, the equation will be.
d1 * h1 = d2 * h2
A unit of pressure abbreviated atm.
1 atm = 760 mm Hg = 760 torr = 101,325 Pa = 101.3 kPa = 1.01 bar
The volume of a gas is inversely proportional to the pressure at constant temperature.
Boyles Law:
PV = k
Where k is a constant for a given gas at a specified temperature.
V = k/P = k(1/P) gives a straight line plot of V vs (1/P), with slope = k, and intercept = 0
Boyle's law is strictly true only at very low pressures
The ideal value of k (at zero pressure) is 22.41 L atm
To use Boyle's law for changes in volume or pressure:
P1V1 = P2V2
The volume of a gas is directly proportional to the absolute temperature at constant pressure.
Absolute temperature is the Kelvin scale.
Charles Law:
V = bT where b is a constant and T is in Kelvin
or V/T = b
To use Charles Law for changes in volume or temperature
V1/T1 = V2/T2
Extrapolation of temperature to zero volume for any gas
gives the same value of
–273.15°C
or 0 Kelvin
Equal volumes of gases at the same temperature and pressure contain the same number of molecules.
Also
Volume is directly proportional to the number of moles at constant temperature
and pressure
V = kn where k is a constant and n is number of moles
V1/n1 = V2/n2
This is obeyed closely at low pressures
If all the properties of a gas allowed to change except for the number of moles. The combined gas law can be used to calculate the final properties from an initial set of properties.
P1V1/T1 = P2V2/T2
If two of the new conditions are known, the third can be calculated.
PV = nRT
Ideal gas constant = R
R = 0.08206 atm·L/mol·K
The Ideal gas law is an equation of state, it describes the state of the gas without regards of how it got to that state. Knowledge of three of the four properties is sufficient to describe the state of the gas since the fourth property can be calculated
The ideal gas law is referred to as "ideal" because it is not perfect. Real gases obey the ideal gas law to a good approximation under standard conditions and at low pressures and high temperatures. Deviations from the ideal gas law occur at high pressures and low temperatures.
If three of the properties of a gas are known, the ideal gas law can be used to determine the value of the forth property.
0°C (273.15 K) and 1 atm
These conditions are a common reference conditions for gases and is referred to as STP
One mole of an ideal gas occupies 22.41 L of volume
This is termed the molar volume and is often written 22.41 L/mol
Real gases deviate some from this value.
Using Molar Volume
Stoichiometry with volume
mass to volume
moles to volume
volume to volume (STP)
volume to volume (not STP)
volume to moles
volume to mass
If one set of conditions is known about a gas, and then two or three properties are allowed to change, with one of the new values being known, then the other property can be calculated.
If pressure and volume vary with moles and temperature held constant, the ideal gas law can become Boyles law. Rearrange the ideal gas equation with all variables on one side and all constants on other side.
P1V1 = P2V2 = nRT = constant
If volume and temperature are allowed to vary with moles and pressure held constant, the ideal gas law becomes equivalent to Charles Law.
V1/T1 = V2/T2 = nR/P = constant
If all the properties of a gas allowed to change except for the number of moles. the ideal gas equation takes the form referred to as the combined gas law.
P1V1/T1 = P2V2/T2 = nR = constant
If two of the new conditions are known, the third can be calculated.
Molar mass of a gas can be calculated from the gas density.
P = nRT/V = mnRT/mV = (n/m)(m/V)RT = dRT/M where M is molar mass
PM = dRT where M is molar mass
Where m/V = d = gas density
and molar mass = M = m/n
alternatively
n = mass/molar mass = m/molar mass
P = [mRT]/[V molar mass]
P = [dRT]/molar mass
molar mass = dRT/P
A gas in a mixture will exert the same pressure it would if it were the only gas present. The pressure the gas exerts is not influenced by the presence of other gases.
Dalton's law of partial pressures, The total pressure of a gas mixture is the sum of the individual pressures of each gas in the mixture.
Ptotal = SPi
The pressure each gas exerts in a mixture is termed the partial pressure.
The ratio of the number of moles of one component of a mixture to the total moles of all components of the mixture.
c1 = n1/nTotal
since n = P(V/RT)
c1 = P1/PTotal
c1 = n1/nTotal = P1/PTotal
This also means that
P1 = c1 (PTotal)
The partial pressure of a component of a gas mixture is the mole fraction times the total pressure.
Gases can be collected over water to determine volume or quantity generated. However, this wet gas contains the gas of interest and water vapor. To determine the quantity of the gas of interest, the vapor pressure of water, at the collection temperature, must be subtracted off of the collection pressure, generally 1 atm.
Ptotal = Pgas + PH2O
This model speculates on the behavior of individual gas molecules to explain the properties of an ideal gas.
·
Gases are made up of
very tiny molecule compared to the distance between thems. Gases are mostly empty space. The volume of the gas
molecules is negligible. [Real gases have volume]
·
Gas molecules are in
constant motion. They move in straight lines and random directions. The collisions of the molecules with the container
walls is the source of the gas pressure.
·
Gas molecules exert
no force on each other. Gas molecules
are not attracted to nor repel each other.
·
Gas molecules have
elastic collisions. Two molecules can
exchange energy when they collide but the sum of their energies does not change
from the collision. [One of the main differences between ideal and real gases]
·
The average kinetic
energy of gas molecules is proportional to the Kelvin Temperature. At the same temperature, all gas molecules,
regardless of type, have the same average kinetic energy.
As volume decreases, the number of collisions between molecules and walls will increase increasing pressure.
As temperature increases, the molecules will travel faster and exert more force in collisions with the walls.
As temperature increases, the molecules will travel faster and exert more force in collisions with the walls, to keep pressure constant, the volume must decrease reducing the frequency of collisions.
Increasing number of moles will increase frequency of collisions and therefore pressure unless volume is increased to decrease frequency of collisions.
The total pressure is equal to the sum of the individual pressures since the molecules are independent of each other and their volumes are negligible.
Deriving the Ideal Gas Law
The average kinetic energy of a gas particle is (1/2)mu2
where u2 is the average square of the particle velocities
and m is the mass of a single particle in kilograms.
Multiplying by Avogadro's number, NA, will give the average kinetic energy of a mole of gas
(KE)avg = NA((1/2)mu2)
Using the assumptions of the kinetic molecular theory (KMT) the pressure of a gas can be derived to be
P = (2/3)[nNA((1/2)mu2)]/V
Substituting in (KE)avg gives
P = (2/3)[n(KE)avg]/V
or PV/n = (2/3)(KE)avg
But the ideal gas law gives us
PV/n = RT
so
PV/n = RT = (2/3)(KE)avg
or
(KE)avg = (3/2)RT
The form of the Gas Constant useful for this equation is:
R = 8.3145 J/K mol
u2 is the average of the squares of the velocity
The square root of u2 is called the root mean square velocity
The root-mean-square (rms) speed, u, of molecules is the speed of a molecule possessing the average kinetic energy, e, of the gas.
e = (1/2)mu2
The rms speed is different from the most probable speed or the average speed.
This is related to temperature by:
(KE)avg = (3/2)RT
Difference between average speed and rms speed: If we have 4 objects with speeds of 4.0, 6.0, 10.0 and 12.0 m/s, the average speed is (4.0 + 6.0 + 10.0 + 12.0)/4 = 8.0 m/s. The rms speed is Ö(4.02 + 6.02 + 10.02 + 12.02)/4) = 8.6 m/s.
Since, the energy of a molecule is
e = (1/2)mu2
The energy of a mole of gas gas is this energy multiplied by Avogadro's number, NA
(KE)avg = NA((1/2)mu2) and (KE)avg = (3/2)RT
NA((1/2)mu2) = (3/2)RT or u2 = 3RT/[NAm]
Since NAm is equal to the mass (in kg) of a mole of particles, i.e. the molar mass, M, then
u2 = 3RT/[M]
The rms speed can be derived from the kinetic molecular theory.
urms = Ö3RT/M)
The rms velocity is inversely proportional to molar mass, M, at constant temperature. Light molecules travel faster than larger molecules.
Using the gas constant in the form of R = 8.3145 J/mol K mol will give urms in terms of m/s, since the joule is defined as J = kg m2/s2, so the units of R are kg m2/(mol K s2). M is the molar mass with units of kg/mole.
The kinetic energy and rms velocity are both averages. The gas molecules are in constant collisions within a short travel distance. This transfers energy with each collisions having molecules constantly increasing and decreasing their energies.
Effusion is the process of a gas passing through a hole into a vacuum.
The rate of effusion measures the speed at which the gas is transferred into the evacuated chamber.
The rate of effusion is inversely proportional to the square root of the mass of its particles (or molar mass)
Rate of effusion a 1/ÖM)
The ratio of rates of effusion of two gases is equal to the inverse ratio of the square roots of the masses of the gas particles
Rate
of effusion(1)
= ÖM2)
Rate of effusion(2) ÖM1)
Low mass molecules effuse faster than high mass molecules.
Diffusion is the mixing or spread of one gas in another.
This is the mixing of gases by molecular motion.
Gases at room temperature have a velocity close to half a kilometer per second. But they also have collisions on the order of 1010 times per second. The average distance between collisions is called the mean free path of the molecule. The mean free path of air at sea level is about 60 nm.
The ideal gas equation only approximates the behavior of real gases.
Real gases deviate from the ideal gas equation. This deviation increases at high pressures and at low temperatures.
The main differences between real gases and the ideal gas
equation and the kinetic molecular theory is that
Real gases occupy volume, and
Real gases have attractive forces between them
The volume of gases requires an adjustment to the volume term of the gas equation
The attractive forces between molecules reduces the pressure the gas exerts, so an adjustment is required to the presser term in the gas equation.
[P + a(n/V)2][V – nb] = nRT
another way it can be written is:
P = (nRT/[V – nb]) - a(n/V)2
The values of a and b are determined experimentally for each gas. Table 10.3 p. 429.
The two correction factors have opposing effects on the pressure. The volume correction factor increases pressure, while the pressure correction factor (based on attractions between molecules) reduces pressure. At a low pressure, the pressure correction factor dominates, reducing the observed pressure below that predicted by the ideal gas law. At high pressures, the volume term dominated creating a higher pressure than predicted by the ideal gas law. Low temperature magnifies the effect of both terms.
Boyles Law
P1V1 = P2V2
Charles Law
V1/T1 = V2/T2
Avogadro's Law
V1/n1 = V2/n2
Combined Gas Law
P1V1/T1 = P2V2/T2
Ideal Gas Law
PV = nRT
Dalton's Law of Partial Pressure
Ptotal = SPi
P1 = c1 (PTotal)
Mole Freaction
c1 = n1/nTotal
Gas law arranged with density and molar mass
PM = dRT where M is molar mass
Energy/Temperature of a gas
(KE)avg = (3/2)RT
Root mean Square velocity
e = (1/2)mu2
urms = Ö3RT/M)
Grahams Law of Effusion
Rate of effusion(1) = ÖM2)
Rate of effusion(2) ÖM1)
Van der Waals Equation
[P + a(n/V)2][V – nb] = nRT
or
P = (nRT/[V – nb]) - a(n/V)2
Ideal Gas Constant
R = 0.08206 atm·L/mol·K