AP Chemistry - Gases: The Empirical Gas Laws


Quantitative measurements on gases were first made by the English chemist Robert Boyle (1627 - 1691). Boyle used two instruments to measure pressure: the manometer, which measures differences in pressure, and the barometer, which measures the total pressure of the atmosphere.

The operation of a manometer, which is simply a bent piece of tubing, preferably glass with one end closed. When the fluid level in both arms is the same, the pressure of the sample of gas inside the closed end must equal the pressure of the external atmosphere since the downward force on the two columns of liquid is then equal. When the liquid levels are unequal, the pressures must differ. The difference in pressure can be measured in units of length of the vertical column of liquid. The mmHg, or its modern version the torr, originated in this use of the manometer. Mercury is particularly convenient for use in manometers (and barometers) because at room temperature it has low vapor pressure, does not wet glass, and has a high density. Other liquids such as linseed oil or water have also been used in manometers.

The barometer was invented by Torricelli, one of Galileo's students. It is a device for measuring the total pressure of the atmosphere. A Torricellian barometer can easily be constructed by taking a glass tube about a meter long, sealing one end, filling the tube completely with mercury, placing your thumb firmly over the open end, and carefully inverting the tube into an open dish filled with mercury. The mercury will fall to a height independent of the diameter of the tube and a vacuum will be created above it. The height of the mercury column will be the height which the atmospheric pressure can support. The standard atmospheric pressure, one atmosphere (atm), is 760 mmHg but the actual atmospheric pressure varies depending upon altitude and local weather conditions. For this reason barometers can be used to help predict the weather. A falling barometer indicates the arrival of a low pressure air system, which often means stormy weather. A rising barometer indicates the arrival of a high pressure air system, and that often means clear weather.

While mercury is again the most convenient liquid for use in barometers it is by no means the only liquid which can be used. Preparation of a water barometer, and many of the early barometers did use water, requires use of a vacuum pump (or arms 13 meters long).

Units of Pressure

Units of pressure were originally all based on the length of the column of liquid, usually mercury, supported in a manometer or barometer. By far the most common of these units was the mmHg, however, the modern SI unit of pressure is derived from the fundamental units of the SI.  Pressure is force per unit area, and force is the product of mass times acceleration, so the SI unit of pressure is the kg m s-2/m2 or newton/m2, which is called the pascal (Pa).

All of the older units of pressure have now been redefined in terms of the pascal. One standard atmosphere, the pressure of the atmosphere at sea level, is by definition exactly 101,325 Pa. The torr, named in honor of Torricelli, is defined as 1/760 of a standard atmosphere or as 101,325/760 Pa. The mmHg can be considered identical to the torr. The term bar is used for 100000 Pa, which is slightly belowone standard atmosphere.

Robert Boyle and his Law
Boyle used the manometer and barometer to study the pressures and volumes of different samples of different gases. The results of his studies can be summarized in a simple statement which has come to be known as Boyle's law:

At any constant temperature, the product of the pressure and the volume of any size sample of any gas is a constant.

For a particular sample of any gas, Boyle's law can be shown graphically as is done in the Figure below. It is more common to express it mathematically as P1V1 = P2V2.

The pressure and the volume vary inversely; as the pressure increases the volume of the sample of gas must decrease.
Example:  A sample of gas occupies a volume of 47.3 cm3 at 20oC when the pressure is 30 cm of mercury.  If the pressure is increased to 75 cm of mercury, the sample will occupy a volume of 47.3 cm3 (30 cmHg/75 cmHg) = 18.9 cm3.

Go to the Boyle's Law Worksheet

Celcius and Kelvin Temperatures

In SI Metric the temperature scale is defined as Kelvoin temperature scale. The degree unit is the kelvin (K).  The symbol for the uint is K, not oK.  Kelvin temperatures must be used in many gas law equations in which temperature enters directly into the calculations.

The Celcius and kelvin scale are related unit for unit. One degree unit on the Celcius scale is equivalent to one degree unit on the kelvin scale.  The only difference between these two scales is the zero point.  The zero point on the Celcius scale was defined as the freezing point of water, which means that there are higher and lower temperatures around it.  The zero point on the kelvin scale - called absolute zero - corresponds to the lowest temperature that is possible.  It is 273.15 units lower than the zero point on the Celcius scale.  So this means that 0 K equals -273.15oC and 0oC equals 273.15 K. Thermometers are never marked in the kelvin scale. If we need degrees in kelvin the following relationships are to be used.

 TK =  tc  +  273.15             or       tc  =  TK  -  273.15 

Go to the Celcius-Kelvin Temperature Conversions Worksheet
Charles' Law and Temperature
The conventional liquid-in-glass thermometer was invented in the seventeenth century. This bulb-and-tube device is still in use; it is shown in the Figure below. In these thermometers the diameter of the bulb is much greater than the diameter of the tube so that a small change in the volume of liquid in the bulb will produce a large change in the height of the liquid in the tube.Two things were not clear about the thermometer at this time. The first question was what it was that the thermometer measured. As the temperature or "degree of hotness" apparent to one's fingers increased, the height of the liquid obviously did also, and this was useful in medicine for checking fevers, but there was no quantitative measurement made, merely the relative degree of hotness between this and that.The second question was whether the degree of hotness of any particular thing was a constant everywhere so that the temperatures of other things could be measured relative to it. Suggested fixed temperatures included that of boiling water, that of melting butter, and the apparently uniform temperature of deep cellars.

Robert Boyle knew of the thermometer, and also was aware that a gas expands when heated, but since no quantitative temperature scale existed he could not, and did not, determine the relationship between degree of hotness (temperature) and volume of a gas quantitatively. Boyle did propose a scale of temperature, suggesting that use of a specific fluid in a standardized thermometer bulb with a capacity of 10,000 units filled at the boiling point of water would give a proper scale if changes were at the one-unit level; that is, one degree would have a volume of 10,001 units. His scale was not adopted.

Guillaume Amontons (d. 1705) developed the air thermometer, which uses the increase in the volume of a gas with temperature rather than the volume of a liquid. The air thermometer is an excellent demonstration of Charles' law because the atmosphere maintains a fixed downward pressure above a small mercury plug of constant mass. The volume of a trapped sample of air increases on heating until the pressure of the trapped air equals the pressure of the atmosphere plus the small pressure due to the plug. Nevertheless, Amontons failed to achieve formulation of Charles' law for the same reason as did Boyle: a quantitative scale of temperature was needed.

A quantitative scale of temperature could only be developed after it was realized that at a fixed pressure any pure substance undergoes a phase change at a single fixed temperature which is characteristic of that substance. The melting point of ice to water was taken as 0oC and the boiling point of water was taken as 100oC to give our common Celsius scale of temperature.

The study of the effect of temperature upon the properties of gases took considerably longer to achieve a simple quantitative relation than did study of the effect of pressure, primarily because the development of a quantitative scale of temperature was a difficult process. However, once such a scale was developed, the appropriate measurements were made, primarily by the French chemist Jacques Charles (1746 - 1823).

The experimental data were formulated into a general law which became known as the law of Charles or Charles' law:
At any constant pressure, the volume of any sample of any gas is directly proportional to the temperature.

However, as the graph above shows, the volume extrapolates to zero at a temperature of -273.15oC. If this temperature were taken as the zero of a temperature scale then all negative temperatures could be eliminated.  Such a temperature scale is now the fundamental scale of temperature in the SI.  It is called the absolute scale, the thermodynamic scale, and the Kelvin scale. Temperature on the Kelvin scale, and only on the Kelvin scale, is symbolized by T.

A useful formula when the volume of one particular sample of gas changes with temperature, is:

V1T2= V2T1
Example: The volume of a sample of gas is 23.2 cm3 at 20oC. If the gas is ideal and the pressure remains unchanged, its volume at 80oC will be given by 23.2 cm3 (353.15 K/293.15 K) = 27.95 cm3.
Go to the Charles' Law Worksheet
Guy-Lussac's Law
This law also contains a third law that is implied but not explicitly stated.  The pressure-temperature law was discovered by Jacques Charles and Joseph Gay-Lussac (1778-1850), a French scientist.  It was not given a name because it is easily obtained by using Boyle's and Charles' Laws properly, however, it is given below for your enjoyment.  If the volume of a gas is fixed during an experiment then the pressure and temperature vary in a specific way.
This law, known as Gay-Lussac's Law is:         P1T2 = P2T1
Avogadro's Law

Chemical reactions which consumed and produced gases were studied carefully by many chemists at the beginning of the nineteenth century. In 1809, the French chemist Joseph-Louis Gay-Lussac summarized the results of many experiments into what we now call Gay-Lussac's law of combining volumes.

When measured under the same conditions of temperature and pressure, the volumes of gases which react together are in the ratio of small whole numbers.

The measurements of volume made in 1809 were sufficiently accurate to show that the volume relationships were in fact integers. For example, one volume of hydrogen reacts with one volume of chlorine to produce one volume of hydrogen chloride; two volumes of hydrogen react with one volume of oxygen to produce two volumes of water vapor; and three volumes of hydrogen react with one volume of nitrogen to produce three volumes of ammonia.

The law of combining volumes was interpreted by the Italian chemist Amedeo Avogadro in 1811,  using what was then known as the Avogadro hypothesis. We would now properly refer to it as Avogadro's law:

Equal volumes of gases under the same conditions of temperature and pressure contain equal numbers of molecules.

Avogadro's interpretation was not accepted for some forty years, during which confusion prevailed in distinguishing atoms from molecules. Elemental gases were assumed to be monatomic. We now know that most of the common gaseous elements actually exist as diatomic molecules: hydrogen, nitrogen, oxygen, fluorine, and chlorine. Avogadro's interpretation cleared up many of these discrepancies; it enabled explanation of the empirical results of Gay-Lussac in terms of simple molecular reactions:
  H2(g) + Cl2(g) --> 2 HCl(g) 
2 H2(g) + O2(g) --> 2 H2O(g)
3 H2(g) + N2(g) --> 2 NH3(g)
The Avogadro law is equivalent to the statement that volume is directly proportional to number of atoms or molecules. Since the fundamental unit of amount of substance, the mole, is equal to Avogadro's number of atoms or molecules, the amount of substance in moles.
In order to make proper calculations the following standards have been instituted world wide.

STP: Standard Temperature and Pressure 
( 0oC and 1 atmosphere )
1 mole of any gas at STP has a volume of 22.4 L

But not many laboratories around the world operate at a temperature of 0oC.  A more conventional standard has been introduced and is gradually being accepted.

SATP: Standard Ambient Temperature and Pressure 
(20oC and 1 atmosphere )
1 mole of any gas at SATP has a volume of 24.8 L

Go to the Avogadro's Gas Law Worksheet

The Combined Gas Law
The combined gas law is not a new law but a combination of Boyle's and Charles' laws, hence the name the combined gas law.   In short, this combined gas law is used when it is difficult to keep either the temperature or pressure constant.  In many experiments with gases, keeping either the pressure or temperature constant is not even attempted.  The combined gas law equation is:
P1V1 T2 =  P2V2T1
The temperature must be in kelvins, but the pressure and volume can be in any units as long as they are used consistently; that is, both pressure values must be in the same pressure units and both volume values must be in the same volume units.

Go to the Combined Gas Law Worksheet

The Ideal Gas Law

Combination of the three empirical gas laws, (Boyle's, Charles', and Avogadro's) described in the preceding three sections leads to the ideal gas law which is usually written as:
PV = nRT
where P = pressure, V = volume, n = moles, T = kelvin temperature and R. 
The constant R in this equation is known as the universal gas constant. It arises from a combination of the proportionality constants in the three empirical gas laws. The universal gas constant has a value which depends only upon the units in which the pressure and volume are measured. The best available value of the universal gas constant is:
8.3143510  J/mol K   or   8.3143510  kPa dm3/mol K
Another value which is sometimes convenient is 0.08206 dm3 atm/mol K
This equation is used to determine molecular mass from gas data.
Example:  A liquid can be decomposed by electricity into two gases.   In one experiment, one of the gases was collected. The sample had a mass of 1.090 g, a volume of 850 mL, a pressure of 746 torr, and a temperature of 25oC.  Calculate its molecular mass.
To calculate the molecular mass we need the number of grams and the number of moles.  We can get the number of grams directly from the information in the question. We can calculate the moles from the rest of the information and the ideal gas equation.
        V = 850 mL = 0.850 L = 0.850 dm3
        P = 746 torr/760 torr = 0.982 atm
        T = 25.0oC + 273.15 =  298.15 K

        PV = nRT
       (0.982 atm)(0.850 L) = (n)(0.0821 L atm mol-1 K-1)(298.15 K)
       n = 0.0341 mol

molecular mass = g/mol = 1.090 g/ 0.0341 mol = 31.96 g/mol.  The gas is oxygen.

Go to the Ideal Gas Worksheet

Van der Waal's Equation
A real gas deviates from ideal behavior in two important ways.  First, the model of an ideal gas, as postualted by the kinetic molecular theory, assumes that gas molecules individually have no volume, ie. KMT treats them as point sources, but of course they do.  In the aggregate, they take up part of the total space, a portion called the excluded volume.

It is because the excluded volume at ordinary pressures is such a tiny fraction of the total volume, the value of V in the gas laws -  that real gases follow the gas laws so well.  Because of the excluded volume, however we cannot indefinitely reduce a gas volume by half each time we double the pressure (Boyle's law).  This is why we have said that real gases behave most like an ideal gasd when the pressure is relatively low.

The second reason for deviations from ideal gas behavior is that the particles of a real gas do attract each other, unlike the assumption in the model of the gas laws. 

J.D. van der Waal's (1837-1923), a Dutch scientist, was one of many scientists who tried to modify the gas law equations to get a better fit to the observed data.  He first corrected for the excluded volume.  He reasoned that the excluded volume should be subtracted from the measured volume of the real gas, Vr, before the latter is used in a gas law calculation.  This correction would be given the symbol, Vi, for the gas as if it were actually ideal.

Let n be the number of moles of gas in the sample and b stand for the actual correction  -  the excluded volume per mole  -- the ideal gas volume is then found using the following equation.

Vi = Vrnb

As expected gases with large molecules  have larger values of b than those with small molecules.
Substance a
(L2 atm/mol2)
Nobel Gases

Helium, He 0.03412 0.02370
Neon, Ne 0.2107 0.01709
Argon, Ar 1.345 0.03219
Krypton, Kr 2.318 0.03978
Xenon, Xe 4.194 0.05105

Other Gases

Hydrogen, H2 0.02444 0.02661
Oxygen, O2 1.360 0.03183
Nitrogen, N2 1.390 0.03913
Methane, CH4 2.253 0.04278
Carbon dioxide, CO2 3.592 0.04267
Ammonia, NH3 4.170 0.03707
Water, H2O 5.464 0.03049
Ethyl alcohol, C2H5OH 12.02 0.08407

Van der Waals next reasoned that the measured pressure of the real gas, Pr, under the influences of attraction between molecules, must be less than the pressure, Pi, if the gas were ideal.  We can understand this by reflecting on what the forces of attraction must do to the paths taken by the moving gas molecules.  The particles of gas do move in straight lines after a collision - a postulate of the KMT.  But if the gas molecules attract each other in a real gas, when they get close they must change direction and move in curved paths as they pass each other.  This would make the real gas travel further to reach the walls, so they would take more time and would therefore hit any unit area of  the walls less frequently than if they were ideal.  The result of a decreased collision frequency in a real gas would mean a lower pressure.  To correct for this, van der Waals added a term to the measured pressure, Pr, as follows.

                    Pi    =   Pr  + n2a

In this equation, n, is the number of moles in the gas sample, Vr,  is the measured volume of the real gas, and a  is the specific correction term.  The quantity a is proportional to the  strengths of the attractive forces between the gas molecules.   The larger the value of 'a'  the larger the attractive force.

This leads us to the final equation:  The Ideal Gas Law is PV=nRT.  When we substitute the corrections into this Ideal Gas Law equation we get  van der Waals equation:

         ( P  + n2a )(V - nb) = nRT 

P and V must be the measured values, a and b are called the van der Waals constants, n is the number of moles, R is the Ideal Gas Law constant and T is the temperature in Kelvins.

Go to the van der Waals Worksheet

Dalton's Law on the Partial Pressures of Gases

When Dalton was conducting his studies, which led him to the atomic-molecular theory of matter, he also included studies of the behavior of gases.  These led him to propose, in 1803, what is now called Dalton's Law of Partial Pressures:
For a mixture of gases in a container, the total pressure exerted is the sum of the pressures that each gas would exert if it were alone.
This law can be expressed in equation form as:   p = p1 + p2 + p3 + ...
where p is the total or measured pressure and p1, p2, ... are the partial pressures of the individual gases. For air, an appropriate form of Dalton's law would be:
p(air) = p(N2) + p(O2) + p(CO2) + ...
At temperatures near ordinary room temperature, the partial pressures of each of the components of air is directly proportional to the number of moles of that component in any volume of air.  When the total pressure of air is 100 kPa or one bar, the partial pressures of each of its components (in kPa) are numerically equal to the mole per cent of that component (Table). Thus the partial pressures of the major components of dry air at 100 kPa are nitrogen, 78 kPa; oxygen, 21 kPa; argon, 0.9 kPa; and carbon dioxide, 0.03 kPa.
The same substance may be found in different physical states under different conditions. Water, for example, can exist as a solid phase (ice), a liquid phase (water), and a gas phase (steam or water vapor) at different temperatures. The processes by which a substance is converted from one phase to another are called by specific names. The conversion from solid to liquid is melting and the reverse conversion from liquid to solid is freezing. The conversion from liquid to gas is called boiling or vaporization and the reverse conversion from gas to liquid is called condensation. The conversion from solid to gas, when it occurs directly without going through a liquid state as in the case of iodine and carbon dioxide, is called sublimation; the reverse conversion from gas to solid shares the name of condensation. 
Table: Composition of Dry Air at Sea Level
Component Mole Percent Molar Mass
N2 78.08  28.013
 O2 20.948 31.998
Ar 0.934 29.948
CO2  0.0314 44.010
Ne  0.001818 20.183
He  0.000524 4.003
CH4 0.002 16.043
Kr 0.000114 83.80
H2 0.00005 2.016
N2O 0.00005 44.013
Xe 0.0000087 131.30
Table Footnotes: The amounts of water vapor and of trace gases such as ozone, carbon monoxide, sulfur dioxide, nitrogen dioxide, and ammonia in air are variable under natural conditions. Unusually high concentrations of these gases are often found in urban air as a result of human activities. The values given as mole percents in this table are numerically equal to the partial pressures of the gases, in kPa, when the total atmospheric pressure is 100 kPa or one bar.
Go to the Partial Pressures Worksheet
Vapor Pressures
All solids and liquids, that is, all substances in condensed phases, exhibit a vapor pressure. This is a pressure of the substance in the gas phase which is established at a particular temperature. The vapor pressure of a substance deends upon the temperature. A table of vapor pressures at 25oC for a few selected substances is given below. 

Table: Vapor Pressures and Densities of Pure Substances at 25oC
Substance Vapor Pressure Density

(kPa) (kg/m3)
H2O(l) 3.1691  0.99702
CH3OH(l) 16.8511 0.791
C2H5OH(l) 7.8279  0.785
C6H6(l) 12.6893 0.899
Hg(l) 0.2460 13.5340
I2(s)  0.1889 4.93

If a dish of water is placed in a chamber which is evacuated by means of a vacuum pump, the pressure will drop only until the vapor pressure of water at that temperature is reached. The liquid water will then boil, replacing the water vapor as the pump removes it, until either the liquid water has all boiled away or the pump is shut off. The pressure in the chamber while this is going on will remain constant at the vapor pressure of water.
The vapor pressure of any pure substance is characteristic of that substance and of the temperature. At room temperature the vapor pressure of ethanol is much higher, and the vapor pressure of mercury or potassium chloride is much lower, than is the vapor pressure of water. Since water is the most important as well as one of the most common substances on earth, its vapor pressure at different temperatures is given in the Table below. 
Table: Vapor Pressure and Density of Water at Different Temperatures

 Temperature Vapor Pressure Density
(oC) (kPa) (kg/m3)
0.01 0.61173 0.99978
1 0.65716 0.99985
 4 0.81359 0.99995
5 0.87260 0.99994
10 1.2281 0.99969
15 1.7056 0.99909
20 2.3388 0.99819
25 3.1691 0.99702
30 4.2455 0.99561
35 5.6267  0.99399
40 7.3814 0.99217
45 9.5898 0.99017
50 12.344 0.98799
55 15.752 0.98565
60 19.932 0.98316
65 25.022 0.98053
70 31.176 0.97775
75 38.563 0.97484
80 47.373 0.97179
85 57.815 0.96991
90 70.117 0.96533
95 84.529 0.96192
100 101.325 101.32
105 120.79 0.95475

The vapor pressure of all substances increases, and in many cases increases very significantly, with temperature as shown in the Figure below. The bond structure of solids is in general stronger than that in liquids, and as a general rule the vapor pressuresof solids are much lower than those of similar liquids at the same temperature.
Vapor Pressure As a Partial Pressure
Collection of gases over any liquid which has a vapor pressure, such as water, is carried out as shown in the Figure below. The total pressure measured, p, is the sum of the partial pressure of the collected gas and the partial pressure of the liquid, which is its vapor pressure. If the liquid is water, as is most often the case, the pressure of the gas collected must be:
P(gas) = P - P(H2O)
The measured or total pressure is obtained using a barometer and manometer. The vapor pressure of water at the appropriate temperature is obtained from a table such as that given above.
Example. A volume of 546.3 cm3 of hydrogen is collected over water at 30oC when the atmospheric pressure is 100.45 kPa. The actual pressure of the hydrogen and the amount of hydrogen present are calculated as follows.
The vapor pressure of water at 30oC is 4.246 kPa so the actual pressure of the hydrogen is 96.20 kPa. The amount of hydrogen present is (96.20 kPa)(0.5463 dm3)/(8.314 kPa dm3/mol K)(303.15 K) = 0.0209 mol H2.
Go to the Vapor Pressure Worksheet

Graham's Law of Effusion
Diffusion is the spontaneous intermingling of one substance with another. This occurs with perfumes and aftershaves and also with the fragrance of irate skunks.  Diffusion is over when the molecules of the fragrance are evenly spread within a container, be it a room or a gas flask.  Another way to look at it is to describe the process using partial pressures.  Diffusion is over when the partial pressures of all the gases involved become identical in all parts of the container.  When a fragrance is more or less concentrated in a corner of a room, its partial pressure is higher there.
Another necessary term is the gradient.  A gradient is used to describe how the concentration or partial pressures change from one place to another.  When sugar is added to coffee it sinks to the bottom.  If left alone, (and the coffee stayed hot), the sugar would dissolve and gradially spread on its own accord throughout the coffee.  ( Of course, we stir it because the time involved here is quite long.)   At the bottom of the cup we have a lot of sugar.  At the top of the cup we have zero dissolved sugar.  A concentration gradient is set up from bottom to top.  Sugar will gradually move down this gradient until it is evenly spread out.  When a freshly skunked pet comes into the room, it brings with it a high concentration of skunk perfume molecules.  Where the pet enters the house there will be a high conentration.  Everywhere else in the house there will be a zero concentration of skunk molecules. (Unless of course, this is not your pets first time, or your a skunk rancher!)   The point is, one of the great facts about natural processes is that "Nature abhors a vacuum."  Another way to say this is that nature abhors gradients.  Given the chance, gradients in nature tend to disappear, some rapidly and some only over eons of time, until there is as much an evenness as possible.

The effusion of a gas is its movement through an extremely tiny opening into a region of lower pressure.   The term diffusion really only speaks to the direction of gas movement. Effusion speaks for not only the direction but the rate that a change occurs.

An English scienetist, Thomas Graham (1805-1869), studied the rates at which various gases effuse, and he found that the more dense the gas is,  the slower it effuses.  The exact relationship between rate and gas density, d, is called Graham's Law of Effusion.

(effusion rate)A  X  (dA)1/2  = (effusion rate)B X (dB)1/2

Finding the densities of gases at various temperatures is often difficult to do.  With a little chemical slight of hand we can get a formula for a much simpler answer.

(effusion rate)A  X  (molecular massA)1/2  = (effusion rate)B X (molecular massB)1/2

Example:  Under the same conditions of temperature and pressure, does hydrogen iodide or ammonia effuse faster? Calculate the relative rates at which they effuse.
 molecular mass of HI = 128           molecular mass of NH3 = 17.04

 ( effusion rate of NH3 )  X  (17.04)1/2  = ( effusion rate of HI ) X (127.91)1/2

This rearranges to    rate for NH3(127.91)1/2
                                   rate for HI         (17.01)1/2

This rearrange into  rate for NH3 = (127.91/17.01)1/2
                                             rate for HI
                                                       = 2.74
Thus ammonia effuses almost three times as fast as hydrogen iodide.

As a general rule the more a molecule masses the slower it moves.  To find a mathematical value to this rate use Graham's Law.
Go to the Effusion Law Worksheet