CHAPTER 5 LECTURES: GASES

 

I. Nature of Gases

            Unlike liquids and solids, gases are diffuse rather than condensed.  One of the most important properties of gases is that they can be compressed when the pressure acting on them increases.  By simple deductive logic, we arrive at the idea that the molecules or atoms of gases must be far apart relative to the distance between particles in solids or liquids, whose volumes remain nearly constant when the pressure acting on them increases.

            To fully understand gases, it is necessary to look at the variables that control the behavior of gases.  One variable is the volume the gas occupies.  Volume is an extensive variable, because if other variables are held constant, the volume depends on how much of the gas is present.  Compare this to the density of a gas which is does not change as the sample of the gas increases as long as the pressure and temperature remain unchanged. 

 

The temperature is another variable and as we shall see, the temperature is a measure of the average kinetic energy of the gas.  If the temperature increases, the average kinetic energy increases and therefore the average velocity of the particles of the gas also increases.  This can be seen from the following relationship:

         

 

Finally the pressure that the gas exerts also affects behavior.  Of these four variables, volume, mass, temperature, and pressure, the pressure is the hardest to define.  Pressure results when particles of gas strike the side of the container.  Each collision with the wall of the container results in a force being exerted on the wall of the container.  The total force exerted on the wall of the container is the sum of all of the collisions of the gas particles with the wall of the container.

 

   Pressure like temperature is an intensive variable.  The pressure does not depend on the amount of gas—it depends only on the number of collisions with the wall.  If we take a cylinder of cross sectional area A and fill it with mercury and then inverted into a pan of mercury, we will see that the mercury does not all flow into the pan.  Instead the mercury level reaches a constant value.  This value rises or falls depending on the atmospheric pressure.  The weight of the column of mercury must equal the pressure of the atmosphere.

The pressure exerted by the gas is given by force   exerted on the wall divided by the area of the wall.  We can write this as .  The force is defined as the mass times the acceleration of gravity or mg.     Therefore,  .   Now if we multiply the top and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


bottom of the equation by h, we get  , where d is the density

 of the mercury.  For a cylinder the volume is the product of the area and the height of mercury column.  From this expression we can see that the denser the liquid, the smaller height of the column will be.  A column of water which has a density of about 1.00 g/mL will be 13.6 times that of a mercury column since the density of mercury is 13.6 g/mL.  An interesting question occurs regarding the height of two columns of mercury with different cross sectional areas.  I will give you this as an outside of class exercise.  Experiments have shown that at sea level the average height of a column of mercury is about 76 cm.  This number varies since the actual atmospheric pressure varies.  If we use the expression dgh for mercury we see that we have

 

If you remember we said that pressure is force per unit area.  The unit of force is called a Newton, which is .  If we use the value derived above we see that we get

Hence we see that we get units of pressure.  We now define a standard atmosphere as being 101.325 kiloPascal.  This is equivalent to 75.99624 cm Hg.  In most calculations we simply use 76.0 cm Hg as 1 standard atmosphere.

 

II. Gas Laws

A.   Boyle’s Law

 

Now Robert Boyle began experimenting with fixed masses of gas and holding the temperature constant.  Boyle used a rather simple apparatus.  His apparatus consisted of a tube closed at one end and bent into a J shape.  Into the open end of the tube, he inserted a sample of mercury.  The mercury filled one end of the tube, which caused some air to be trapped in the closed end.  By adding different amounts of mercury to the open end, he could vary the pressure acting on the gas in the closed end.  Boyle used careful measurements of the length of the column of gas and the length of the column of mercury to determine any relationship between the volume occupied by the gas and the pressure acting on the gas.  He was able to plot (not using a computer graphing program) pressure vs. volume. 

 

Text Box: Pressure
 

 


This expression shows that there is an inverse relationship between the pressure and the volume of a gas when the amount of gas and the temperature are held constant.  This can be expressed mathematically as

The plot on the previous page reflects a P vs. Volume plot.  It demonstrated clearly that as the pressure increases the volume decreases.  For example if we double the pressure, the volume will be reduced to one-half its initial value.  What Boyle discovered was that the nature of the gas (or type) did not matter. The same relationship was found for all gases.    In fact, if the same numbers of moles of each gas are used and the temperature is the same, the value of the constant is found to be nearly the same (within experimental error).  For example, if 1.00 moles are used and the temperature is held at 0oC, the volume of the gas when the pressure is 1 atmosphere will be found to be 22.4 liters for any gas.  In some cases we might plot P vs. (1/V).  This plot will look as follows

 

 

Text Box: Pressure (Atm)
 

 

 

 

 

 

 

 

 

 

 

 



                                               

 

The advantage of this plot is to more clearly see the inverse relationship with a liner plot.

If we change the pressure on a fixed amount of gas we can use the relationship above to determine the final volume.  This can be done with a simple recognition that the pressure volume product initially and finally are equal to the same constant as long as the amount of gas and the temperature remains constant.  We can write this as

 

 

Now one can always manipulate this equation, but you might find it easier to thin through logically what must occur.  If the final pressure is greater than the initial pressure, then the final volume must be less than the initial volume.  This means that the final volume will be equal to the initial volume times a pressure ratio that reflects physically what must occur.  For example suppose we have 150. mLs of a gas at 600. mm Hg.  If the pressure increases to 800. mm Hg, then we know that the final volume must be less than 150. mLs and we need to multiply the initial volume by a pressure ratio of less than one.  We do this as follows:

 

 

 

II.  Gas Laws

B  Charles’s Law

Charles’ began investigating the relationship between the temperature and the volume of a fixed amount of gas at a fixed pressure.  Charles’ found that the volume was a linear function of temperature.  The volume of different gases varies in a linear fashion, but they differed by their degree of thermal expansion.  This data is shown in the graph below

Text Box: Volume (L)
 


                                                                           

 

 

 

 

 

 

 

 

                                        

 

 

 

From this we can write .  The extrapolation of the lines for all the gases tested by Charles’ leads to a zero volume at –273.15oC.  From this the ‘alpha’ constant is found to be 1/273.15. 

 

By defining T(K) = t (oC) + 273.15, we find that T = 0 K when t = -273.15 oC.  When substituted into the volume temperature equation discovered by Charles’. We find that .  This direct proportion between the Kelvin temperature and the volume is now called Charles’s Law

 

C. Combined gas Law/Ideal Gas Law

   Since we know that pressure and volume vary inversely and V and Temperature vary directly, we also need to explore pressure and temperature.  Since increasing the temperature increases the kinetic energy of the molecules, thereby increasing the velocity of the gas particles, pressure will also vary directly with the temperature.  Finally we can also explain the effect of the number of moles on the behavior of gases.. The greater the number of moles, the higher the pressure or volume will be   We can see this if we think about the effect of more gas particles striking the walls of the container. 

   The greater the number of collisions, the more force is exerted on the walls of the container.  If the container has a fixed volume, then the pressure increases.  If the container has a movable piston so that the pressure remains constant, the larger the volume will be.  This can be summarized as follows:

 

This equation is called the Ideal Gas Equation or Ideal Gas Law.

 

The Ideal gas equation can be manipulated to provide a variety of in formation.  Since the number of moles is given by: , we can write the equation as :

 

We could also solve for the mass of the gas or the temperature of the gas.  STP is called standard temperature and pressure, which means 0oC and 1.00 atmospheres.  Under these conditions, one mole of a gas occupies 22.4 Liters.  This is often called the molar volume and is expressed as 22.4 L/mole.

 

D. Dalton’s Law of Partial Pressure.

   Suppose we have 1.00 moles of He in a container with a fixed volume and temperature and with the pressure of the gas is found to be 800 Torr.  Suppose we now add 0.75 moles of argon into the flask.  What pressure does the argon exert in the container?  What is the total pressure?   These questions can be answered if we remember that gases act independently unless a chemical reaction can occur.

  Since one mole of gas exerts a pressure of 800 Torr under these condition, 0.75 moles should exert a pressure of 0.75 that of one mole.  In this case this pressure should be

  0.75 (800 Torr) = 600 Torr.  Hence we can write PAr = 600 Torr.  The total pressure must the sum of the pressures exerted by the two gases.  This can be summarized in what is called Dalton’s Law of Partial Pressure (Dalton’s Law for short)

 

This expression is another part of Dalton’s Law.  You need to remember both parts of this Law.  Consider a mixture at room temperature that contains 10.0 moles of CO and 12.5 moles of O2.  The pressure is 600. Torr.  What is the mole fraction of CO in the mixture?  What are the partial pressures of each gas?  Now the temperature is increased until the following reaction occurs:      .  The reaction is allowed to proceed until 3.0 moles of CO2 are produced.  The temperature is then lowered to the initial temperature.  What is the mole fraction of CO in the mixture and what are the total pressure and the partial pressures of each gas in the mixture?

 

III. Kinetic Theory of Gases

   The kinetic theory of gases was developed by several researchers during the 19th century.  Among these are James Clerk Maxwell and Ludwig Boltzmann.  This model is one of the most important that has been developed since it is simple, predicts behavior quite well and gives evidence for the atomic theory of matter.  The Kinetic Molecular Theory of Gases assumes that gas particles are very small in relationship to the container in which they are place.  This model also assumes no interaction between gas particles and that they are in rapid chaotic motion.  Further when a gas particle collides with another gas particle or the wall of the container, the collision s completely elastic. (energy is conserved).

    Now we know that temperature is a measure of the average kinetic energy.  In fact (we will not prove this), the relationship between kinetic energy and temperature is given by

. 

Now the pressure is a measure of the force/unit area that the gas particles exert on the wall of the container.  This pressure is therefore, dependent on the number of collisions made between the wall of the container and the gas particles.  For a fixed amount of gas, (say 1.00 mole for this discussion), the number of collisions depends on the velocity of the molecules.  The higher the velocity of the molecules the greater the number of collisions.  This means that the pressure is proportional to the velocity of the gas

particles.  This can be written as .  This relationship is not completely accurate since not all of the gas particles are necessarily moving at the same speed.  In addition not all of them collide with the wall of the container at right angles to the wall.  To account for these different speeds, we use the average of the squares of the velocity, which is called the mean square speed.  The expression above becomes

.  Now a gas molecule is typically moving in random directions in the x, y, and z directions.    To account for this we resort to a trigonometric relationship between the axes.  This requires the use of spherical coordinates, which is well above the level of this course.  From this relationship we are able to say that on the average, one-third of the molecules are moving in the x direction at any one time.  From this we can write that

 

In this equation m represents the mass of one gas particle.  By multiplying by Avogadro’s number we get one mole of particles.  Therefore we can also write

  .  Now this means that we can actually find the mean square speed of the gas particles.

 

 

This represents the root mean square speed.  This is not the true average speed but is close to it.  In order for the units of the speed to be in meters/second, we must convert the gas constant from liter-atmospheres to energy units.  The basic unit of energy is called a joule.  However, a joule is given as follows:.  Let us now calculate the root-mean-square speed of SO2 at 298K

 

Substituting the values from above into the equation and solving gives the following:

 

Now suppose we wish to compare the velocity of SO2 with that of helium.  We could solve for the speed of helium and then find the ratio of the speeds of the two gases.  However, there is an easier way.  Using the rms equation derived above, we can write the following:

This means that the root mean square speed of He is about 4 times that of SO2. 

This formula is used extensively to compare the velocities of two gases--not for calculating the actual speed of either gas.

 

At this point, we have discussed the root mean square speed of gases and mentioned that in any sample of gas, the particles have various speeds.  Both Ludwig Boltzmann and James Clerk Maxwell studied this problem and derived a functional relationship  This is given below:

 

 

 

 

 

Text Box: Number of Molecules 

 

 


As you can see, the velocity distribution becomes broader and lower as the temperature increases.  Note that at the higher temperatures, there are more molecules with a higher We would also find different distributions for gases with different molar masses.  For example, if we compared helium and sulfur dioxide, we would see that far more of the helium atoms would have speeds in the higher range than the SO2 molecule at given temperature. 

 There are two closely related phenomena to what we discovered about the speeds of gas particles.  If two gases are contained in a closed vessel inside a vacuum and then a small pinhole is made in the vessel, gas particles effuse through the pinhole.  It is found that the rate of effusion follows the same inverse GMM law that root mean square speed do:

 

 The second phenomenon is the diffusion of a gas.  If two gases are placed in side a container their rates of diffusion follows the same law.  This can be seen in the diagram below.

 

 

     Gas A                                                                      Gas B

 

 

 

 

 

 


If gas A has a higher molecular mass than gas B, gas A will diffuse down the tube more slowly than gas B . The rates of diffusion of the gases follows the same relationship: