1. 2.1 Mixtures of Ideal Gases The partial pressure of a gas in a mixture of gases is the pressure that gas would have if it occupied the total volume of the container at some temperature? Conservation of mass (moles): n total = n A + n B and the ideal gas law: P T V / R T = P A V / R T + P B V / R T lead under this view to Dalton’s law of partial pressures: P T = P A + P B A similar derivation defines the mole fraction of an ideal gas as the ratio of its partial pressure to the total pressure in the mixture: X A = n A / n T = (P A V / R T ) / (P T V / R T ) = P A / P T

2. 2.2 A different view defines the partial volume of a gas in a mixture as the volume that a gas would have if it had a pressure equal to the total pressure at some temperature. This viewpoint and a derivation similar to the previous one leads to Amagat’s law of partial volumes: V T = V A + V B Could you derive Amagat’s law of partial volumes? How is the mole fraction of a gas in a mixture of ideal gases related to its partial volume? Can a gas simultaneously follow both Dalton’s law of partial pressures and Amagat’s law of partial volumes? The density of a mixture of ideal gases, A and B, can be expressed as: density = m total / V = (m A + m B ) / V = (n A M A + n B M B ) / (n T R T / P T ) = (X A M A + X B M B ) / (R T / P T ) = M avg P T / R T where M avg is the mole fraction weighted average molecular weight of the gas mixture: M avg = (X A M A + X B M B )

3. 2.3 Heliox 20/80 is a gas mixture that is 20 volume % O 2 and 80 volume % He. Heliox is used by divers to avoid nitrogen narcosis, an intoxication or drunkeness resulting from high levels of N 2 that have absorbed into the diver’s tissue. What is the density of heliox 20/80 at 4.0 o C and 5.00 atm (roughly the pressure on a diver at a depth of 150 ft)? A WEB site explaining the biophysics of scuba diving can be found at http://www.mtsinai.org/pulmonary/books/scuba/sectionl.htmhttp://www.mtsinai.org/pulmonary/books/scuba/sectionl.htm

4. 2.4 A gas or mixture of gases is a fluid and as such obeys the hydrostatic equation, which relates fluid pressure to height or depth in a fluid column. The differential form of the hydrostatic equation applied to the mixture of gases that comprise the atmosphere is: dP = -  g dh Here  is the density of the atmospheric gases and g is gravitational acceleration. Why is there a minus sign in this equation? Sinces gases are compressible, the density of a mixture of gases will vary with pressure. If we make the reasonable assumption that the gases in the atmosphere are ideal, we can write the hydrostatic equation as: dP = - [( P total M average ) / (R T)] g dh Solving this differential equation by separating variables and integrating: P 1  P 2 dP / P = - h 1  h 2 [ M g / ( R T ) ] dh gives, assuming an isothermal atmosphere and after rearrangement, the isothermal barometric distribution law: P 2 = P 1 e - M g ( h 2 - h 1 ) / ( R T )

5. 2.5 According to the isothermal barometric distribution law the pressure of the atmosphere should decrease exponentially with altitude as shown in the plot below: A web sites that discuss the symptoms and treatment of altitude induced hypoxia are: http://www.nols.edu/Publications/FirstAid/AltitudeIllness.html Does the pressure of wet air decrease more or less than that of dry air in an isothermal atmosphere? Is the pressure decrease greater in summer or greater in winter? Calculate the difference in O 2 (g) partial pressure at 25.0 o C between Butte at 5,740 feet and Missoula at 3,167 feet.

6. 2.6 As you will learn later in this course in the section on Adiabatic Processes, the assumption of an isothermal atmosphere is not a particularly good assumption and the temperature of the atmosphere actually falls linearly with height according to the adiabatic lapse rate: T = T reference - M avg g / C p, avg ( h - h reference ) Derive the barometric distribution law in a non-isothermal atmosphere, where the temperature decreases with elevation according to the adiabatic lapse rate. What is the difference in partial pressure of O 2 (g) in Butte at 5,740 feet of elevation and Missoula at 3,167 feet in a non- isothermal atmosphere when the temperature in Missoula is 25.0 o C ? Assume that the constant pressure heat capacity of O 2 (g) is not a function of temperature and is 29.36 J / (mol K).