Prof Gilbert LECTURE 17 CHEM 1211 10 20 10 Last time Chapter 6 Calculations incorporating the molar masses and the densities of gases Gas mixtures and partial pressures This time Partial pressure Px and the mole fraction Xx of gas x in a mixture Ex at 1 00 atm the values of PN2 and PO2 are 0 79 and 0 21 atm because the mole fractions XN2 and XO2 are 0 79 and 0 21 Px Xx Ptotal that they atoms or molecules m u that they are the molecules Kinetic molecular theory how to explain the behavior of gases 1 Gases and gas mixtures are different from liquids solids in a Occupy the entire volume of their container b Have much lower densities large distances between molecules c Are compressible 2 Atoms and molecules of gases are in random motion a Pressure is the result of collisions of these atoms and with surface of their containers b The greater the in momentum mass velocity or occurs during these collisions and the more frequent greater the pressure c Collision frequency depends on the of atoms or per unit volume n V speed u is defined by the equation K E mu2 K E is proportional to absolute temperature T 5 Combining equations from 3 and 4 P T n V and inserting a constant of proportionality R P R T n V which rearranges to P V n R T About molecular speeds u 1 2 Not all molecules of a gas have the same speed They vary by an amount that increases with increasing temperature as does the average speed There are different kinds of averages the most probable speeds are indicated by the dashed lines the one connected to pressure is called the root mean square speed urms which is slightly higher d Collision frequency also increases when the speed u of the atoms or molecules 3 Therefore P m u u n V 4 The kinetic energy K E of a an atom or molecule of a gas of mass m and the square of the average 3 urms is proportional to the square root of absolute temperature as shown above and inversely proportional to the square root of molecular mass as shown to the right To use the urms equation we need appropriate units on R 8 3145 J mol K 4 urms values for different gases at the same temperature can be used to predict their relative rates of effusion of gases A and B Inquiry atmosphere in this room which gas has the highest urms value N2 O2 Ar or CO2 In the Real gases The ideal gas law can be modified to accommodate the behavior of real gases particularly at graph or low temperature high pressure see Ideally the ratio is a constant for a given value of n no matter what the pressure In reality at very high pressure you get these trends The van der Waals equation incorporates a and b correction factors to the pressure and volume terms in PV nRT the ideal gas equation The correction factor nb is subtracted from the volume term to account for the volume of the molecules that is not compressible empty space A correction factor a n V 2 is added to the pressure term to account for fewer wall collisions and less pressure due to intermolecular interactions These interactions are between pairs of molecules and so depend on the concentration or each As a result the correction factor scales as the square of n V With values of a and b from a reference book the pressure of a real gas can be calculated using van der Waal s equation P a n 2 V nb nRT V2
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