| Chem 109 |
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Spring 2009 |
| Lecture Notes:: 25 February |
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Effusion refers to the passage of a substance through a small orifice.
This law states that the effusion of a gas through a small orifice is inversely proportional tothe square root of its density.
or, since the density of a gas is proportional to its molecular weight
(n = mass/MW; from PV = nRT, n/V = const. = (mass/MW)/V; multiplying both sides by MW gives (MW)(const.) = mass/V = density.)
Equivalently, the relative rates of effusion of two gases at the same pressure and temperature is given by the inverse square roots of their densities.
Example: What is the relative rate of effusion of H2 vs. O2?
Diffusion refers to the passage of one substance through another. An example for gases would be the passage of an aroma, such as a perfume or skunk smell, through still air. Given that gases are mostly empty space this interpenetration is not surprising. What we want to look at now is the rate of this process:
This law states that "The rate of diffusion of a gas is inversely proportional to the square root of its density."
or, since the density of a gas is proportional to its molecular weight
Unlike in effusion, this turns out to be not quite the case for diffusion. That is, the ratios of rates of diffusion of different gases will not quite fit prediction. The problem is that, although the average velocities of the molecules follow the inverse proportionality, as in effusion, the molecules are impeded by collisions with the gas they are passing through. Not surprisingly, the description of this more complex process is not quite the simple law originally postulated by Graham. It does still give a useful first order picture however.
We have been looking at the various properties of gases, now we want to look at a theory to explain those behaviors. A simple model is the kinetic-molecular theory. There are four basic postulates:
public domain image via Wikipedia Creative Commons‡ |
Consequences/predictions:
Note that for kinetic energy, KE = 1/2 mV2, so V varies as the square root of the mass (m1/2). Notice also that the energy increases with the square of the velocity. (This is why an accident at 60 mph (88 ft/s) is much worse that one at 30 mph - four times as much energy is involved!)
van der Waals Equation: Real gas data, such as the fact that many real gases cool upon expansion, or the molar volumes tabulated in Table 5.2 (p 191 of Zumdahl, they do not all occupy 22.42 L) is not always as PV = nRT would predict (Zumdahl overheads 44 & 45, figures 5.25, & 5.26, pp 208). Thus the question arises: How can we model the behavior of real gases? The best known equation for real gases is the van der Waals equation. This equation attempts to model real gases by including two obvious differences between real and ideal gases:
To account for attraction van der Waals made the simple assumption that in order to attract, the particles must come close, and in fact collide. The pressure will then be reduced by a constant describing the degree of attraction (specific to each kind of gas) multiplied by the probability of collision which is known to be = (n/V)2 (statistically calculated probability of occupying the same place at the same time). Thus:
Pobs = P - a(n/V)2 is the corrected pressure To account for the actual volume available to a real gas particle we simply have to subtract the volume of the particles in the container, n (vol/mol). If we know the molar volume = b we get:
Vavail = V - bn as the corrected volume Substituting these corrected values into PV = nRT gives the van der Waals Equation:
This gives a better approximation than PV = nRT for most real gases, though it is still just an approximation. The constant a & b are determined empirically. They are sometimes taken as giving an indication of attraction (a) and particle volume (b).
‡ In the animation He atoms are shown with the atoms at the same scale as the distance between atoms at a pressure of 1950 atm.
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© R A Paselk
Last modified 23 February 2009
