Show that the variation of atmospheric pressure with...

Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha} y,$ where $\alpha=\rho_{0} g / P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0,$ and $\rho_{0}$ is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 as $d P=-\rho g d y .$ Also assume the density of air is proportional to the pressure, which, as we will see in Chapter 18 , is equivalent to assuming the temperature of the air is the same at all altitudes.

Transcript

00:01 Okay, so let's look at some little infinitesimal change in height in the atmosphere, d -y, and look how the pressure at the top, or how does the pressure change as we go through this little region.

00:15 So we have two relationships here.

00:18 One, let's assume that the area of this little sliver we've drawn as a, then the pressure through this region or the infinitesimal pressure through this region times.

00:30 The area is like the infinitesimal change in mass.

00:34 We'll write that as d .m.

00:35 Times g because this will be basically the change in force or the change in weight over this region is like the change in pressure times this area.

00:43 And d .m.

00:45 Is just the density of air at this region times the volume or times the area times d .y.

00:52 We'll just write it this way.

00:53 So we have a, d .p equals.

00:59 And really the pressure is going to be decreased.

01:01 As we go up so we should probably put a negative sign here because we are like looking at the the weight is pointing downwards right the force points downwards so we're looking at the pressure differences that'll be relevant so then we have negative row a g d y on this side the a is cancel and so we have d p equals negative row g d y and now let's look at the ideal gas loss really quickly.

01:32 It says that the pressure times the volume is equal to the number of particles times bolts and it's constant times the temperature.

01:38 And if we divide up by the volume on both sides, we can define something called the particle density, which is not to be confused with the number of moles.

01:45 Let's just define as in.

01:47 N over v...

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