The Barometric Formula: Pressure vs. Altitude
(a) Derive a general formula for the pressure P(z) as a function of
the altitude z, if the pressure at sea level (i.e., at z = 0). Your
formula may involve the temperature T and the molar mass of air
(Mair = 0.0290 kg/mol). We're talking about reasonably small
changes in altitude here, so you can assume that temperature is
essentially independent of z.
Hint: the differential form of the pressuredepth relation might be useful.
(b) If the pressure at sea level is
P0 = 1 atm at T = 25° C, what
would the pressure be at this
same temperature in
Colorado Springs, where the
elevation is roughly 1839 m
above sea level?
Hot-Air Balloon
The gondola of a hot-air balloon has a mass of 50 kg. If it carries two
passengers, each of whom has a mass of 75 kg. The radius of the balloon
itself is 10 m and it's a sunny, 30° C day where the atmospheric pressure is
roughly 1 atm at ground level. What temperature must the air inside the
balloon be in order for it to attain a height of 500 m above the ground?
Gas Expansion
(a) Find the fractional change ΔV/V in the volume of an ideal gas
when it is raised from 20° C to 80° C at constant pressure.
(b) Find the fractional change ΔV/V in the volume of a block of iron
when it is raised from 20° C to 80° C at constant pressure. How
does this compare to your result in part (a)?
Energy Conservation
For each of the situations described on the next slide, use your intuition to
explain the energy flow – i.e. where the energy comes from and where it is
going. Then identify a “system” and use your intuitive explanation to set
up a conservation of energy equation in words (such as “work,” “heat,” or
“change in internal energy”) for that system.
●
Example: Hitting a nail with a hammer
The energy from your body is transferred into the movement of the
hammer. The kinetic energy of the moving hammer is used to move the
nail into the wood. After everything is stationary, the energy is stored in
the increased temperature of the wood and nail. Thus if “wood and nail”
is our system, we can write:
(chemical energy used by your body to swing the hammer) =
(kinetic energy of hammer just before it hits the nail) =
(work done on the system moving nail into wood) =
(energy stored in the increased internal energy of the wood and nail)
●
Situations:
(a) You compress the air in a bicycle pump by pushing down on the
handle very rapidly.
(b) You hold a nail over a bunsen burner.
(c)
High pressure gas in a cylinder pushes a piston outward very
rapidly.
(d) Steam contacts a cold surface and condenses.