PHYSICS 313 Thermodynamics
Midterm exam, Friday October
20 2000
Answer
all questions. All questions have equal value. Allowed aids: two sided formula
sheet, calculator.
1:
At the bottom of a 50m deep lake the pressure is approximately 6atm. Assume the temperature throughout the lake is constant, T=300K. Two air bubbles each of volume 1mm3 are formed near the bottom and rise to the surface.
a: One bubble rises slowly enough that the rise can be taken to be isothermal.
What is the volume of the bubble when it reaches the surface?
b: The other bubble rises fast enough that the process can be taken to be adiabatic.
How big is it when it reaches the surface? (A formula for adiabatic expansion is PVg=const., with g=1.4 for air)
c: Can you give a simple physical explanation for why the isothermal bubble is bigger than the adiabatic bubble?
d: The Sackur-Tetrode formula for the entropy of a monatomic ideal gas is
S=Nk[ln(V/(NvQ) )+5/2]
where
vQ=(2pmkT/h2)-3/2
use this result to find g for the monatomic ideal gas.
2:
A mercury thermometer has a roughly spherical bulb of radius r and a thin cylindrical tube of diameter d in which the mercury column expands and contracts as the temperature changes. Assume that the volume thermal expansion coefficient b for mercury is much larger than the coefficient for the surrounding glass. Also assume that the amount of mercury in the column is much less than the amount in the bulb.
a: It is required that if the temperatures increases by 1oC the height increases by h. Find a
formula for the column diameter d as a function of the bulb radius r.
b: Find the column diameter d numerically if r=2.5mm, b=1.81 10-4 K-1, h=1mm
c: Is the simplifying assumption justified that the amount of mercury in the column is much less than the amount in the bulb?
3:
A hot air balloon has mass M (exclusive of the air inside), air volume V. The average weight of a mole of air is m. The outside temperature is TC. Neglect the volume of the mass M.
a: What must the inside air temperature TH be for the balloon to be able to rise?
b: Give a numerical estimate for the minimum inside temperature assuming M=500kg, V=2000 m3, m=29 g mol-1, TC=290K, 1 atm=105Nm-2, R=8.315 J mol-1K-1.
c: How would you modify result obtained in a:
if the volume Vp of the pay-load cannot be neglected?