CALC Earth’s Atmosphere. In t he troposphere, the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km, the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship T = T 0 − α y where T 0 is the temperature at the earth’s surface and T temperature at height y , the pressure p at height y is ln ( p p 0 ) = M g R α ln ( T 0 − α y T 0 ) where P 0 is the pressure at the earth’s surface and M is the molar mass for air. The coefficient α is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 C°/100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that α → 0. (c) With α = 0 6 C°/100 m, calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take T 0 = 288 K and p 0 = 1.00 atm.
CALC Earth’s Atmosphere. In t he troposphere, the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km, the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship T = T 0 − α y where T 0 is the temperature at the earth’s surface and T temperature at height y , the pressure p at height y is ln ( p p 0 ) = M g R α ln ( T 0 − α y T 0 ) where P 0 is the pressure at the earth’s surface and M is the molar mass for air. The coefficient α is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 C°/100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that α → 0. (c) With α = 0 6 C°/100 m, calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take T 0 = 288 K and p 0 = 1.00 atm.
CALC Earth’s Atmosphere. In t he troposphere, the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km, the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship
T
=
T
0
−
α
y
where T0 is the temperature at the earth’s surface and T temperature at height y, the pressure p at height y is
ln
(
p
p
0
)
=
M
g
R
α
ln
(
T
0
−
α
y
T
0
)
where P0 is the pressure at the earth’s surface and M is the molar mass for air. The coefficient α is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 C°/100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that α → 0. (c) With α = 0 6 C°/100 m, calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take T0 = 288 K and p0 = 1.00 atm.
A student foolishly attempts to stop a steel bar, of length L = 1 m and at a temperature of 20ºC, from thermally expanding by attaching it to a wooden support with a nail at each end. Steel's Young's modulus is Y = 1.1 × 1011 N/m2 and it's linear thermal expansion coefficient is α = 13 × 10-6 1/C. Randomized Variables
Y = 1.1 × 1011 N/m2α = 13 × 10-6 1/C
What is the volue of the stress, in pascals, that develops due to a rise of temprature to 21 C?
Assuming the nails have a cross- sectional area of A= 10^-5 m^2 all of which is perpendicular to the stress force from the bar, what is the force acting on each due to that temperature rise?
A diver is at 5 meter depth in a fresh water lake. At that depth, the temperature is 15◦C. The pressure in the water at a depth d is given by p(d) = psurface + ρgd, where ρ is the liquid density, and g = 9.81m s−2is the gravitational constant.(a) The diver releases an air bubble of 1cm diameter. What is the diameter of the bubble when it reaches the surface where the temperature is 20◦C?Assume that the bubble temperature is always the same as the surroundingwater.(b) The diver stayed 1h at a 5m depth. The pressure in the diving tank went down from 250 bars to 50 bars. We assume that the volume of the lungs is 5L. Assuming that the water temperature is also 15◦C, how long could the diver have stayed underwater at 20m depth using the same amount of air?(c) The diver is at 5m depth and decides to surface. The air in the lungs is at a constant temperature of 37◦C. What fraction of the air in her/his lungs should the diver inhale/release to maintain a constant lung volume?
The atmospheric pressure on the top of Mt. Everest is 0.333 atm and the average temperature is -25.0 oC. What is the density of air at the top of Mt. Everest in units of mol/L?
Chapter 18 Solutions
University Physics with Modern Physics (14th Edition)
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