Concept explainers
An air bubble starts rising from the bottom of a lake. Its diameter is 3.60 mm at the bottom and 4.00 mm at the surface. The depth of the lake is 2.50 m, and the temperature at the surface is 40.0°C. What is the temperature at the bottom of the lake? Consider the atmospheric pressure to be 1.01 × 105 Pa and the density of water to be 1.00 × 103 kg/m3. Model the air as an ideal gas.
53. Use the
Use the ideal gas law (Eq. 19.17).
The volume ratio at the bottom and top of the lake can be calculated with the diameters given.
The temperature at the bottom of the lake.
Answer to Problem 53PQ
The temperature at the bottom of the lake is
Explanation of Solution
The pressure at the surface of the lake is the atmospheric pressure and the pressure at the bottom of the lake is the sum of atmospheric pressure and pressure due to the water column.
Write an expression for the pressure at the bottom of the lake.
Here,
Write an expression for the pressure due to the water column.
Here,
Substitute equation (II) in equation (I).
Write an expression for the pressure at the top of the lake.
Here,
Write an expression for the temperature at the bottom of the lake.
Here,
Write an expression for the volume at the bottom of the lake.
Here,
Write an expression for the volume at the surface of the lake.
Here,
Substitute equation (III), (IV), (VI) and (VII) in equation (V).
Conclusion:
Substitute
Thus, the temperature at the bottom of the lake is
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Chapter 19 Solutions
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