13.6 Humidity, evaporation, and boiling  (Page 5/9)

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(a) At what temperature does water boil at an altitude of 1500 m (about 5000 ft) on a day when atmospheric pressure is $8\text{.}\text{59}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\text{?}$ (b) What about at an altitude of 3000 m (about 10,000 ft) when atmospheric pressure is $7\text{.}\text{00}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\text{?}$

What is the atmospheric pressure on top of Mt. Everest on a day when water boils there at a temperature of $\text{70}\text{.}0\text{º}\text{C?}$

$3\text{.}\text{12}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{Pa}$

At a spot in the high Andes, water boils at $\text{80}\text{.}0\text{º}\text{C}$ , greatly reducing the cooking speed of potatoes, for example. What is atmospheric pressure at this location?

What is the relative humidity on a $\text{25}\text{.}0\text{º}\text{C}$ day when the air contains $\text{18}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{g/m}}^{3}$ of water vapor?

78.3%

What is the density of water vapor in ${\text{g/m}}^{3}$ on a hot dry day in the desert when the temperature is $\text{40}\text{.}0\text{º}\text{C}$ and the relative humidity is 6.00%?

A deep-sea diver should breathe a gas mixture that has the same oxygen partial pressure as at sea level, where dry air contains 20.9% oxygen and has a total pressure of $1\text{.}\text{01}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ . (a) What is the partial pressure of oxygen at sea level? (b) If the diver breathes a gas mixture at a pressure of $2\text{.}\text{00}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ , what percent oxygen should it be to have the same oxygen partial pressure as at sea level?

(a) $2\text{.}\text{12}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{Pa}$

(b) $1\text{.}\text{06}\phantom{\rule{0.25em}{0ex}}\text{%}$

The vapor pressure of water at $\text{40}\text{.}0\text{º}\text{C}$ is $7\text{.}\text{34}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ . Using the ideal gas law, calculate the density of water vapor in ${\text{g/m}}^{3}$ that creates a partial pressure equal to this vapor pressure. The result should be the same as the saturation vapor density at that temperature $\left(\text{51}\text{.}{\text{1 g/m}}^{3}\right)\text{.}$

Air in human lungs has a temperature of $\text{37}\text{.}0\text{º}\text{C}$ and a saturation vapor density of $\text{44}\text{.}{\text{0 g/m}}^{3}$ . (a) If 2.00 L of air is exhaled and very dry air inhaled, what is the maximum loss of water vapor by the person? (b) Calculate the partial pressure of water vapor having this density, and compare it with the vapor pressure of $6\text{.}\text{31}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ .

(a) $8\text{.}\text{80}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{g}$

(b) $6\text{.}\text{30}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{Pa}$ ; the two values are nearly identical.

If the relative humidity is 90.0% on a muggy summer morning when the temperature is $\text{20}\text{.}0\text{º}\text{C}$ , what will it be later in the day when the temperature is $\text{30}\text{.}0\text{º}\text{C}$ , assuming the water vapor density remains constant?

Late on an autumn day, the relative humidity is 45.0% and the temperature is $\text{20}\text{.}0\text{º}\text{C}$ . What will the relative humidity be that evening when the temperature has dropped to $\text{10}\text{.}0\text{º}\text{C}$ , assuming constant water vapor density?

82.3%

Atmospheric pressure atop Mt. Everest is $3\text{.}\text{30}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ . (a) What is the partial pressure of oxygen there if it is 20.9% of the air? (b) What percent oxygen should a mountain climber breathe so that its partial pressure is the same as at sea level, where atmospheric pressure is $1\text{.}\text{01}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\text{?}$ (c) One of the most severe problems for those climbing very high mountains is the extreme drying of breathing passages. Why does this drying occur?

What is the dew point (the temperature at which 100% relative humidity would occur) on a day when relative humidity is 39.0% at a temperature of $\text{20}\text{.}0\text{º}\text{C}$ ?

$4\text{.}\text{77}\text{º}\text{C}$

On a certain day, the temperature is $\text{25}\text{.}0\text{º}\text{C}$ and the relative humidity is 90.0%. How many grams of water must condense out of each cubic meter of air if the temperature falls to $\text{15}\text{.}0\text{º}\text{C}$ ? Such a drop in temperature can, thus, produce heavy dew or fog.

Integrated Concepts

The boiling point of water increases with depth because pressure increases with depth. At what depth will fresh water have a boiling point of $\text{150}\text{º}\text{C}$ , if the surface of the water is at sea level?

$\text{38}\text{.}3\phantom{\rule{0.25em}{0ex}}\text{m}$

Integrated Concepts

(a) At what depth in fresh water is the critical pressure of water reached, given that the surface is at sea level? (b) At what temperature will this water boil? (c) Is a significantly higher temperature needed to boil water at a greater depth?

Integrated Concepts

To get an idea of the small effect that temperature has on Archimedes’ principle, calculate the fraction of a copper block’s weight that is supported by the buoyant force in $0\text{º}\text{C}$ water and compare this fraction with the fraction supported in $\text{95}\text{.}0\text{º}\text{C}$ water.

$\frac{\left({F}_{\text{B}}/{w}_{\text{Cu}}\right)}{{\left({F}_{\text{B}}/{w}_{\text{Cu}}\right)}^{\prime }}=1\text{.}\text{02}$ . The buoyant force supports nearly the exact same amount of force on the copper block in both circumstances.

Integrated Concepts

If you want to cook in water at $\text{150}\text{º}\text{C}$ , you need a pressure cooker that can withstand the necessary pressure. (a) What pressure is required for the boiling point of water to be this high? (b) If the lid of the pressure cooker is a disk 25.0 cm in diameter, what force must it be able to withstand at this pressure?

Unreasonable Results

(a) How many moles per cubic meter of an ideal gas are there at a pressure of $1\text{.}\text{00}×{\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ and at $0\text{º}\text{C}$ ? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?

(a) $4\text{.}\text{41}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}{\text{mol/m}}^{3}$

(b) It’s unreasonably large.

(c) At high pressures such as these, the ideal gas law can no longer be applied. As a result, unreasonable answers come up when it is used.

Unreasonable Results

(a) An automobile mechanic claims that an aluminum rod fits loosely into its hole on an aluminum engine block because the engine is hot and the rod is cold. If the hole is 10.0% bigger in diameter than the $\text{22}\text{.}0\text{º}\text{C}$ rod, at what temperature will the rod be the same size as the hole? (b) What is unreasonable about this temperature? (c) Which premise is responsible?

Unreasonable Results

The temperature inside a supernova explosion is said to be $2\text{.}\text{00}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{K}$ . (a) What would the average velocity ${v}_{\text{rms}}$ of hydrogen atoms be? (b) What is unreasonable about this velocity? (c) Which premise or assumption is responsible?

(a) $7\text{.}\text{03}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(b) The velocity is too high—it’s greater than the speed of light.

(c) The assumption that hydrogen inside a supernova behaves as an idea gas is responsible, because of the great temperature and density in the core of a star. Furthermore, when a velocity greater than the speed of light is obtained, classical physics must be replaced by relativity, a subject not yet covered.

Unreasonable Results

Suppose the relative humidity is 80% on a day when the temperature is $\text{30}\text{.}0\text{º}\text{C}$ . (a) What will the relative humidity be if the air cools to $\text{25}\text{.}0\text{º}\text{C}$ and the vapor density remains constant? (b) What is unreasonable about this result? (c) Which premise is responsible?

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