# 13.3 The ideal gas law  (Page 7/11)

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In the text, it was shown that $N/V=2\text{.}\text{68}×{\text{10}}^{\text{25}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-3}$ for gas at STP. (a) Show that this quantity is equivalent to $N/V=2\text{.}\text{68}×{\text{10}}^{\text{19}}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{-3},$ as stated. (b) About how many atoms are there in one ${\text{μm}}^{3}$ (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules?

Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at $\text{37}\text{.}0\text{º}\text{C}$ (body temperature).

$7\text{.}\text{86}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{mol}$

An airplane passenger has $\text{100}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to $7\text{.}\text{50}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}?$

(a) What is the volume (in ${\text{km}}^{3}$ ) of Avogadro’s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.

(a) $6\text{.}\text{02}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{km}}^{3}$

(b) $6\text{.}\text{02}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{km}$

An expensive vacuum system can achieve a pressure as low as $1\text{.}\text{00}×{\text{10}}^{–7}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ at $\text{20}\text{º}\text{C}$ . How many atoms are there in a cubic centimeter at this pressure and temperature?

The number density of gas atoms at a certain location in the space above our planet is about $1\text{.}\text{00}×{\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-3},$ and the pressure is $2\text{.}\text{75}×{\text{10}}^{–\text{10}}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ in this space. What is the temperature there?

$-\text{73}\text{.}9\text{º}\text{C}$

A bicycle tire has a pressure of $7\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ at a temperature of $\text{18}\text{.}0\text{º}\text{C}$ and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of $\text{100}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ at atmospheric pressure? Assume tire temperature and volume remain constant.

A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of $1\text{.}\text{40}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ and a temperature of $\text{25}\text{.}0\text{º}\text{C}$ . Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature $\left(–\text{78}\text{.}5\text{º}\text{C}\right)$ to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution?

(a) $9\text{.}\text{14}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$

(b) $8\text{.}\text{23}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$

(c) 2.16 K

(d) No. The final temperature needed is much too low to be easily achieved for a large object.

Find the number of moles in 2.00 L of gas at $\text{35}\text{.}0\text{º}\text{C}$ and under $7\text{.}\text{41}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ of pressure.

Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.

41 km

(a) What is the gauge pressure in a $\text{25}\text{.}0\text{º}\text{C}$ car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and $\text{25}\text{.}0\text{º}\text{C}$ ? Assume the temperature returns to $\text{25}\text{.}0\text{º}\text{C}$ and the volume remains constant.

(a) In the deep space between galaxies, the density of atoms is as low as ${\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{atoms/m}}^{3},$ and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in ${\text{m}}^{3}$ ) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

(a) $3\text{.}7×{\text{10}}^{-\text{17}}\phantom{\rule{0.25em}{0ex}}\text{Pa}$

(b) $6\text{.}0×{\text{10}}^{\text{17}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$

(c) $8\text{.}4×{\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{km}$

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