Problems & Exercises

Frost damage to most plants occurs at temperatures of $\text{28}\text{.}0\text{º}\text{F}$ or lower. What is this temperature on the Kelvin scale?

A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?

One of the hottest temperatures ever recorded on the surface of Earth was $\text{134}\text{º}\text{F}$ in Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin?

(a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value? (b) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?

How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by $\text{15}\text{º}\text{C}$? Its original height is 321 m and you can assume it is made of steel.

How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature $\text{35}\text{.}0\text{º}\text{C}$ greater than when they were laid? Their original length is 10.0 m.

Global warming will produce rising sea levels partly due to melting ice caps but also due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of $1\text{.}\text{00}\text{º}\text{C}\text{.}$ Note that this calculation is only approximate because ocean warming is not uniform with depth.

(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at $0\text{º}\text{C}$. What is their difference in length at $\text{22}\text{.}0\text{º}\text{C}$? (b) Repeat the calculation for two 30.0-m-long surveyor’s tapes.

Most automobiles have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0-L capacity when at $\text{10}\text{.}\mathrm{0º}\text{C}\text{.}$ What volume of radiator fluid will overflow when the radiator and fluid reach their $\text{95}\text{.}\mathrm{0º}\text{C}$ operating temperature, given that the fluid’s volume coefficient of expansion is $\beta =\text{400}\times {\text{10}}^{–6}/\text{º}\text{C}$? Note that this coefficient is approximate, because most car radiators have operating temperatures of greater than $\text{95}\text{.}0\text{º}\text{C}\text{.}$

(a) The density of water at $0\text{º}\text{C}$ is very nearly $\text{1000}{\text{kg/m}}^3$ (it is actually $9\text{99}\text{.}{\text{84 kg/m}}^3$), whereas the density of ice at $0\text{º}\text{C}$ is $9{\text{17 kg/m}}^3$. Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Convert an absolute pressure of $7\text{.}\text{00}\times {\text{10}}^5{\text{N/m}}^2$ to gauge pressure in ${\text{lb/in}}^2\text{.}$ (This value was stated to be just less than $\text{90}\text{.}{\text{0 lb/in}}^2$ in Example 13.9. Is it?)

To test a balloon, it is placed in a lab and filled with helium. The temperature of the helium is $\text{10}\text{.}0\text{º}\text{C}$ and the pressure is 1.00 atmosphere. The pressure in the lab is maintained. Assume the membrane of the balloon provides a negligible inward pressure, so it is not considered significant. (a) What is the pressure inside the balloon if the helium is replaced with helium that is at $–\text{50}\text{.}0\text{º}\text{C}$? and the balloon is filled until it has a volume of 20.0 times its original volume? (b) What is the gauge pressure? (Assume the pressure in the lab remains at 1.00 atmosphere during the experiment.)

In the text, it was shown that $N/V=2\text{.}\text{68}\times {\text{10}}^{\text{25}}{\text{m}}^{-3}$ for gas at STP. (a) Show that this quantity is equivalent to $N/V=2\text{.}\text{68}\times {\text{10}}^{\text{19}}{\text{cm}}^{-3},$ 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?

An airplane passenger has $\text{100}{\text{cm}}^3$ 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 $7\text{.}\text{50}\times {\text{10}}^4{\text{N/m}}^2?$

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

A bicycle tire has a pressure of $7\text{.}\text{00}\times {\text{10}}^5{\text{N/m}}^2$ 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 $\text{100}{\text{cm}}^3$ at atmospheric pressure? Assume tire temperature and volume remain constant.

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

(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.

Average atomic and molecular speeds $(v_{\text{rms}})$ are large, even at low temperatures. What is $v_{\text{rms}}$ for helium atoms at 5.00 K, just one degree above helium’s liquefaction temperature?

The escape velocity of any object from Earth is 11.2 km/s. (a) Express this speed in m/s and km/h. (b) At what temperature would oxygen molecules (molecular mass is equal to 32.0 g/mol) have an average velocity $v_{\text{rms}}$ equal to Earth’s escape velocity of 11.1 km/s?

Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of $6\text{.}\text{40}\times {\text{10}}^{–\text{14}}\text{J}$. What temperature is needed?

Hydrogen molecules (molecular mass is equal to 2.016 g/mol) have an average velocity $v_{\text{rms}}$ equal to 193 m/s. What is the temperature?

There are two important isotopes of uranium— ${}^{\text{235}}\text{U}$ and ${}^{\text{238}}\text{U}$; these isotopes are nearly identical chemically but have different atomic masses. Only ${}^{\text{235}}\text{U}$ is very useful in nuclear reactors. One of the techniques for separating them (gas diffusion) is based on the different average velocities $v_{\text{rms}}$ of uranium hexafluoride gas, ${\text{UF}}_6$. (a) The molecular masses for ${}^{\text{235}}\text{U}$${\text{UF}}_6$ and ${}^{\text{238}}\text{U}$${\text{UF}}_6$ are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their average velocities? (b) At what temperature would their average velocities differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

(a) What is the vapor pressure of water at $\text{20}\text{.}0\text{º}\text{C}$? (b) What percentage of atmospheric pressure does this correspond to? (c) What percent of $\text{20}\text{.}0\text{º}\text{C}$ air is water vapor if it has 100% relative humidity? (The density of dry air at $\text{20}\text{.}0\text{º}\text{C}$ is $1\text{.}\text{20}{\text{kg/m}}^3$.)

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

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 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%?

The vapor pressure of water at $\text{40}\text{.}0\text{º}\text{C}$ is $7\text{.}\text{34}\times {\text{10}}^3{\text{N/m}}^2$. 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 $(\text{51}\text{.}{\text{1 g/m}}^3)\text{.}$

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?

Atmospheric pressure atop Mt. Everest is $3\text{.}\text{30}\times {\text{10}}^4{\text{N/m}}^2$. (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 $1\text{.}\text{01}\times {\text{10}}^5{\text{N/m}}^2\text{?}$ (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?

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

(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

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) 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

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?