Chapter 10

A person running a fever has a body temperature of \(40^{\circ} \mathrm{C} .\) What is this temperature on the Fahrenheit scale?

Convert the following to Celsius readings: (a) \(80^{\circ} \mathrm{F}\), (b) \(0^{\circ} \mathrm{F},\) and \((\mathrm{c})-10^{\circ} \mathrm{F}\).

Convert the following to Fahrenheit readings: (a) \(120^{\circ} \mathrm{C}\) (b) \(12^{\circ} \mathrm{C}\) and \((\mathrm{c})-5^{\circ} \mathrm{C}\).

Which is the lower temperature: (a) \(245^{\circ} \mathrm{C}\) or \(245^{\circ} \mathrm{F}\) ? (b) \(200^{\circ} \mathrm{C}\) or \(375^{\circ} \mathrm{F} ?\)?

The coldest inhabited village in the world is Oymyakon, a town located in eastern Siberia, where it gets as cold as \(-94^{\circ} \mathrm{F}\). What is this temperature on the Celsius scale?

The highest and lowest recorded air temperatures in the world are, respectively, \(58^{\circ} \mathrm{C}\) (Libya, 1922 ) and \(-89^{\circ} \mathrm{C}\) (Antarctica, 1983 ). What are these temperatures on the Fahrenheit scale?

The highest and lowest recorded air temperatures in the United States are, respectively, \(134^{\circ} \mathrm{F}\) (Death Valley, California, 1913 ) and \(-80^{\circ} \mathrm{F}\) (Prospect Creek, Alaska, 1971). What are these temperatures on the Celsius scale?

During open-heart surgery it is common to cool the patient's body down to slow body processes and gain an extra margin of safety. A drop of \(8.5^{\circ} \mathrm{C}\) is typical in these types of operations. If a patient's normal body temperature is \(98.2^{\circ} \mathrm{F}\), what is her final temperature in both Celsius and Fahrenheit?

In the troposphere (the lowest part of the atmosphere), the temperature decreases rather uniformly with altitude at a so-called "lapse" rate of about \(6.5^{\circ} \mathrm{C} / \mathrm{km}\). What are the temperatures (a) near the top of the troposphere (which has an average thickness of \(11 \mathrm{~km}\) ) and (b) outside a commercial aircraft flying at a cruising altitude of \(34000 \mathrm{ft} ?\) (Assume that the ground temperature is normal room temperature.)

The temperature drops from \(60^{\circ} \mathrm{F}\) during the day to \(35^{\circ} \mathrm{F}\) during the night. (a) The corresponding temperature drop on the Celsius scale is (1) greater than, (2) the same as, or (3) less than. Explain. (b) Compute the temperature drop on the Celsius scale.

There is one temperature at which the Celsius and Fahrenheit scales have the same reading. (a) To find that temperature, would you set \((1) 5 T_{\mathrm{F}}=9 T_{\mathrm{C}}(2) 9 T_{\mathrm{F}}=5 T_{C}\) C or (3) \(T_{\mathrm{F}}=T_{\mathrm{C}}\) ? Why? (b) Find the temperature.

Convert the following temperatures to absolute temperatures in kelvins: (a) \(0^{\circ} \mathrm{C}\) (b) \(100^{\circ} \mathrm{C}\) (c) \(20^{\circ} \mathrm{C},\) and (d) \(-35^{\circ} \mathrm{C}\).

Convert the following temperatures to Celsius: (a) \(0 \mathrm{~K}\), (b) \(250 \mathrm{~K},\) (c) \(273 \mathrm{~K},\) and (d) \(325 \mathrm{~K}\).

When lightning strikes, it can heat the air around it to more than \(30000 \mathrm{~K}\), five times the surface temperature of the Sun. (a) What is this temperature on the Fahrenheit and Celsius scales? (b) The temperature is sometimes reported to be \(30000^{\circ} \mathrm{C}\). Assuming that \(30000 \mathrm{~K}\) is correct, what is the percentage error of this Celsius value?

In a constant volume gas thermometer, if the pressure of the gas decreases, has the temperature of the (2) decreased, or (3) remained the same? \(\operatorname{gas}(1)\) increased, Why? (b) The initial absolute pressure of a gas is \(1000 \mathrm{~Pa}\) at room temperature \(\left(20^{\circ} \mathrm{C}\right)\). If the pressure increases to \(1500 \mathrm{~Pa},\) what is the new Celsius temperature?

If the pressure of an ideal gas is doubled while its absolute temperature is halved, what is the ratio of the final volume to the initial volume?

Show that 1.00 mol of ideal gas under STP occupies a volume of \(0.0224 \mathrm{~m}^{3}=22.4 \mathrm{~L}\).

What volume is occupied by \(160 \mathrm{~g}\) of oxygen under a pressure of \(2.00 \mathrm{~atm}\) and a temperature of \(300 \mathrm{~K} ?\)

An athlete has a large lung capacity, 7.0 L. Assuming air to be an ideal gas, how many molecules of air are in the athlete's lungs when the air temperature in the lungs is \(37^{\circ} \mathrm{C}\) under normal atmospheric pressure?

Is there a temperature that has the same numerical value on the Kelvin and the Fahrenheit scales? Justify your answer.

A husband buys a helium-filled anniversary balloon for his wife. The balloon has a volume of \(3.5 \mathrm{~L}\) in the warm store at \(74^{\circ} \mathrm{F}\). When he takes it outside, where the temperature is \(48^{\circ} \mathrm{F}\), he finds it has shrunk. By how much has the volume decreased?

An automobile tire is filled to an absolute pressure of 3.0 atm at a temperature of \(30^{\circ} \mathrm{C}\). Later it is driven to a place where the temperature is only \(-20^{\circ} \mathrm{C} .\) What is the absolute pressure of the tire at the cold place? (Assume that the air in the tire behaves as an ideal gas and the volume is constant.)

On a warm day \(\left(92^{\circ} \mathrm{F}\right),\) an air-filled balloon occupies a volume of \(0.200 \mathrm{~m}^{3}\) and has a pressure of \(20.0 \mathrm{lb} / \mathrm{in}^{2}\). If the balloon is cooled to \(32^{\circ} \mathrm{F}\) in a refrigerator while its pressure is reduced to \(14.7 \mathrm{lb} / \mathrm{in}^{2},\) what is the volume of the air in the container? (Assume that the air behaves as an ideal gas.)

A steel-belted radial automobile tire is inflated to a gauge pressure of \(30.0 \mathrm{lb} / \mathrm{in}^{2}\) when the temperature is \(61^{\circ} \mathrm{F}\). Later in the day, the temperature rises to \(100^{\circ} \mathrm{F}\) Assuming the volume of the tire remains constant, what is the tire's pressure at the elevated temperature? [Hint: Remember that the ideal gas law uses absolute pressure.]

\- A scuba diver takes a tank of air on a deep dive. The tank's volume is 10 Land it is completely filled with air at an absolute pressure of \(232 \mathrm{~atm}\) at the start of the dive. The air temperature at the surface is \(94^{\circ} \mathrm{F}\) and the diver ends up in deep water at \(60^{\circ} \mathrm{F}\). Assuming thermal equilibrium and neglecting air loss, determine the absolute internal pressure of the air when it is cold.

(a) If the temperature of an ideal gas increases and its volume decreases, will the pressure of the gas (1) increase, (2) remain the same, or (3) decrease? Why? (b) The Kelvin temperature of an ideal gas is doubled and its volume is halved. How is the pressure affected?

If \(2.4 \mathrm{~m}^{3}\) of a gas initially at STP is compressed to \(1.6 \mathrm{~m}^{3}\) and its temperature is raised to \(30^{\circ} \mathrm{C}\), what is its final pressure?

The pressure on a low-density gas in a cylinder is kept constant as its temperature is increased. (a) Does (2) decrease, or the volume of the gas (1) increase, (3) remain the same? Why? (b) If the temperature is increased from \(10^{\circ} \mathrm{C}\) to \(40{ }^{\circ} \mathrm{C},\) what is the percentage change in the volume of the gas?

A pie plate is filled up to the brim with pumpkin pie filling. The pie plate is made of Pyrex and its expansion can be neglected. It is a cylinder with an inside depth of \(2.10 \mathrm{~cm}\) and an inside diameter of \(30.0 \mathrm{~cm}\). It is prepared at a room temperature of \(68^{\circ} \mathrm{F}\) and placed in an oven at \(400^{\circ} \mathrm{F}\). When it taken out, 151 cc of the pie filling has flowed out and over the rim. Determine the coefficient of volume expansion of the pie filling, assuming it is a fluid.

A circular piece is cut from an aluminum sheet at room temperature. (a) When the sheet is then placed in an oven, will the hole (1) get larger, (2) get smaller, or (3) remain the same? Why? (b) If the diameter of the hole is \(8.00 \mathrm{~cm}\) at \(20^{\circ} \mathrm{C}\) and the temperature of the oven is \(150^{\circ} \mathrm{C}\), what will be the new area of the hole?

One morning, an employee at a rental car company fills a car's steel gas tank to the top and then parks the car a short distance away. (a) That afternoon, when the temperature increases, will any gas overflow? Why? (b) If the temperatures in the morning and afternoon are, respectively, \(10^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\) and the gas tank can hold 25 gal in the morning, how much gas will be lost? (Neglect the expansion of the tank.)

A Pyrex beaker that has a capacity of \(1000 \mathrm{~cm}^{3}\) at \(20^{\circ} \mathrm{C}\) contains \(990 \mathrm{~cm}^{3}\) of mercury at that temperature. Is there some temperature at which the mercury will completely fill the beaker? Justify your answer. (Assume that no mass is lost by vaporization and include the expansion of the beaker.)

If the average kinetic energy per molecule of a monatomic gas is \(7.0 \times 10^{-21} \mathrm{~J},\) what is the Celsius temperature of the gas?

What is the average kinetic energy per molecule in a monatomic gas at (a) \(10^{\circ} \mathrm{C}\) and (b) \(90^{\circ} \mathrm{C} ?\)

What is the rms speed of the molecules in low-density oxygen gas at \(0^{\circ} \mathrm{C} ?\) (The mass of an oxygen molecule, \(\mathrm{O}_{2},\) is \(\left.5.31 \times 10^{-26} \mathrm{~kg}\right)\).

(a) What is the average kinetic energy per molecule of a monatomic gas at a temperature of \(25^{\circ} \mathrm{C} ?\) (b) What is the rms speed of the molecules if the gas is helium? (A helium molecule consists of a single atom of mass \(6.65 \times 10^{-27} \mathrm{~kg}\).

A quantity of an ideal gas is at \(0^{\circ} \mathrm{C}\). An equal quantity of another ideal gas is at twice the absolute temperature. What is its Celsius temperature?

A sample of oxygen \(\left(\mathrm{O}_{2}\right)\) and another sample of nitrogen \(\left(\mathrm{N}_{2}\right)\) are at the same temperature. (a) The rms speed of the nitrogen sample is (1) greater than, (2) the same as, or (3) less than the rms speed of the oxygen sample. Explain. (b) Calculate the ratio of the rms speed in the nitrogen sample to in the oxygen sample.

If the temperature of an ideal gas increases from \(300 \mathrm{~K}\) to \(600 \mathrm{~K},\) what happens to the rms speed of the gas molecules?

If the temperature of an ideal gas is raised from \(25^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\), how much faster is the new rms speed of the gas molecules?

If the rms speed of the molecules in an ideal gas at \(20^{\circ} \mathrm{C}\) increases by a factor of \(2,\) what is the new Celsius temperature?

During the race to develop the atomic bomb in World War II, it was necessary to separate a lighter isotope of uranium (U-235 was the fissionable one needed for bomb material) from a heavier variety (U-238). The uranium was converted into a gas, uranium hexafluoride (UF \(_{6}\) ), and the two uranium isotopes were separated by gaseous diffusion using the difference in their rms speeds. As a two-component molecular mixture at room temperature, which of the two types of molecules would be moving faster, on average: (1) \({ }^{235} \mathrm{UF}_{6}\) or (2) \(^{238} \mathrm{UF}_{6}\). Or (3) would they move equally fast? Explain. (b) Determine the ratio of their rms speeds, light molecule to heavy molecule. Treat the molecules as ideal gases and neglect rotations and/or vibrations of the molecules. The masses of the three atoms in atomic mass units are 238 and 235 for the two uranium isotopes and 19 for fluorine.

For an average molecule of \(\mathrm{N}_{2}\) gas at \(10^{\circ} \mathrm{C}\), what are its (a) translational kinetic energy, (b) rotational kinetic energy, and (c) total energy? Repeat for He gas at the same temperature.

A diatomic gas has a certain total kinetic energy at \(25^{\circ} \mathrm{C} .\) If a monatomic gas of the same number of molecules as the diatomic gas has the same total kinetic energy, what is the Celsius temperature of the monatomic gas?

An ideal gas sample occupies a container of volume 0.75 Lat STP. Find (a) the number of moles and (b) the number of moleculesin in the sample. (c) If the gas is carbon monoxide (CO), what is the sample's mass?

\(2.00 \mathrm{~mol}\) of a monatomic gas at atmospheric pressure has a total internal energy of \(7.48 \times 10^{3} \mathrm{~J}\). What is the volume occupied a rigid cylinder by the gas?

An ideal gas in a cylinder is at \(20^{\circ} \mathrm{C}\) and \(2.0 \mathrm{~atm}\). If it is heated so its rms speed increases by \(20 \%,\) what is its new pressure?

The escape speed from the Earth is about \(11000 \mathrm{~m} / \mathrm{s}\) (Section 7.5). Assume that for a given type of gas to eventually escape the Earth's atmosphere, its average molecular speed must be about \(10 \%\) of the escape speed. (a) Which gas would be more likely to escape the Earth: (1) oxygen, (2) nitrogen, or (3) helium? (b) Assuming a temperature of \(-40^{\circ} \mathrm{F}\) in the upper atmosphere, determine the rms speed of a molecule of oxygen. Is it enough to escape the Earth? (Data: The mass of an oxygen molecule is \(5.34 \times 10^{-26} \mathrm{~kg}\), that of a nitrogen molecule is \(4.68 \times 10^{-26} \mathrm{~kg},\) and that of a helium molecule is \(6.68 \times 10^{-27} \mathrm{~kg}\).