Problems

Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of $20.0\text{°}\text{C}$. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is $60.0\text{°}\text{C}$ (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. Is this effect significant?

How many moles are there in (a) 0.0500 g of ${\text{N}}_2$ gas $(M=28.0\text{g/mol})\text{?}$ (b) 10.0 g of ${\text{CO}}_2$ gas $(M=44.0\text{g/mol})\text{?}$ (c) How many molecules are present in each case?

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 $37.0\text{°}\text{C}$ (body temperature) and that the total volume in the lungs is several times the amount inhaled in a typical breath as given in Example 2.2.

A company advertises that it delivers helium at a gauge pressure of $1.72\times {10}^7\text{Pa}$ in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is $1.01\times {10}^5\text{Pa}$ and the temperature in the cylinder and the balloons is $25.0\text{°C}$.

An expensive vacuum system can achieve a pressure as low as $1.00\times {10}^{\mathrm{-7}}{\text{N/m}}^2$ at $20.0\text{°}\text{C}\text{.}$ How many molecules are there in a cubic centimeter at this pressure and temperature?

A bicycle tire contains 2.00 L of gas at an absolute pressure of $7.00\times {10}^5{\text{N/m}}^2$ and a temperature of $18.0\text{°}\text{C}$. What will its pressure be if you let out an amount of air that has a volume of $100{\text{cm}}^3$ 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 $1.40\times {10}^7{\text{N/m}}^2$ and a temperature of $25.0\text{°C}$. The cylinder is cooled to dry ice temperature $(\mathrm{-78.5}\text{°}\text{C})$ 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 as in part (c) appear to be a practical solution?

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.

A person hits a tennis ball with a mass of 0.058 kg against a wall. The average component of the ball’s velocity perpendicular to the wall is 11 m/s, and the ball hits the wall every 2.1 s on average, rebounding with the opposite perpendicular velocity component. (a) What is the average force exerted on the wall? (b) If the part of the wall the person hits has an area of $3.0{\text{m}}^2,$ what is the average pressure on that area?

Five bicyclists are riding at the following speeds: 5.4 m/s, 5.7 m/s, 5.8 m/s, 6.0 m/s, and 6.5 m/s. (a) What is their average speed? (b) What is their rms speed?

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

What is the ratio of the average translational kinetic energy of a nitrogen molecule at a temperature of 300 K to the gravitational potential energy of a nitrogen-molecule−Earth system at the ceiling of a 3-m-tall room with respect to the same system with the molecule at the floor?

The product of the pressure and volume of a sample of hydrogen gas at $0.00\text{°}\text{C}$ is 80.0 J. (a) How many moles of hydrogen are present? (b) What is the average translational kinetic energy of the hydrogen molecules? (c) What is the value of the product of pressure and volume at $200\text{°}\text{C?}$

If the rms speed of oxygen molecules inside a refrigerator of volume $22.0\text{ft}{\text{.}}^3$ is 465 m/s, what is the partial pressure of the oxygen? There are 5.71 moles of oxygen in the refrigerator, and the molar mass of oxygen is 32.0 g/mol.

The escape velocity from the Moon is much smaller than that from the Earth, only 2.38 km/s. At what temperature would hydrogen molecules (molar mass is equal to 2.016 g/mol) have a root-mean-square velocity $v_{\text{rms}}$ equal to the Moon’s escape velocity?

Suppose that the typical speed $(v_{\text{rms}})$ of carbon dioxide molecules (molar mass is 44.0 g/mol) in a flame is found to be 1350 m/s. What temperature does this indicate?

There are two important isotopes of uranium, ${}^{235}\text{U}$ and ${}^{238}\text{U}$; these isotopes are nearly identical chemically but have different atomic masses. Only ${}^{235}\text{U}$ is very useful in nuclear reactors. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, ${\text{UF}}_6$. (a) The molar masses of ${}^{235}\text{U}$ and ${}^{238}{\text{UF}}_6$ are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their typical speeds $v_{\text{rms}}$? (b) At what temperature would their typical speeds differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

Dry air consists of approximately $78\text{\%}\text{nitrogen},21\text{\%}\text{oxygen},\text{and}1\text{\%}\text{argon}$ by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of 293 K. How much oxygen does it contain in moles?

(a) Given that air is $21\%$ oxygen, find the minimum atmospheric pressure that gives a relatively safe partial pressure of oxygen of 0.16 atm. (b) What is the minimum pressure that gives a partial pressure of oxygen above the quickly fatal level of 0.06 atm? (c) The air pressure at the summit of Mount Everest (8848 m) is 0.334 atm. Why have a few people climbed it without oxygen, while some who have tried, even though they had trained at high elevation, had to turn back?

To give a helium atom nonzero angular momentum requires about 21.2 eV of energy (that is, 21.2 eV is the difference between the energies of the lowest-energy or ground state and the lowest-energy state with angular momentum). The electron-volt or eV is defined as $1.60\times {10}^{\mathrm{-19}}\text{J}\text{.}$ Find the temperature T where this amount of energy equals $k_{\text{B}}T\text{/}2.$ Does this explain why we can ignore the rotational energy of helium for most purposes? (The results for other monatomic gases, and for diatomic gases rotating around the axis connecting the two atoms, have comparable orders of magnitude.)

A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of $30.0\text{°C}$ is cooled to $\mathrm{-40.0}\text{°C}$. In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?

A steel container of mass 135 g contains 24.0 g of ammonia, ${\text{NH}}_3$, which has a molar mass of 17.0 g/mol. The container and gas are in equilibrium at $12.0\text{°C}$. How much heat has to be removed to reach a temperature of $\mathrm{-20.0}\text{°C}$ ? Ignore the change in volume of the steel and treat the ammonia as an ideal gas with 6 degrees of freedom (3 translational and 3 rotational; at room temperature, the vibrational degree of freedom is inactive.)

Heliox, a mixture of helium and oxygen, is sometimes given to hospital patients who have trouble breathing, because the low mass of helium makes it easier to breathe than air. Suppose helium at $25\text{°C}$ is mixed with oxygen at $35\text{°C}$ to make a mixture that is $70\%$ helium by mole. What is the final temperature? Ignore any heat flow to or from the surroundings, and assume the final volume is the sum of the initial volumes.

In car racing, one advantage of mixing liquid nitrous oxide $({\text{N}}_2\text{O})$ with air is that the boiling of the “nitrous” absorbs latent heat of vaporization and thus cools the air and ultimately the fuel-air mixture, allowing more fuel-air mixture to go into each cylinder. As a very rough look at this process, suppose 1.0 mol of nitrous oxide gas at its boiling point, $\mathrm{-88}\text{°C}$, is mixed with 4.0 mol of air (assumed diatomic) at $30\text{°C}$. What is the final temperature of the mixture? Use the measured heat capacity of ${\text{N}}_2\text{O}$ at $25\text{°C}$, which is $30.4\text{J/mol}\text{°C}$. (The primary advantage of nitrous oxide is that it consists of 1/3 oxygen, which is more than air contains, so it supplies more oxygen to burn the fuel. Another advantage is that its decomposition into nitrogen and oxygen releases energy in the cylinder.)

Using the approximation ${\displaystyle {\int }_{v_1}^{v_1+\text{Δ}v}f(v)dv\approx f(v_1)\text{Δ}v}$ for small $\text{Δ}v$, estimate the fraction of nitrogen molecules at a temperature of $3.00\times {10}^2\text{K}$ that have speeds between 290 m/s and 291 m/s.

By counting squares in the following figure, estimate the fraction of argon atoms at $T=300\text{K}$ that have speeds between 600 m/s and 700 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x-axis times its height as measured on the y-axis, with the units given on those axes.

Find (a) the most probable speed, (b) the average speed, and (c) the rms speed for nitrogen molecules at 295 K.

At what temperature is the average speed of carbon dioxide molecules $(M=44.0\text{g/mol})$ 510 m/s?

a) At what temperature do oxygen molecules have the same average speed as helium atoms $(M=4.00\text{g/mol})$ have at 300 K? b) What is the answer to the same question about most probable speeds? c) What is the answer to the same question about rms speeds?