### Problems

## 2.1 Molecular Model of an Ideal Gas

The gauge pressure in your car tires is $2.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ at a temperature of $35.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to $-40.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ ? Assume the tires have not gained or lost any air.

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 $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is $60.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ (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?

People buying food in sealed bags at high elevations often notice that the bags are puffed up because the air inside has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of $22.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}.$ When opened at a summer picnic in Santa Fe, New Mexico, at a temperature of $32.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C},$ the volume of the air in the bag is 1.38 times its original volume. What is the pressure of the air?

How many moles are there in (a) 0.0500 g of ${\text{N}}_{2}$ gas $(M=28.0\phantom{\rule{0.2em}{0ex}}\text{g/mol})\text{?}$ (b) 10.0 g of ${\text{CO}}_{2}$ gas $(M=44.0\phantom{\rule{0.2em}{0ex}}\text{g/mol})\text{?}$ (c) How many molecules are present in each case?

A cubic container of volume 2.00 L holds 0.500 mol of nitrogen gas at a temperature of $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}\text{.}$ What is the net force due to the nitrogen on one wall of the container? Compare that force to the sample’s weight.

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 $37.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ (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.

An airplane passenger has $100\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{3}$ 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 $7.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ ?

A company advertises that it delivers helium at a gauge pressure of $1.72\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ 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 $1.01\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ and the temperature in the cylinder and the balloons is $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$.

According to http://hyperphysics.phy-astr.gsu.edu/hbase/solar/venusenv.html, the atmosphere of Venus is approximately $96.5\text{\%}\phantom{\rule{0.2em}{0ex}}{\text{CO}}_{2}$ and $3.5\text{\%}\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}$ by volume. On the surface, where the temperature is about 750 K and the pressure is about 90 atm, what is the density of the atmosphere?

An expensive vacuum system can achieve a pressure as low as $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}\text{.}$ How many molecules are there in a cubic centimeter at this pressure and temperature?

The number density *N*/*V* of gas molecules at a certain location in the space above our planet is about $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{11}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-3}},$ and the pressure is $2.75\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-10}}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ in this space. What is the temperature there?

A bicycle tire contains 2.00 L of gas at an absolute pressure of $7.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ and a temperature of $18.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. What will its pressure be if you let out an amount of air that has a volume of $100\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{3}$ at atmospheric pressure? Assume tire temperature and volume remain constant.

In a common demonstration, a bottle is heated and stoppered with a hard-boiled egg that’s a little bigger than the bottle’s neck. When the bottle is cooled, the pressure difference between inside and outside forces the egg into the bottle. Suppose the bottle has a volume of 0.500 L and the temperature inside it is raised to $80.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ while the pressure remains constant at 1.00 atm because the bottle is open. (a) How many moles of air are inside? (b) Now the egg is put in place, sealing the bottle. What is the gauge pressure inside after the air cools back to the ambient temperature of $25\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ but before the egg is forced into the bottle?

A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of $1.40\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ and a temperature of $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. The cylinder is cooled to dry ice temperature $(\mathrm{-78.5}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C})$ 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?

Find the number of moles in 2.00 L of gas at $35.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and under $7.41\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ 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.

(a) What is the gauge pressure in a $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ 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 $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ ? Assume the temperature remains at $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and the volume remains constant.

## 2.2 Pressure, Temperature, and RMS Speed

In the problems in this section, assume all gases are ideal.

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 $3.0\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2},$ what is the average pressure on that area?

A person is in a closed room (a racquetball court) with $V=453\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ hitting a ball $(m=42.0\phantom{\rule{0.2em}{0ex}}\text{g})$ around at random without any pauses. The average kinetic energy of the ball is 2.30 J. (a) What is the average value of ${v}_{x}^{2}?$ Does it matter which direction you take to be *x*? (b) Applying the methods of this chapter, find the average pressure on the walls? (c) Aside from the presence of only one “molecule” in this problem, what is the main assumption in Pressure, Temperature, and RMS Speed that does not apply here?

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?

Some incandescent light bulbs are filled with argon gas. What is ${v}_{\text{rms}}$ for argon atoms near the filament, assuming their temperature is 2500 K?

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 average kinetic energy in joules of hydrogen atoms on the $5500\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ surface of the Sun? (b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is $6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{K}$ ?

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?

What is the total translational kinetic energy of the air molecules in a room of volume $23\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ if the pressure is $9.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ (the room is at fairly high elevation) and the temperature is $21\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ ? Is any item of data unnecessary for the solution?

The product of the pressure and volume of a sample of hydrogen gas at $0.00\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ 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 $200\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C?}$

What is the gauge pressure inside a tank of $4.86\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{mol}$ of compressed nitrogen with a volume of $6.56\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ if the rms speed is 514 m/s?

If the rms speed of oxygen molecules inside a refrigerator of volume $22.0\phantom{\rule{0.2em}{0ex}}\text{ft}{\text{.}}^{3}$ 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 of any object from Earth is 11.1 km/s. At what temperature would oxygen molecules (molar mass is equal to 32.0 g/mol) have root-mean-square velocity ${v}_{\text{rms}}$ equal to Earth’s escape velocity of 11.1 km/s?

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?

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 $6.40\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-14}}\phantom{\rule{0.2em}{0ex}}\text{J}$. What temperature is needed?

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?

(a) Hydrogen molecules (molar mass is equal to 2.016 g/mol) have ${v}_{\text{rms}}$ equal to 193 m/s. What is the temperature? (b) Much of the gas near the Sun is atomic hydrogen (H rather than ${\text{H}}_{2}).$ Its temperature would have to be $1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{K}$ for the rms speed ${v}_{\text{rms}}$ to equal the escape velocity from the Sun. What is that velocity?

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?

The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, about 0.033%.

Dry air consists of approximately $78\text{\%}\phantom{\rule{0.2em}{0ex}}\text{nitrogen},21\text{\%}\phantom{\rule{0.2em}{0ex}}\text{oxygen},\text{and}\phantom{\rule{0.2em}{0ex}}1\text{\%}\phantom{\rule{0.2em}{0ex}}\text{argon}$ 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) Using data from the previous problem, find the mass of nitrogen, oxygen, and argon in 1 mol of dry air. The molar mass of ${\text{N}}_{2}$ is 28.0 g/mol, that of ${\text{O}}_{2}$ is 32.0 g/mol, and that of argon is 39.9 g/mol. (b) Dry air is mixed with pentane $({\text{C}}_{5}{\text{H}}_{12},$ molar mass 72.2 g/mol), an important constituent of gasoline, in an air-fuel ratio of 15:1 by mass (roughly typical for car engines). Find the partial pressure of pentane in this mixture at an overall pressure of 1.00 atm.

(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?

(a) If the partial pressure of water vapor is 8.05 torr, what is the dew point? $(760\phantom{\rule{0.2em}{0ex}}\text{torr}=1\phantom{\rule{0.2em}{0ex}}\text{atm}=101,325\phantom{\rule{0.2em}{0ex}}\text{Pa})$ (b) On a warm day when the air temperature is $35\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and the dew point is $25\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, what are the partial pressure of the water in the air and the relative humidity?

## 2.3 Heat Capacity and Equipartition of Energy

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 $1.60\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\phantom{\rule{0.2em}{0ex}}\text{J}\text{.}$ 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) How much heat must be added to raise the temperature of 1.5 mol of air from $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ to $33.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ at constant volume? Assume air is completely diatomic. (b) Repeat the problem for the same number of moles of xenon, Xe.

A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of $30.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is cooled to $\mathrm{-40.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?

A sample of neon gas (Ne, molar mass $M=20.2\phantom{\rule{0.2em}{0ex}}\text{g/mol})$ at a temperature of $13.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is put into a steel container of mass 47.2 g that’s at a temperature of $\mathrm{-40.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. The final temperature is$\mathrm{-28.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon?

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 $12.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. How much heat has to be removed to reach a temperature of $\mathrm{-20.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ ? 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.)

A sealed room has a volume of $24\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$. It’s filled with air, which may be assumed to be diatomic, at a temperature of $24\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and a pressure of $9.83\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{Pa}\text{.}$ A 1.00-kg block of ice at its melting point is placed in the room. Assume the walls of the room transfer no heat. What is the equilibrium temperature?

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 $25\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is mixed with oxygen at $35\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ 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.

Professional divers sometimes use heliox, consisting of $79\%$ helium and $21\%$ oxygen by mole. Suppose a perfectly rigid scuba tank with a volume of 11 L contains heliox at an absolute pressure of $2.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ at a temperature of $31\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (a) How many moles of helium and how many moles of oxygen are in the tank? (b) The diver goes down to a point where the sea temperature is $27\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ while using a negligible amount of the mixture. As the gas in the tank reaches this new temperature, how much heat is removed from it?

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, $\mathrm{-88}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, is mixed with 4.0 mol of air (assumed diatomic) at $30\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. What is the final temperature of the mixture? Use the measured heat capacity of ${\text{N}}_{2}\text{O}$ at $25\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, which is $30.4\phantom{\rule{0.2em}{0ex}}\text{J/mol}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (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.)

## 2.4 Distribution of Molecular Speeds

In a sample of hydrogen sulfide $(M=34.1\phantom{\rule{0.2em}{0ex}}\text{g/mol})$ at a temperature of $3.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}\text{K},$ estimate the ratio of the number of molecules that have speeds very close to ${v}_{\text{rms}}$ to the number that have speeds very close to $2{v}_{\text{rms}}.$

Using the approximation ${\int}_{{v}_{1}}^{{v}_{1}+\text{\Delta}v}\phantom{\rule{0.2em}{0ex}}f(v)dv\approx f({v}_{1})\text{\Delta}v$ for small $\text{\Delta}v$, estimate the fraction of nitrogen molecules at a temperature of $3.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}\text{K}$ that have speeds between 290 m/s and 291 m/s.

Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between $3.45\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-21}}\phantom{\rule{0.2em}{0ex}}\text{J}$ and$3.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-21}}\phantom{\rule{0.2em}{0ex}}\text{J}$.

By counting squares in the following figure, estimate the fraction of argon atoms at $T=300\phantom{\rule{0.2em}{0ex}}\text{K}$ 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.

Using a numerical integration method such as Simpson’s rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen $\left({\text{O}}_{2}\right)$ is 32.0 g/mol. A precision to two significant digits is enough.

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

Repeat the preceding problem for nitrogen molecules at 2950 K.

At what temperature is the average speed of carbon dioxide molecules $(M=44.0\phantom{\rule{0.2em}{0ex}}\text{g/mol})$ 510 m/s?

The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)

a) At what temperature do oxygen molecules have the same average speed as helium atoms $(M=4.00\phantom{\rule{0.2em}{0ex}}\text{g/mol})$ 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?