### Chemistry End of Chapter Exercises

Using the postulates of the kinetic molecular theory, explain why a gas uniformly fills a container of any shape.

Can the speed of a given molecule in a gas double at constant temperature? Explain your answer.

## Solution

Yes. At any given instant, there are a range of values of molecular speeds in a sample of gas. Any single molecule can speed up or slow down as it collides with other molecules. The average velocity of all the molecules is constant at constant temperature.

Describe what happens to the average kinetic energy of ideal gas molecules when the conditions are changed as follows:

(a) The pressure of the gas is increased by reducing the volume at constant temperature.

(b) The pressure of the gas is increased by increasing the temperature at constant volume.

(c) The average velocity of the molecules is increased by a factor of 2.

The distribution of molecular velocities in a sample of helium is shown in [link]. If the sample is cooled, will the distribution of velocities look more like that of H_{2} or of H_{2}O? Explain your answer.

## Solution

H_{2}O. Cooling slows the velocities of the He atoms, causing them to behave as though they were heavier.

What is the ratio of the average kinetic energy of a SO_{2} molecule to that of an O_{2} molecule in a mixture of two gases? What is the ratio of the root mean square speeds, *u*_{rms}, of the two gases?

A 1-L sample of CO initially at STP is heated to 546 K, and its volume is increased to 2 L.

(a) What effect do these changes have on the number of collisions of the molecules of the gas per unit area of the container wall?

(b) What is the effect on the average kinetic energy of the molecules?

(c) What is the effect on the root mean square speed of the molecules?

## Solution

(a) The number of collisions per unit area of the container wall is constant.

(b) The average kinetic energy doubles.

(c) The root mean square speed increases to $\sqrt{2}$ times its initial value; $u_{rms}$ is proportional to $\sqrt{KE_{avg}}$

The root mean square speed of H_{2} molecules at 25 °C is about 1.6 km/s. What is the root mean square speed of a N_{2} molecule at 25 °C?

Answer the following questions:

(a) Is the pressure of the gas in the hot-air balloon shown at the opening of this chapter greater than, less than, or equal to that of the atmosphere outside the balloon?

(b) Is the density of the gas in the hot-air balloon shown at the opening of this chapter greater than, less than, or equal to that of the atmosphere outside the balloon?

(c) At a pressure of 1 atm and a temperature of 20 °C, dry air has a density of 1.2256 g/L. What is the (average) molar mass of dry air?

(d) The average temperature of the gas in a hot-air balloon is $1.30×10^2$ °F. Calculate its density, assuming the molar mass equals that of dry air.

(e) The lifting capacity of a hot-air balloon is equal to the difference in the mass of the cool air displaced by the balloon and the mass of the gas in the balloon. What is the difference in the mass of 1.00 L of the cool air in part (c) and the hot air in part (d)?

(f) An average balloon has a diameter of 60 feet and a volume of $1.1×10^5\;ft^3$. What is the lifting power of such a balloon? If the weight of the balloon and its rigging is 500 pounds, what is its capacity for carrying passengers and cargo?

(g) A balloon carries 40.0 gallons of liquid propane (density 0.5005 g/L). What volume of CO_{2} and H_{2}O gas is produced by the combustion of this propane?

(h) A balloon flight can last about 90 minutes. If all of the fuel is burned during this time, what is the approximate rate of heat loss (in kJ/min) from the hot air in the bag during the flight?

## Solution

(a) equal; (b) less than; (c) 29.48 g mol^{−1}; (d) 1.0966 g L^{−1}; (e) 0.129 g/L; (f) $4.01×10^5$ g; net lifting capacity = 384 lb; (g) 270 L; (h) 39.1 kJ min^{−1}

Show that the ratio of the rate of diffusion of Gas 1 to the rate of diffusion of Gas 2,$\frac{R_1}{R_2}$ is the same at 0 °C and 100 °C.