Show that the bulk modulus (Section 12–5) for an ideal gas held at constant temperature is B = P, where P is the pressure.

Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. Assume the atmosphere is about 10 km high and of constant density. What other assumptions did you make?

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant temperature, pressure is inversely proportional to the square of the volume.

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant pressure, the volume varies directly with the 2/3 power of the temperature.

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At 273.15 K and 1.00 atm pressure, 1.00 mole of an ideal gas is found to occupy 22.4 L. Obtain the form of the ideal gas law in this alternate universe, including the value of the gas constant R.

From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth’s atmosphere.

A brass lid screws tightly onto a glass jar at 15°C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 55°C. The inside diameter of the lid is 7.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

A helium balloon has volume V₀ and temperature T₀ at sea level where the pressure is P₀ and the air density is ρ₀. The balloon is allowed to float up in the air to altitude y where the temperature is T₁. Show that the buoyant force does not depend on altitude y. Assume that the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure. [Hint: Assume that the pressure change with altitude is P = P₀ e⁻ᶜʸ , Eq. 13–6c in Chapter 13.]

A copper wire sags 54.0 cm between two utility poles 30.0 m apart when the temperature is -15° C. Estimate the amount of sag when the temperature is + 35° C. [Hint: An estimate can be made by assuming the shape of the wire is approximately an arc of a circle; hard equations can sometimes be solved by guessing values.]

Why snorkels are not 4 feet long. Snorkelers breathe through short tubular “snorkels” while swimming under water very near the surface (Fig. 17–24). One end of the snorkel is in the snorkeler’s mouth and the other end protrudes just above the water’s surface. Unfortunately, snorkels cannot support breathing to any great depth: it is said that a typical snorkeler below a water depth of only about 30 cm cannot draw a breath through a snorkel. Based on this observation, what is the approximate change in a typical person’s lung pressure (in atm) when drawing a breath? (Note that your diaphragm muscles, which expand your lungs, must work also against the extra water pressure.)

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