21. Kinetic Theory of Ideal Gases

Equation 20.3 is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius. Electrons travel 3.0 km through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least 50 km. What is the maximum possible pressure inside the accelerator tube, assuming T = 20℃? Give your answer in both P_a and atm.

What is the pressure inside a 41.0-L container holding 105.0 kg of argon gas at 21.6°C?

A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and volume = 14L (0.014m³) at T = 295 K . How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.0 L/min?

If a scuba diver fills his lungs to full capacity of 4.8 L when 9.0 m below the surface of sea water, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

A scuba tank when fully charged has an absolute pressure of 185 atm at 18°C. The volume of the tank is 11.3 L. What would the volume of the air be at 1.00 atm and at the same temperature?

How well does the ideal gas law describe the pressurized air in a scuba tank? To fill a typical scuba tank, an air compressor intakes about 2300 L of air at 1.0 atm and compresses this gas into the tank’s 12-L internal volume. If the filling process occurs at 20°C, show that a tank holds about 96 mol of air.

Assuming a typical nitrogen or oxygen molecule is about 0.3 nm in diameter, what percent of the room you are sitting in is taken up by the volume of the molecules themselves?

What is the partial pressure of water on a day when the temperature is 25°C and the relative humidity is 75%?

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 volume occupied by the balloon is then $V=V_0\left(T_1/T_0\right)e^{+cy}$ where $c={\rho }_{0}g/P_0=1.25\times {10}^{-4}{\text{ m}}^{-1}$.

Using the ideal gas law, find an expression for the mean free path ℓ_M that involves pressure and temperature instead of (N/V). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K.

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.]

Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length ℓ at temperature T and pressure P.

(a) Show that the frequency f with which gas molecules strike a wall is ƒ = ($\overline{{\upsilon }_x}$ /2)(P/kT) ℓ² where $\overline{{\upsilon }_x}$ is the average x component of the molecule’s velocity.

(b) Show that the equation can then be written as ƒ≈ Pℓ² /$\sqrt{4mkT}$ where m is the mass of a gas molecule.

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.