2.1. A vessel of volume V = 30 l contains ideal gas at the temperature 0 °C. After a portion of the gas has been let out, the pressure in the vessel decreased by Δp = 0.78 atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions ρ = 1.3 g/l.
2.2. Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the other if the pressure difference Δp ≥ 1.10 atm. Initially there was a vacuum in one vessel while the other contained ideal gas at a temperature t1 = 27 °C and pressure p1 = 1.00 atm. Then both vessels were heated to a temperature t2 = 107 °C. Up to what value will the pressure in the first vessel (which had vacuum initially) increase?
2.3. A vessel of volume V = 20 l contains a mixture of hydrogen and helium at a temperature t = 20 °C and pressure p = 2.0 atm. The mass of the mixture is equal to m = 5.0 g. Find the ratio of the mass of hydrogen to that of helium in the given mixture.
2.4. A vessel contains a mixture of nitrogen (m1 = 7.0 g) and carbon dioxide (m2 = 11 g) at a temperature T = 290 K and pressure p0 = 1.0 atm. Find the density of this mixture, assuming the gases to be ideal.
2.6. A vertical cylinder closed from both ends is equipped with an easily moving piston dividing the volume into two parts, each containing one mole of air. In equilibrium at T0 = 300 K the volume of the upper part is η = 4.0 times greater than that of the lower part. At what temperature will the ratio of these volumes be equal to η' = 3.0?
2.7. A vessel of volume V is evacuated by means of a piston air pump. One piston stroke captures the volume ΔV. How many strokes are needed to reduce the pressure in the vessel η times? The process is assumed to be isothermal, and the gas ideal.
2.8. Find the pressure of air in a vessel being evacuated as a function of evacuation time t. The vessel volume is V, the initial pressure is p0. The process is assumed to be isothermal, and the evacuation rate equal to C and independent of pressure.
2.9. A chamber of volume V = 87 l is evacuated by a pump whose evacuation rate (see Note to the foregoing problem) equals C = 10 l/s. How soon will the pressure in the chamber decrease by η = 1000 times?
2.10. A smooth vertical tube having two different sections is open from both ends and equipped with two pistons of different areas (Fig. 2.1). Each piston slides within a respective tube section. One mole of ideal gas is enclosed between the pistons tied with a non-stretchable thread. The cross-sectional area of the upper piston is ΔS = 10 cm2 greater than that of the lower one. The combined mass of the two pistons is equal to m = 5.0 kg. The outside air pressure is p0 = 1.0 atm. By how many kelvins must the gas between the pistons be heated to shift the pistons through l = 5.0 cm?
2.11. Find the maximum attainable temperature of ideal gas in each of the following processes:
2.12. Find the minimum attainable pressure of ideal gas in the process T = T0 + αV2, where T0 and α are positive constants, and V is the volume of one mole of gas. Draw the approximate p vs V plot of this process.
2.13. A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free-fall acceleration is equal to g. The temperature of the nitrogen varies along the height h so that its density is the same throughout the volume. Find the temperature gradient dT/dh.
2.17. An ideal gas of molar mass M is contained in a tall vertical cylindrical vessel whose base area is S and height h. The temperature of the gas is T, its pressure on the bottom base is p0. Assuming the temperature and the free-fall acceleration g to be independent of the height, find the mass of gas in the vessel.
2.18. An ideal gas of molar mass M is contained in a very tall vertical cylindrical vessel in the uniform gravitational field in which the free-fall acceleration equals g. Assuming the gas temperature to be the same and equal to T, find the height at which the centre of gravity of the gas is located.
2.19. An ideal gas of molar mass M is located in the uniform gravitational field in which the free-fall acceleration is equal to g. Find the gas pressure as a function of height h, if p = p0 at h = 0, and the temperature varies with height as
2.20. A horizontal cylinder closed from one end is rotated with a constant angular velocity ω about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to p0, the temperature to T, and the molar mass of air to M. Find the air pressure as a function of the distance r from the rotation axis. The molar mass is assumed to be independent of r.
2.21. Under what pressure will carbon dioxide have the density ρ = 500 g/l at the temperature T = 300 K? Carry out the calculations both for an ideal and for a Van der Waals gas.
2.22. One mole of nitrogen is contained in a vessel of volume V = 1.00 l. Find:
2.23. One mole of a certain gas is contained in a vessel of volume V = 0.250 l. At a temperature T1 = 300 K the gas pressure is p1 = 90 atm, and at a temperature T2 = 350 K the pressure is p2 = 110 atm. Find the Van der Waals parameters for this gas.