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Condition 
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401  A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the freefall 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. 

402  Suppose the pressure p and the density p of air are related as p/r^n = const regardless of height (n is a constant here). Find the corresponding temperature gradient. 

403  Let us assume that air is under standard conditions close to the Earth's surface. Presuming that the temperature and the molar mass of air are independent of height, find the air pressure at the height 5.0 km over the surface and in a mine at the depth 5.0 km below the surface. 

404  Assuming the temperature and the molar mass of air, as well as the freefall acceleration, to be independent of the height, find the difference in heights at which the air densities at the temperature 0 °C differ (a) e times; (b) by = 1.0%. 

405  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 freefall acceleration g to be independent of the height, find the mass of gas in the vessel. 

406  An ideal gas of molar mass M is contained in a very tall vertical cylindrical vessel in the uniform gravitational field in which the freefall 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. 

407  An ideal gas of molar mass M is located in the uniform gravitational field in which the freefall 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 (a) T = T0 (1  ah); (b) T = T0 (1 + ah), where a is a positive constant. 

408  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. 

409  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. 

410  One mole of nitrogen is contained in a vessel of volume V = 1.00 l. Find: (a) the temperature of the nitrogen at which the pressure can be calculated from an ideal gas law with an error η = 10% (as compared with the pressure calculated from the Van der Waals equation of state); (b) the gas pressure at this temperature. 

411  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. 

412  Find the isothermal compressibility x of a Van der Waals gas as a function of volume V at temperature T. Note. By definition, x = ... 

413  Making use of the result obtained in the foregoing problem, find at what temperature the isothermal compressibility x of a Van der Waals gas is greater than that of an ideal gas. Examine the case when the molar volume is much greater than the parameter b. 

414  Demonstrate that the interval energy U of the air in a room is independent of temperature provided the outside pressure p is constant. Calculate U, if p is equal to the normal atmospheric pressure and the room's volume is equal to V = 40 m3. 

415  A thermally insulated vessel containing a gas whose molar mass is equal to M and the ratio of specific heats Cp/Cv = y moves with a velocity v. Find the gas temperature increment resulting from the sudden stoppage of the vessel. 

416  Two thermally insulated vessels 1 and 2 are filled with air and connected by a short tube equipped with a valve. The volumes of the vessels, the pressures and temperatures of air in them are known (V1, p1, T1 and V2, p2, T2). Find the air temperature and pressure established after the opening of the valve. 

417  Gaseous hydrogen contained initially under standard conditions in a sealed vessel of volume V = 5.0 l was cooled by ΔT = 55 K. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas. 

418  What amount of heat is to be transferred to nitrogen in the isobaric heating process for that gas to perform the work A = 2.0 J? 

419  As a result of the isobaric heating by ΔT = 72 K one mole of a certain ideal gas obtains an amount of heat Q = 1.60 kJ. Find the work performed by the gas, the increment of its internal energy, and the value of γ = Cp/CV. 

420  Two moles of a certain ideal gas at a temperature T0 = 300 K were cooled isochorically so that the gas pressure reduced n = 2.0 times. Then, as a result of the isobaric process, the gas expanded till its temperature got back to the initial value. Find the total amount of heat absorbed by the gas in this process. 
