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(a) the combined kinetic energy T of both neutrons in the frame of their centre of inertia and the momentum ofof each neutron in that frame; (b) the velocity of the centre of inertia of this system of particles. Instruction. Make use of the invariant E2— p2c2 remaining con- stant on transition from one inertial reference frame to another (E is the total energy of the system, p is its composite momentum). 1.385. A particle of rest mass mowith kinetic energy T strikes a stationary particle of the same rest mass. Find the rest mass and the velocity of the compound particle formed as a result of the collision. 1.386. How high must be the kinetic energy of a proton striking another, stationary, proton for their combined kinetic energy in the frame of the centre of inertia to be equal to the total kinetic energy of two protons moving toward each other with individual kinetic energies T = 25.0 GeV? 1.387. A stationary particle of rest mass modisintegrates into three particles with rest masses m1, m2, and m3. Find the maximum total energy that, for example, the particle m1may possess. 1.388. A relativistic rocket emits a gas jet with non-relativistic velocity u constant relative to the rocket. Find how the velocity v of the rocket depends on its rest mass (^) m if the initial rest mass of the rocket equals mo.
PART TWO
THERMODYNAMICS
AND MOLECULAR PHYSICS
2.1. EQUATION OF THE GAS STATE. PROCESSES
pV = M RT,
where M is the molar mass.
- Barometric formula: Poe MghIRT where Po is the pressure at the height h = 0.
- Van der Weals equation of gas state (for a mole):
(p -HTI- a 1 ) (V m—b)-=RT,
where VMis the molar volume under given p and T.
2.1. A vessel of volume V = 30 1 contains ideal gas at the tempera-
ture 0 °C. After a portion of the gas has been let out, the pressure in the vessel decreased by Op = 0.78 atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions p = 1.3 WI. 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 differ- ence Op 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 pi= 1.00 atm. Then both vessels were heated to a tem-
perature t 2 = 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 1 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 pres-
sure pc, = 1.0 atm. Find the density of this mixture, assuming the gases to be ideal.
2.5. A vessel of volume V = 7.5 1 contains a mixture of ideal gases
at a temperature T = 300 K: v1= 0.10 mole of oxygen, v2= 0.
mole of nitrogen, and v3= 0.30 mole of carbon dioxide. Assuming the gases to be ideal, find: (a) the pressure of the mixture;
(2.1a)
(2.1b)
(2.1c)
75
h (^) so that its density is the same throughout the volume. Find the temperature gradient dT/dh. 2.14. Suppose the pressure p and the density p of air are related as plpn = const regardless of height (n is a constant here). Find the corresponding temperature gradient. 2.15. 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. 2.16. Assuming the temperature and the molar mass of air, as well as the free-fall acceleration, to be independent of the height, find the difference in heights at which the air densities at the tempe- rature 0 °C differ (a) e times; (b) by = 1.0%. 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 Po. 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 /If is located in the uniform gravi- tational field in which the free-fall acceleration is equal to g. Find the gas pressure as a function of height h, if p = Poat h = 0, and the temperature varies with height as
(a) T = T o(1 — ah); (b) T = T o(1 ah),
where a is a positive constant. 2.20. A horizontal cylinder closed from one end is rotated with a constant angular velocity (0 about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to Po, 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 p = 500 g/1 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 1. Find:
(a) the temperature of the nitrogen at which the pressure can be calculated from an ideal gas law with an error = 10% (as compared with the pressure calculated from the Van der Waals equation of state); (b) the gas pressure at this temperature. 2.23. One mole of a certain gas is contained in a vessel of volume
V = 0.250 1. At a temperature Ti= 300 K the gas pressure is pi
77
= 90 atm, and at a temperature T2 = 350 K the pressure is p, =
= 110 atm. Find the Van der Waals parameters for this gas. 2.24. Find the isothermal compressibility x of a Van der Waals gas as a function of volume V at temperature T.
Note. By definition, x = — I.^ ay 2.25. 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.
2.2. THE FIRST LAW OF THERMODYNAMICS. HEAT CAPACITY
- The first law of thermodynamics: Q= + A ,
where AU is the increment of the internal energy of the system.
- Work performed by gas: A= p dV.
- Internal energy of an ideal gas: m RT pV U C vT =—M M y-1 — y - 1 •
- Molar heat capacity in a polytropic process (p Vn = const): (n— R C — — 1 n — 1 (n — 1) (17 — 1) '
- Internal energy of one mole of a Van der Waals gas:
U=CvT— (^) V ma
2.26. 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 pres- sure and the room's volume is equal to V = 40 m3. 2.27. A thermally insulated vessel containing a gas whose molar mass is equal to M and the ratio of specific heats CpICv = y moves with a velocity v. Find the gas temperature increment resulting from the sudden stoppage of the vessel. 2.28. 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, pi, T1and V 2, p2, 7'2). Find the air temperature and pressure established after the opening of the valve. 2.29. Gaseous hydrogen contained initially under standard con- ditions in a sealed vessel of volume V = 5.0 1 was cooled by AT =
(2.2a)
(2.2b)
(2.2c)
(2.2d)
(2.2e)
