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= 55 K. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas. 2.30. 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?
2.31. As a result of the isobaric heating by AT = 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 y = Cp/Cv.
2.32. Two moles of a certain ideal gas at a temperature To= 300 K were cooled isochorically so that the gas pressure reduced n = 2. 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.
2.33. Calculate the value of y = Cp/Cv for a gaseous mixture con-
sisting of vi= 2.0 moles of oxygen and v2= 3.0 moles of carbon dioxide. The gases are assumed to be ideal. 2.34. Find the specific heat capacities cvand cpfor a gaseous mix- ture consisting of 7.0 g of nitrogen and 20 g of argon. The gases are assumed to be ideal. 2.35. One mole of a certain ideal gas is contained under a weight-
less piston of a vertical cylinder at a temperature T. The space over
the piston opens into the atmosphere. What work has to be performed in order to increase isothermally the gas volume under the piston it times by slowly raising the piston? The friction of the piston against the cylinder walls is negligibly small. 2.36. A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volume Vo, in which an
ideal gas is contained under the same pressure Poand at the same tem-
perature. What work has to be performed in order to increase isother- mally the volume of one part of gas i1times compared to that of the other by slowly moving the piston? 2.37. Three moles of an ideal gas being initially at a temperature
T o = 273 K were isothermally expanded n = 5.0 times its initial
volume and then isochorically heated so that the pressure in the final state became equal to that in the initial state. The total amount of
heat transferred to the gas during the process equals Q =^ 80 kJ.
Find the ratio y = Cp/Cv for this gas.
2.38. Draw the approximate plots of isochoric, isobaric, isother- mal, and adiabatic processes for the case of an ideal gas, using the following variables:
(a) p, T; (b) V, T.
2.39. One mole of oxygen being initially at a temperature To = = 290 K is adiabatically compressed to increase its pressure = 10.0 times. Find: (a) the gas temperature after the compression; (b) the work that has been performed on the gas. 2.40. A certain mass of nitrogen was compressed ii = 5.0 times 79
(in terms of volume), first adiabatically, and then isothermally. In both cases the initial state of the gas was the same. Find the ratio of the respective works expended in each compression. 2.41. A heat-conducting piston can freely move inside a closed thermally insulated cylinder with an ideal gas. In equilibrium the piston divides the cylinder into two equal parts, the gas temperature being equal to To. The piston is slowly displaced. Find the gas tem- perature as a function of the ratio of the volumes of the greater and smaller sections. The adiabatic exponent of the gas is equal to y. 2.42. Find the rate v with which helium flows out of a thermally insulated vessel into vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these con- ditions. The temperature of helium in the vessel is T = 1,000 K. 2.43. The volume of one mole of an ideal gas with the adiabatic exponent y is varied according to the law V = alT, where a is a con- stant. Find the amount of heat obtained by the gas in this process if the gas temperature increased by AT. 2.44. Demonstrate that the process in which the work performed by an ideal gas is proportional to the corresponding increment of its internal energy is described by the equation pVn = const, where n is a constant. 2.45. Find the molar heat capacity of an ideal gas in a polytropic process pVn = const if the adiabatic exponent of the gas is equal to '. At what values of the polytropic constant n will the heat capacity of the gas be negative? 2.46. In a certain polytropic process the volume of argon was in- creased a = 4.0 times. Simultaneously, the pressure decreased = 8.0 times. Find the molar heat capacity of argon in this process, assuming the gas to be ideal. 2.47. One mole of argon is expanded polytropically, the polytrop- ic constant being n = 1.50. In the process, the gas temperature changes by AT = — 26 K. Find: (a) the amount of heat obtained by the gas; (b) the work performed by the gas. 2.48. An ideal gas whose adiabatic exponent equals y is expanded according to the law p = aV , where a is a constant. The initial vol- ume of the gas is equal to V0. As a result of expansion the volume in- creases itimes. Find: (a) the increment of the internal energy of the gas; (b) the work performed by the gas; (c) the molar heat capacity of the gas in the process. 2.49. An ideal gas whose adiabatic exponent equals y is expanded so that the amount of heat transferred to the gas is equal to the de- crease of its internal energy. Find: (a) the molar heat capacity of the gas in this process; (b) the equation of the process in the variables T, V; (c) the work performed by one mole of the gas when its volume increases 11 times if the initial temperature of the gas is To.
(b) the amount of the absorbed heat. The gas is assumed to be a Van der Waals gas. 2.59. For a Van der Waals gas find:
(a) the equation of the adiabatic curve in the variables T, V;
(b) the difference of the molar heat capacities CI, — Cv as a func-
tion of T and V.
2.60. Two thermally insulated vessels are interconnected by a tube equipped with a valve. One vessel of volume V1= 10 1 contains
v = 2.5 moles of carbon dioxide. The other vessel of volume V 2 =
100 1 is evacuated. The valve having been opened, the gas adiabatic-
ally expanded. Assuming the gas to obey the Van der Waals equation, find its temperature change accompanying the expansion. 2.61. What amount of heat has to be transferred to v = 3.0 moles of carbon dioxide to keep its temperature constant while it ex-
pands into vacuum from the volume V1= 5.0 1 to V 2 = 10 1? The
gas is assumed to be a Van der Waals gas.
2.3. KINETIC THEORY OF GASES. BOLTZMANN' S LAW AND MAXWELL'S DISTRIBUTION
- Number of collisions exercised by gas molecules on a unit area of the wall surface per unit time:
v=^1 4
— n (v),
where n is the concentration of molecules, and (v) is their mean velocity.
- Equation of an ideal gas state: p = nkT.
- Mean energy of molecules:
(e) = 2
— kT, (^) (2.3c)
where i is the sum of translational, rotational, and the double number of vibra- tional degrees of freedom.
- Maxwellian distribution: m (^1) 1/2 -mv2/21a dN (vx)=-- N ( 2nkT ) e s dvx, (2.3d)
dN (v)= N (^) ( e-mv2/2kT^ 4:tv2^ dv. (2.3e)
2 m \ 3/ 2nkT )
- Maxwellian distribution in a reduced form:
dN (u)= N^4 e-u2 U2 du, (2.3f)
where u = v/vp, vpis the most probable velocity.
- (^) The most probable, the mean, and the root mean square velocities of molecules:
(2.3a)
(2.3b)
Vp^ n^ kT^ (V) = kT^ kT Vsq = -. (^) (2.3g)
- Boltzmann's formula: n=noe—(u—uo/hT, (^) (2.3h)
where U is the potential energy of a molecule.
2.62. Modern vacuum pumps permit the pressures down to p =
= 4.10-13atm to be reached at room temperatures. Assuming that
the gas exhausted is nitrogen, find the number of its molecules per 1 cm3and the mean distance between them at this pressure.
2.63. A vessel of volume V .=^ 5.0 1 contains m = 1.4 g of nitrogen
at a temperature T = 1800 K. Find the gas pressure, taking into
account that 11 = 30% of molecules are disassociated into atoms at this temperature. 2.64. Under standard conditions the density of the helium and nitrogen mixture equals p = 0.60 g/l. Find the concentration of helium atoms in the given mixture. 2.65. A parallel beam of nitrogen molecules moving with velocity v = 400 m/s impinges on a wall at an angle 0 = 30° to its normal. The concentration of molecules in the beam n = 0.9.1019 cm-3. Find the pressure exerted by the beam on the wall assuming the mo- lecules to scatter in accordance with the perfectly elastic collision law. 2.66. How many degrees of freedom have the gas molecules, if under standard conditions the gas density is p = 1.3 mg/cm3and the velocity of sound propagation in it is v = 330 m/s. 2.67. Determine the ratio of the sonic velocity v in a gas to the root mean square velocity of molecules of this gas, if the molecules are (a) monatomic; (b) rigid diatomic. 2.68. A gas consisting of N-atomic molecules has the temperature
T at which all degrees of freedom (translational, rotational, and vi-
brational) are excited. Find the mean energy of molecules in such a gas. What fraction of this energy corresponds to that of transla- tional motion?
2.69. Suppose a gas is heated up to a temperature at which all
degrees of freedom (translational, rotational, and vibrational) of its molecules are excited. Find the molar heat capacity of such a gas in the isochoric process, as well as the adiabatic exponent y, if the
g as consists of
(a) diatomic; (b) linear N-atomic; (c) network N-atomic molecules. 2.70. An ideal gas consisting of N-atomic molecules is expanded isobarically. Assuming that all degrees of freedom (translational, rotational, and vibrational) of the molecules are excited, find what fraction of heat transferred to the gas in this process is spent to perform the work of expansion. How high is this fraction in the case of a monatomic gas?
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