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1 PROBLEMS Thermodynamic Variables and the Eqation of State 1. Compter (a) the nmber of moles and (b) the nmber of molecles in 1.00 cm of an ideal gas at a pressre of 100 Pa and a temperatre of 0 K.. Oxygen gas having a volme of 1000 cm at 40 C and Pa expands ntil its volme is 1500 cm and its pressre is Pa. Find (a) the nmber of moles of oxygen present and (b) the final temperatre of the sample.. A qantity of ideal gas at 10.0 C and 100 kpa occpies a volme of.50 m. (a) How many moles of the gas are present? (b) If the pressre is now raised to 00 kpa and the temperatre is raised to 0.0 C, how mch volme does the gas occpy? Assme no leaks. 4. An air bbble of 0 cm volme is at the bottom of a lake 40 m deep where the temperatre is 4.0 C. The bbble rises to the srface, which is at a temperatre of 0 C. take the temperatre of the bbble s air to be the same as that of the srronding water. Jst as the bbble reaches the srface, what is its volme? 5. Taking the normal lng capacity to be 500 cm eqivalent of 761 mm Hg (which is still the way the medical texts are listing it), estimate the nmber of molecles per breath? 6. At an altitde of abot 1.5 km, the temperatre of the Earth s atmosphere is roghly 55 C and the pressre is arond 19.4 kpa. How many kilograms of hydrogen gas (H) shold be pt in a balloon to fill it to 000 m at that altitde and the pressre therein to be the 7. Prove that for an ideal gas p V V T T p = Pressre and Moleclar Motion 8. Calclate (a) the rms speed of an oxygen molecle at 0 C and (b) determine how many times per second it wold move back and forth across a 7.0-m-long room on the average, assming it made very few collisions with other molecles. 9. The temperatre and pressre in the Sn s atmosphere are K and Pa. Calclate the rms speed of free electrons (mass = ) there, assming they are an ideal gas. 10. A beam of hydrogen molecles (H) is directed toward a wall, at an angle of 55 with the normal to the wall. Each molecle in the beam has a speed of 1.0 km/s and a mass of g. The beam strikes the wall over an area of.0 cm, at the rate of 10 molecles per second. What is the beam s pressre on the wall? 7-4 The Microscopic Interpretation of Temperatre 11. What is the average translational kinetic energy of nitrogen molecles at 1600 K? 1. A gas is heated from 7 K to 7 K. What is the change in the rms speed? 1. A relativistic gas is one in which a significant fraction of the constitents have speeds that
2 are some finite fraction of the speed of light, c 10 8 m/s. Sppose that the rms speed of the constitents of a gas of atomic hydrogen is jst 1 percent of c. What is the temperatre? The energy that binds the electron and the proton together into a hydrogen atom is J. Wold this gas break into a gas of electrons and protons? In the interior of stars, where temperatres comfortably exceed the temperatre yo have calclated, atoms cannot exist; they are broken into their components. 7-5 The Eqipartition of Energy 14. One mole of monatomic ideal gas is placed in chamber nder 10 atm pressre. The volme of the chamber is 100 cm. (a) What is the internal energy of the gas? (b) What is the temperatre of the gas? (c) Assme that the mass of a molecle of the gas is kg, what is the v of a gas molecle? (d) What is the root-mean-sqare speed? 15. The rms speed of 1 mole of argon atoms (atomic weight 40 g/mole) in a box is 400 m/s. (a) What is the temperatre inside the box? (b) What is the internal energy? (c) If the box has a volme of 0.1 m, what is the pressre? Treat the gas as ideal. 7-6 Mean Free Path 16. At abot what pressre wold the mean free path of air molecles be (a) 1.0 m and (b) eqal to the diameter of air molecles, m. Assme that the temperatre of air is 00 K. 17. A very small amont of hydrogen gas is released into the air. If the air is at 1.0 atm and 5 C, estimate the mean free path for a H molecle. What assmptions did yo make? The diameter of hydrogen atoms is m. 18. At 00 km above Earth s srface, the nmber density of molecles of the atmosphere is abot /m. (a) What mean free path is predicated by Eq and (b) What is its significance nder these conditions? 19. In a certain particle accelerator, protons travel arond a circlar path of diameter.0 m in 6 an evacated chamber, whose residal gas is at 95 K and torr pressre (1 torr = 1 mmhg). (a) Calclate the nmber of gas molecles per cbic centimeter at this pressre. (b) What is the mean free path of the gas molecles if the moleclar diameter is cm? 7-7 The Distribtion of Moleclar Speeds 0. The speed of 10 molecles are.0,.0, 4.0,, 11 km/s. (a) What is their average speed? (b) What is their root-mean-sqare speed? 1. (a) Compte the temperatre at which the rms speed for (a) moleclar hydrogen and (b) moleclar oxygen is eqal to the speed of escape from Earth. (c) Do the same for the speed of escape from the Moon, assming the gravitational acceleration on its srface is abot 0.16g. The radis of the Moon is m. (d) The temperatre high in Earth s pper atmosphere is abot 1000 K. Wold yo expect to find mch hydrogen there? Mch oxygen? Explain.. It is fond that the most probable speed of molecles in a gas has (niform) temperatre T is the same as the rms speed of the molecles in this gas when it has (niform) temperatre T 1. Calclate T /T 1.
3 . Figre 7-8 shows a hypothetical speed distribtion for a sample of N gas particles (note f (v) = 0 for v > v 0 ). (a) Express a in terms of N and v 0. (b) How many of particles have speeds between 1.5v 0 and.0v 0? (c) Express the average speed of the particles in terms of v 0. (d) Find v rms. f (v) a O v 0 v 0 speed Fig. 7-8 Problem. 4. Helim gas is in eqilibrim at a temperatre of 5 K and is slowly raised to 10 K. Write an expression for the fraction of molecles with speeds than 1000 m/s for each of the two temperatres, and state the ratio of these fractions. Problems 1. (a) mole; (b) (a) mole; (b) 0 C mol; (b) 0.89 m cm molecles, for T= 7 C kg. 7. proof problem. 8. (a) 461 m/s; (b) s m/s kpa J. 1. vrms = 1.17v rms K; yes, becase K = J, which is mch greater than the binding energy. 14. (a) 15 J; (b) 1 K; (c) (m/s) ; (d).8 10 m/s. 15. (a) 57 K; (b). 10 J; (c) Pa (a) 10 atm; (b) 00 atm m. 18. (a) 0 km. 19. (a) ; (b) 17 m. 0. (a) 6.5 km/s; (b) 7.1 km/s. 1. (a) K; (b) K; (c) 440 K, 7040 K; (d) hydrogen, yes; oxygen, no.
4 . /.. (a) /v 0 ; (b) N/; (c) 1.v 0 ; (d) 1.1v N N 5 10 = e e d, where d mv =. kt
5
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