From the van der Waals equation of state, show that the critical temperature and pressure are given by Tcr = 8a / 27bR , Pcr = a / 27b². [Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.]
21. Kinetic Theory of Ideal Gases
The Ideal Gas Law
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Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to β = 1/ T, where T is the kelvin temperature. Compare to Table 17–1 for gases at T = 293 K.
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A sauna has 7.8 m³ of air volume, and the temperature is 85°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18–2.)
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Why snorkels are not 4 feet long. Snorkelers breathe through short tubular “snorkels” while swimming under water very near the surface (Fig. 17–24). One end of the snorkel is in the snorkeler’s mouth and the other end protrudes just above the water’s surface. Unfortunately, snorkels cannot support breathing to any great depth: it is said that a typical snorkeler below a water depth of only about 30 cm cannot draw a breath through a snorkel. Based on this observation, what is the approximate change in a typical person’s lung pressure (in atm) when drawing a breath? (Note that your diaphragm muscles, which expand your lungs, must work also against the extra water pressure.)
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A heat engine takes a diatomic gas around the cycle shown in Fig. 20–23. Using the ideal gas law, determine how many moles of gas are in this engine.
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A heat engine takes a diatomic gas around the cycle shown in Fig. 20–23. Determine the temperature at point c.
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1.00 mole of an ideal monatomic gas at STP first undergoes an isothermal expansion so that the volume at b is 2.5 times the volume at a (Fig. 20–25). Next, heat is extracted at a constant volume so that the pressure drops. The gas is then compressed adiabatically back to the original state. Calculate the pressures at b and c.
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1.00 mole of an ideal monatomic gas at STP first undergoes an isothermal expansion so that the volume at b is 2.5 times the volume at a (Fig. 20–25). Next, heat is extracted at a constant volume so that the pressure drops. The gas is then compressed adiabatically back to the original state. Determine the temperature at c.
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(III) A 0.5-mol sample of O2 gas is in a large cylinder with a movable piston on one end so it can be compressed (as in Fig. 17–14a). The initial volume is large enough that there is not a significant difference between the pressure given by the ideal gas law and that given by the van der Waals equation. As the gas is slowly compressed at constant temperature (use 300 K), at what volume does the van der Waals equation give a pressure that is 5% different than the ideal gas law pressure? Let a = 0.14 N m4/mol2 and b = 3.2 x 10-5 m3/mol.
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We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at °C while the gas in box B is at °C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning.
(a) The pressure in A is higher than in B.
(b) There are more molecules in A than in B.
(c) A and B do not contain the same type of gas.
(d) The molecules in A have more average kinetic energy per molecule than those in B.
(e) The molecules in A are moving faster than those in B.
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