Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem solving technique.

| 58 | Concept Questions Suppose you know the mass per mole (in units of g/mol) of a substance. (a) Without doing any calculations, how can you determine the mass of one of its atoms (expressed in atomic mass units)? (b) What information about the sample is obtained when you divide its mass by the mass per mole?
Problem A certain element has a mass per mole of 196.967 g/mol. What is the mass of a single atom in (a) atomic mass units and (b) kilograms? (c) How many moles of atoms are in a 285-g sample?

| 59 | Concept Questions The temperature of an ideal gas is doubled while the volume is kept constant. Does the absolute pressure of the gas double when the temperature that doubles is (a) the Kelvin temperature and (b) the Celsius temperature? Explain.
Problem Determine the ratio P2/P1 of the final pressure P2 to the initial pressure P1 when the temperature rises (a) from 35.0 to 70.0 K and (b) from 35.0 to 70.0 °C. Check to see that your answers are consistent with your answers to the Concept Questions.

| 60 | Concept Question Four closed tanks, A, B, C, and D, each contain identical numbers of moles of an ideal gas. The table gives the absolute pressure and volume of the gas in each tank. Which tanks (if any) have the same temperature? Account for your answer.

| A | B | C | D
Absolute pressure (Pa) | 25.0 | 30.0 | 20.0 | 2.0
Volume (m3) | 4.0 | 5.0 | 5.0 | 75

Problem Each of the tanks contains 0.10 mol of gas. Using this number and the data in the table, compute the temperature of each gas. Verify that your answers are consistent with your answer to the Concept Question.

| 61 | Concept Question Four containers are filled with monatomic ideal gases. For each container, the mass of an individual atom and the rms speed of the atoms are expressed in terms of m and vrms, respectively (see the table). Rank the gases according to their temperatures, highest first. Justify your ranking.

| A | B | C | D
Mass | m | m | 2m | 2m
Rms speed | vrms | 2vrms | vrms | 2vrms

Problem Suppose that , and . Find the temperature of the gas in each tank. Be sure that your answers are consistent with your answer to the Concept Question.

| 62 | Concept Questions Imagine a gas diffusing through air in an arrangement like that in Figure 14-16a. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why?
Problem The atomic mass of a hydrogen atom is 1.0 u and that of a deuterium atom is 2.0 u. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.

| 63 | Concept Questions Consider a mixture of three different gases: 1.20 g of argon , 2.60 g of neon , and 3.20 g of helium . (a) Explain how to calculate the number of moles of each species in terms of its mass and its molecular mass. (b) Of the total number of molecules in the mixture, each component is a certain percentage. Explain how to calculate that percentage. (c) Without doing any detailed calculations, determine which component has the greatest percentage. Give your reasoning. (d) Which component has the smallest percentage? Account for your answer.
Problem For the mixture described in the Concept Questions, determine the percentage of the total number of molecules that corresponds to each of the components. Verify that your answers are consistent with your answers to the Concept Questions.

| 64 | Concept Questions The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is A. The initial pressure, volume, and temperature of the gas are, respectively, P0, V0, and T0, and the spring is initially stretched by an amount x0 with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are Pf, Vf, and Tf and the spring is stretched by an amount xf with respect to its unstrained length. Assume that x0 and xf are positive variables. (a) What is the relation between the magnitude of the force required to stretch an ideal spring and the amount of the stretch with respect to the unstrained length of the spring? (b) What are the magnitudes, F0 and FF, of the forces that the initial and final pressures apply to the piston and, hence, to the spring? Express your answers in terms of the pressures and the cross-sectional area of the piston. (c) According to the ideal gas law, how are the initial pressure, volume, and temperature related to the final pressure, volume, and temperature? (d) How is the final volume related to the initial volume, the cross-sectional area of the piston, and the initial and final amounts by which the spring is stretched? Account for your answer.
Problem The initial temperature and volume of the gas described in the Concept Questions are 273 K and . The initial and final amounts by which the spring is stretched are, respectively, 0.0800 and 0.1000 m. The cross-sectional area of the piston is . What is the final temperature of the gas?

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