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An ideal gas is at a temperature of 300 K. To double the average speed of its molecules, what does the temperature need to be changed to?

1200 K

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Problems

In a sample of hydrogen sulfide ( M = 34.1 g/mol ) at a temperature of 3.00 × 10 2 K , estimate the ratio of the number of molecules that have speeds very close to v rms to the number that have speeds very close to 2 v rms .

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Using the approximation v 1 v 1 + Δ v f ( v ) d v f ( v 1 ) Δ v for small Δ v , estimate the fraction of nitrogen molecules at a temperature of 3.00 × 10 2 K that have speeds between 290 m/s and 291 m/s.

0.00157

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Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between 3.45 × 10 −21 J and 3.50 × 10 −21 J .

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By counting squares in the following figure, estimate the fraction of argon atoms at T = 300 K that have speeds between 600 m/s and 800 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x -axis times its height as measured on the y -axis, with the units given on those axes.

The figure is a plot of f of v in seconds per meter as a function of v in meters per second. The horizontal scale is 0 to 1200 seconds per meter, with major grid lines every 0.0005 and with minor grid lines every 0.0001. The vertical scale is 0 to 0.0025 meters per second, with major grid lines every 200 and with minor grid lines every 20. The function peaks at v equal to about 350 with a value of f of about 0.00235. Additional values of the function over the full range shown are as follows, in ordered pairs of v and f: 0, 0; 100, 0.0005; 200, 0.0015; 300, 0.0022; 400, 0.0023; 500, 0.00152; 600, 0.001; 700, 0.0005; 800, 0.0002; 900, 0.0001; 1000 and higher, 0. From 600 to 800, the function has approximate coordinates of: 600, 0.001; 620, 0.0009; 640, 0.0008; 660, 0.0007; 680, 0.0007; 700, 0.0005; 720, 0.0004; 740, 0.00035; 760, 0.0003; 780, 0.00023; 800, 0.0002.

About 0.072. Answers may vary slightly. A more accurate answer is 0.074.

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Using a numerical integration method such as Simpson’s rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen ( O 2 ) is 32.0 g/mol. A precision to two significant digits is enough.

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Find (a) the most probable speed, (b) the average speed, and (c) the rms speed for nitrogen molecules at 295 K.

a. 419 m/s; b. 472 m/s; c. 513 m/s

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Repeat the preceding problem for nitrogen molecules at 2950 K.

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At what temperature is the average speed of carbon dioxide molecules ( M = 44.0 g/mol ) 510 m/s?

541 K

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The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)

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a) At what temperature do oxygen molecules have the same average speed as helium atoms ( M = 4.00 g/mol ) have at 300 K? b) What is the answer to the same question about most probable speeds? c) What is the answer to the same question about rms speeds?

2400 K for all three parts

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Additional problems

In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as 10 6 atoms/m 3 , and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m 3 ) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

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(a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of 20 °C , assuming that air is 78 % N 2 , 21 % O 2 , and 1 % Ar , (b) Find the density of the atmosphere on Venus, assuming that it’s 96 % CO 2 and 4 % N 2 , with a temperature of 737 K and a pressure of 92.0 atm.

a. 1.20 kg/m 3 ; b. 65.9 kg/m 3

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The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as 78 % N 2 , 21 % O 2 , and 1 % Ar , find the density of the air inside the balloon.

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When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80 % . If the temperatures at the bottom and the top of the lake are 4.0 and 10 ° C , respectively, how deep is the lake?

7.9 m

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Practice Key Terms 3

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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