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Chapter 19: Problem 5

Which of the following gases has the highest rootmean-square speed? a) nitrogen at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) b) argon at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) c) argon at \(2 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) d) oxygen at 2 atm and \(30^{\circ} \mathrm{C}\) e) nitrogen at \(2 \mathrm{~atm}\) and \(15^{\circ} \mathrm{C}\)

Short Answer

Expert verified
a) Nitrogen at 1 atm and 30°C b) Argon at 1 atm and 30°C c) Argon at 2 atm and 30°C d) Oxygen at 2 atm and 30°C e) Nitrogen at 2 atm and 15°C Answer: (a) Nitrogen at 1 atm and 30°C

Step by step solution

01

Convert temperatures to Kelvin

To convert temperatures from Celsius to Kelvin, we add 273.15. \(T_1 = 30 + 273.15 = 303.15 \mathrm{~K}\) \(T_2 = 15 + 273.15 = 288.15 \mathrm{~K}\)
02

Find the molar masses of the gases

Molar Mass of Nitrogen (N\(_2\)) = \(28.02 \mathrm{~g/mol}\) Molar Mass of Argon (Ar) = \(39.95 \mathrm{~g/mol}\) Molar Mass of Oxygen (O\(_2\)) = \(32.00 \mathrm{~g/mol}\)
03

Calculate the root-mean-square speeds

We use the ideal gas constant in J/(mol K): \(R = 8.314 \mathrm{~J/(mol\cdot K)}\) a) Nitrogen at 1 atm and 303.15 K: \(v_\mathrm{rms,a} = \sqrt{\frac{3(8.314)(303.15)}{28.02}} = 515.3 \mathrm{~m/s}\) b) Argon at 1 atm and 303.15 K: \(v_\mathrm{rms,b} = \sqrt{\frac{3(8.314)(303.15)}{39.95}} = 431.6 \mathrm{~m/s}\) c) Argon at 2 atm and 303.15 K: \(v_\mathrm{rms,c} = \sqrt{\frac{3(8.314)(303.15)}{39.95}} = 431.6 \mathrm{~m/s}\) d) Oxygen at 2 atm and 303.15 K: \(v_\mathrm{rms,d} = \sqrt{\frac{3(8.314)(303.15)}{32.00}} = 482.4 \mathrm{~m/s}\) e) Nitrogen at 2 atm and 288.15 K: \(v_\mathrm{rms,e} = \sqrt{\frac{3(8.314)(288.15)}{28.02}} = 501.5 \mathrm{~m/s}\)
04

Compare the root-mean-square speeds

Comparing the root-mean-square speeds, we find that: \(v_\mathrm{rms,a} > v_\mathrm{rms,b}\), \(v_\mathrm{rms,c}\), \(v_\mathrm{rms,d}\), and \(v_\mathrm{rms,e}\) Thus, the gas with the highest root-mean-square speed is option (a) nitrogen at 1 atm and 30°C.

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Most popular questions from this chapter

Air at 1.00 atm is inside a cylinder \(20.0 \mathrm{~cm}\) in radius and \(20.0 \mathrm{~cm}\) in length that sits on a table. The top of the cylinder is sealed with a movable piston. A \(20.0-\mathrm{kg}\) block is dropped onto the piston. From what height above the piston must the block be dropped to compress the piston by \(1.00 \mathrm{~mm} ? 2.00 \mathrm{~mm} ? 1.00 \mathrm{~cm} ?\)

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