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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

A tank of compressed helium for inflating balloons is advertised as containing helium at a pressure of 2400 psi, which, when allowed to expand at atmospheric pressure, will occupy a volume of \(244 \mathrm{ft}^{3}\). Assuming no temperature change takes place during the expansion, what is the volume of the tank in cubic feet?

A sample of gas at \(p=1000 . \mathrm{Pa}, V=1.00 \mathrm{~L},\) and \(T=300 . \mathrm{K}\) is confined in a cylinder. a) Find the new pressure if the volume is reduced to half of the original volume at the same temperature. b) If the temperature is raised to \(400 . \mathrm{K}\) in the process of part (a), what is the new pressure? c) If the gas is then heated to \(600 . \mathrm{K}\) from the initial value and the pressure of the gas becomes \(3000 . \mathrm{Pa},\) what is the new volume?

Consider a box filled with an ideal gas. The box undergoes a sudden free expansion from \(V_{1}\) to \(V_{2}\). Which of the following correctly describes this process? a) Work done by the gas during the expansion is equal to \(n R T \ln \left(V_{2} / V_{1}\right)\) b) Heat is added to the box. c) Final temperature equals initial temperature times \(\left(V_{2} / V_{1}\right)\). d) The internal energy of the gas remains constant.

1 .00 mol of molecular nitrogen gas expands in volume very quickly, so no heat is exchanged with the environment during the process. If the volume increases from \(1.00 \mathrm{~L}\) to \(1.50 \mathrm{~L},\) determine the work done on the environment if the gas's temperature dropped from \(22.0^{\circ} \mathrm{C}\) to \(18.0^{\circ} \mathrm{C}\). Assume ideal gas behavior.

Treating air as an ideal gas of diatomic molecules, calculate how much heat is required to raise the temperature of the air in an \(8.00 \mathrm{~m}\) by \(10.0 \mathrm{~m}\) by \(3.00 \mathrm{~m}\) room from \(20.0^{\circ} \mathrm{C}\) to \(22.0^{\circ} \mathrm{C}\) at \(101 \mathrm{kPa}\). Neglect the change in the number of moles of air in the room.
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