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Chapter 8: Problem 149

Consider separate \(1.0-\mathrm{L}\) gaseous samples of \(\mathrm{He}, \mathrm{N}_{2},\) and \(\mathrm{O}_{2}\) all at \(\mathrm{STP}\) and all acting ideally. Rank the gases in order of increasing average kinetic energy and in order of increasing average velocity.

Short Answer

Expert verified
The average kinetic energy for all three gases (He, N2, and O2) at STP is the same. For the average velocity, the gases can be ranked in order of increasing average velocity as: \(O_2 < N_2 < He\).

Step by step solution

01

Define Relevant Equations

To rank the gases based on average kinetic energy and average velocity, we need two equations. For average kinetic energy (KE), we can use the following equation: \( KE = \dfrac{3}{2}nRT \) where n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature (in Kelvin). For the average velocity (root-mean-square velocity, v_rms), we can use the following equation: \( v_{rms} = \sqrt{\dfrac{3RT}{M}} \) where M is the molar mass of the gas (in kg/mol).
02

Calculate the Average Kinetic Energy

Since all gases are at the same temperature (STP, which corresponds to 273.15 K) and contain the same number of moles (1.0 L), the average kinetic energy will be the same for all gases as KE only depends on \(n\) and \(T\) in the equation. So, He, N2, and O2 will have the same average kinetic energy at STP.
03

Calculate the Average Velocity (v_rms) for Each Gas

Using the equation for the average velocity (v_rms), we need to know the molar mass (M) of each gas. Here are the molar masses: He: 4.00 g/mol (0.004 kg/mol) N2: 28.02 g/mol (0.028 kg/mol) O2: 32.00 g/mol (0.032 kg/mol) Now, we will calculate the average velocity (v_rms) for each gas at STP: He: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.004}} \approx 1446.89 \, m/s \) N2: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.028}} \approx 517.37 \, m/s \) O2: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.032}} \approx 482.52 \, m/s \)
04

Rank the Gases

Now that we have calculated the average velocity (v_rms) for each gas, we can rank them in order of increasing average velocity: O2 < N2 < He As for the average kinetic energy, we already determined that they are the same for all gases at STP. #Conclusion# Thus, the gases can be ranked in order of increasing average kinetic energy as "Same for all gases" since all are at STP and contain the same number of moles (1.0 L). In terms of increasing average velocity, the order is O2 < N2 < He.

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Most popular questions from this chapter

A student adds \(4.00 \mathrm{g}\) of dry ice (solid \(\mathrm{CO}_{2}\) ) to an empty balloon. What will be the volume of the balloon at STP after all the dry ice sublimes (converts to gaseous \(\mathrm{CO}_{2}\) )?

A hot-air balloon is filled with air to a volume of \(4.00 \times\) \(10^{3} \mathrm{m}^{3}\) at \(745\) torr and \(21^{\circ} \mathrm{C}\). The air in the balloon is then heated to \(62^{\circ} \mathrm{C},\) causing the balloon to expand to a volume of \(4.20 \times 10^{3} \mathrm{m}^{3} .\) What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon? (Hint: Openings in the balloon allow air to flow in and out. Thus the pressure in the balloon is always the same as that of the atmosphere.)

At \(0^{\circ} \mathrm{C}\) a \(1.0\)-\(\mathrm{L}\) flask contains \(5.0 \times 10^{-2}\) mole of \(\mathrm{N}_{2}, 1.5 \times\) \(10^{2} \mathrm{mg} \mathrm{O}_{2},\) and \(5.0 \times 10^{21}\) molecules of \(\mathrm{NH}_{3} .\) What is the partial pressure of each gas, and what is the total pressurelin the flask?

A compound has the empirical formula \(\mathrm{CHCI}\) A \(256-\mathrm{mL}\) flask, at \(373 \mathrm{K}\) and \(750 .\) torr, contains \(0.800 \mathrm{g}\) of the gaseous compound. Give the molecular formula.

A steel cylinder contains \(5.00\) moles of graphite (pure carbon) and \(5.00\) moles of \(\mathrm{O}_{2}\). The mixture is ignited and all the graphite reacts. Combustion produces a mixture of \(\mathrm{CO}\) gas and \(\mathrm{CO}_{2}\) gas. After the cylinder has cooled to its original temperature, it is found that the pressure of the cylinder has increased by \(17.0 \% .\) Calculate the mole fractions of \(\mathrm{CO}, \mathrm{CO}_{2},\) and \(\mathrm{O}_{2}\) in the final gaseous mixture.
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