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Chapter 5: Problem 155

Consider separate \(2.5-\) L gaseous samples of \(\mathrm{He}\), \(\mathrm{N}_{2},\) and \(\mathrm{F}_{2},\) 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 of the gases He, N2, and F2 is the same since they are all at the same temperature (STP). Therefore, the ranking by increasing average kinetic energy is: He = N2 = F2. The ranking of the gases by increasing average velocity, calculated using the root-mean-square velocity formula, is: F2 < N2 < He.

Step by step solution

01

Determine the temperature at STP

At standard temperature and pressure (STP), the temperature is 0°C, which is equivalent to 273.15 K. Since all three gases are at STP, they all have the same temperature: T = 273.15 K.
02

Calculate the average kinetic energy of the gases

Since all three gases are at the same temperature, their average kinetic energies are also the same, as the average kinetic energy of an ideal gas depends solely on its temperature. According to the kinetic molecular theory, the average kinetic energy (KE) can be described by the equation: KE = \( \frac{3}{2} \)kT, where k is Boltzmann's constant (1.38 x 10^(-23) J/K) and T is the temperature in Kelvin. For all three gases, the average kinetic energy will be the same, as the temperature is equal: KE = \( \frac{3}{2} \)kT = \( \frac{3}{2} \) (1.38 x 10^(-23) J/K)(273.15 K) = KE. Therefore, the ranking of the gases by increasing average kinetic energy is: He = N2 = F2.
03

Calculate the molar mass of the gases

In order to calculate the average velocity of the gases, we first need to determine their molar masses. The molar masses of the gases are: He: 4.00 g/mol N2: 28.02 g/mol F2: 38.00 g/mol
04

Calculate the root-mean-square velocity of the gases

The root-mean-square (rms) velocity (v_rms) of an ideal gas is given by the equation: v_rms = \( \sqrt{\frac{3RT}{M}} \), where R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and M is the molar mass of the gas in kg/mol. We can calculate the rms velocities for each gas: v_rms (He) = \( \sqrt{\frac{3(8.314 \frac{J}{mol \cdot K})(273.15 K)}{0.004 kg/mol}} \) v_rms (N2) = \( \sqrt{\frac{3(8.314 \frac{J}{mol \cdot K})(273.15 K)}{0.02802 kg/mol}} \) v_rms (F2) = \( \sqrt{\frac{3(8.314 \frac{J}{mol \cdot K})(273.15 K)}{0.038 kg/mol}} \)
05

Rank the gases in order of increasing average velocity

By comparing the rms velocities calculated in Step 4, we can determine the order of increasing average velocities: v_rms (F2) < v_rms (N2) < v_rms (He) Therefore, the ranking of the gases by increasing average velocity is: F2 < N2 < He.

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Most popular questions from this chapter

A tank contains a mixture of 52.5 g oxygen gas and 65.1 \(\mathrm{g}\) carbon dioxide gas at \(27^{\circ} \mathrm{C}\) . The total pressure in the tank is 9.21 atm. Calculate the partial pressures of each gas in the container.

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