Question

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

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

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.

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.

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

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) = \( \sqrt{\frac{3(8.314 \frac{J}{mol \cdot K})(273.15 K)}{0.038 kg/mol}} \)

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.

Key concepts

STP (Standard Temperature and Pressure)

Standard Temperature and Pressure (STP) is a reference point used often in chemistry to allow for meaningful comparisons and calculations regarding gases. At STP, the conditions are standardized to a temperature of 0°C or 273.15 Kelvin (K) and a pressure of 1 atmosphere (atm).
These conditions simplify many chemical calculations because gases behave more predictably at STP, allowing chemists to apply the ideal gas laws easily.

• Standard pressure: 1 atm is equivalent to 101.325 kPa.
• Standard temperature: 273.15 K is equivalent to 0°C.

For the problem at hand, we'll use these conditions to determine the average kinetic energy and the root-mean-square velocity of gaseous samples, given their behavior at STP.

root-mean-square velocity

The root-mean-square (rms) velocity is an important concept related to the velocities of gas molecules in a sample. It is defined as the square root of the average of the squares of the velocities of the molecules. In mathematical terms, for an ideal gas, it is given by the equation:\[v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]where:

• \( R \) is the gas constant, which is 8.314 J/(mol·K).
• \( T \) represents the temperature in Kelvin.
• \( M \) is the molar mass of the gas in kg/mol.

This equation shows that the rms velocity increases with temperature and decreases with increasing molar mass. Hence, lighter molecules like helium will move faster than heavier ones like nitrogen or fluorine at the same temperature. In our exercise, the ranking by increasing average velocity shows that the lightest gas, He, has the highest rms velocity.

average kinetic energy

The concept of average kinetic energy is pivotal in understanding the behavior of gases according to the Kinetic Molecular Theory. This theory assumes that the particles of an ideal gas are in random motion, and it defines the average kinetic energy of gas molecules using the formula:\[KE = \frac{3}{2} kT\]Here, \( k \) is Boltzmann's constant (1.38 x 10^{-23} J/K) and \( T \) is the temperature in Kelvin. From the equation, you can see that the average kinetic energy of gas molecules is directly proportional to the temperature. This means that gases at the same temperature, such as our gases at STP (273.15 K), will have the same average kinetic energy, regardless of their molar mass or identity.

• At STP, the average kinetic energy of He, N₂, and F₂ are the same due to identical temperatures.
• This uniformity in kinetic energy at the same temperature is a key feature of gases, supporting the idea that temperature is a measure of the average kinetic energy of particles.