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A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

Short Answer

Expert verified
(a) Average kinetic energies are equal. (b) Ne has the highest \( v_{rms} \), Rn the lowest.

Step by step solution

01

Understanding Kinetic Energy

According to kinetic molecular theory, the average kinetic energy of gas particles is given by \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is Boltzmann's constant and \( T \) is the temperature in Kelvin. Importantly, the average kinetic energy is dependent only on temperature, not the type of gas. Therefore, all three gases (Ne, Kr, and Rn) have the same average kinetic energy at the same temperature.
02

Finding Molar Masses

Check Appendix D for the molar masses: Neon (Ne) is approximately 20 g/mol, Krypton (Kr) is approximately 84 g/mol, and Radon (Rn) is approximately 222 g/mol.
03

Calculating Root-Mean-Square Speed

The root-mean-square speed for a gas is calculated using the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass in kg/mol. Since Neon has the smallest molar mass, it will have the highest root-mean-square speed, followed by Krypton, and then Radon.
04

Summary

For part (a), the average kinetic energies of Ne, Kr, and Rn are equal at constant temperature. For part (b), the root-mean-square speed varies inversely with the square root of the molar mass, with Ne having the highest speed and Rn the lowest.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Kinetic Energy
In the realm of physics and chemistry, kinetic molecular theory plays a vital role in understanding how gas particles behave. One of its fundamental aspects is the concept of average kinetic energy. This is determined by the equation \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is Boltzmann's constant and \( T \) is the temperature in Kelvin.
This equation tells us that average kinetic energy is influenced solely by temperature, not by the nature of the gas itself.
So, in our example, even though neon, krypton, and radon are different gases, they have the same average kinetic energy if they are at the same temperature. This concept underscores how temperature is a universal measure influencing the energy exhibited by particles in a gas.
Thus, regardless of the gas type, temperature holds the key to determining their kinetic energy dynamics.
Root-Mean-Square Speed
Root-mean-square speed, often abbreviated as \( v_{rms} \), is a mathematical way of representing the speed of gas particles in a container. It accounts for the fact that particles in a gas don't move uniformly but cover a range of speeds.
The formula for calculating the root-mean-square speed is \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass. Of the three gases we discussed, neon, krypton, and radon, the speed differences arise from their varying molar masses.
Since Neon has the smallest molar mass, it ends up having the highest \( v_{rms} \), meaning its particles move the fastest. On the contrary, Radon, with the largest molar mass, has the slowest particles.
  • High molar mass (Rn): slower speeds
  • Low molar mass (Ne): faster speeds
This shows how inversely the root-mean-square speed of particles relates to the square root of their molar mass, making each gas particle's speed distinct.
Molar Mass
Molar mass is a fundamental concept in understanding the properties of different substances, especially gases. It is defined as the mass of one mole of a given substance, and it is typically expressed in grams per mole (g/mol).
In our example, to find the molar mass of each gas, one should refer to a periodic table or specific appendices like Appendix D in textbooks. The values are as follows:
  • Neon (Ne): approximately 20 g/mol
  • Krypton (Kr): approximately 84 g/mol
  • Radon (Rn): approximately 222 g/mol
These values are crucial for calculating properties such as the root-mean-square speed of gases, as seen earlier. Understanding molar mass helps us comprehend how different gases behave under equivalent temperature and pressure conditions.
Moreover, it reflects how mass factors contribute to the energy and speed of gas particles, linking back to the kinetic molecular theory. This interconnectedness highlights the importance of molar mass in gas dynamics and other areas of chemistry and physics.

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Most popular questions from this chapter

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.300 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0\(^\circ\)C? (b) What is the force when the temperature of the gas is increased to 100.0\(^\circ\)C?

Oxygen (O\(_2\)) has a molar mass of 32.0 g/mol. What is (a) the average translational kinetic energy of an oxygen molecule at a temperature of 300 K; (b) the average value of the square of its speed; (c) the root-mean-square speed; (d) the momentum of an oxygen molecule traveling at this speed? (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.10 m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.) (f) What is the average force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 atm? (h) Compute the number of oxygen molecules that are contained in a vessel of this size at 300 K and atmospheric pressure. (i) Your answer for part (h) should be three times as large as the answer for part (g). Where does this discrepancy arise?

An empty cylindrical canister 1.50 m long and 90.0 cm in diameter is to be filled with pure oxygen at 22.0\(^\circ\)C to store in a space station. To hold as much gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 g/mol. (a) How many moles of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilograms does this gas increase the mass to be lifted?

We have two equal-size boxes, \(A\) and \(B\). Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box \(A\) is at 50\(^\circ\)C while the gas in box \(B\) is at 10\(^\circ\)C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning. (a) The pressure in \(A\) is higher than in \(B\). (b) There are more molecules in \(A\) than in \(B\). (c) A and B do not contain the same type of gas. (d) The molecules in \(A\) have more average kinetic energy per molecule than those in \(B\). (e) The molecules in \(A\) are moving faster than those in \(B\).

A 20.0-L tank contains \(4.86 \times 10{^-}{^4}\) kg of helium at 18.0\(^\circ\)C. The molar mass of helium is 4.00 g/mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

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