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Wize University Chemistry Textbook > Gases and their Properties

Kinetic Energy, Temperature, and Speed of Gas Particles

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Relationships Between Kinetic Energy, Temperature, and Speeds!

Just focus on understanding 💡 the following lesson.

Scenario #1:

We have two different gases at the same temperature in two different containers. What can we say about the average kinetic energy of the two gases? (are they the same, different, if different how?)
Since both gases are at the same temperature, they'll have the same AVERAGE kinetic energy

Hint: This was the part of the kinetic molecular theory!

We can make sense of this using the following equation:

KEavg=32RTKE_{avg}=\frac{3}{2}RT
KEavg is the average kinetic energy of a mole of gas particles
R here is 8.314 J/mol K
How are KEavg and T related?
Directly proportional

Watch Out!
The molar masses of the gases are not important when it comes to the average kinetic energy so if you're ever asked about the average kinetic energy, you just need to think about how it relates to temperature!


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Scenario #2:

We have two different gases at the same temperature in two different containers. What can we say about the speeds of the two gases? (are they the same, different, if different how?)

The gases are at the same temperature, so have the same average kinetic energy
Recall: KEavg=1/2mv2avg
The gases are different so their masses will be different!
The lighter gas (smaller mass) will have a faster average speed so that it will have the same average kinetic energy! :)


Photo by CNS OpenStax / CC BY

Therefore, a heavier gas will move at a (faster/slower)
slower
average speeds than a lighter gas at the same temperature.
Note: The lighter the gas molecule, the more (broad/narrow)
broad
the speed distribution
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Scenario #3:

If we just consider one gas in a container, all of the gas particles in the container don't move at the exact same speed!
There is actually a distribution of speeds.

If temperature gets increased, then the average speed (increases/decreases)
increases
and the average kinetic energy (increases/decreases)
increases
and the spread in the graph below (increases/decreases)
increases
.


Photo by CNS OpenStax / CC BY




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Root Mean Square Speed (urms)


urms=3RTM\boxed{u_{rms}=\sqrt{\frac{3RT}{M}}}


R is the ideal gas constant (8.314 J/mol K)
T is temperature in K
M is molar mass in kg/mol


Watch Out!
To get the correct units for urms (m/s) we need to use units kg/mol for molar mass in this equation.
The good thing is that for all other equations, we use the units g/mol for molar mass.



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Example: Applications of Kinetic Molecular Theory

Consider the following two curves showing distributions of the number of molecules vs. their molecular speeds for two different gas samples, System 1 and System 2. If the area under both curves is equal, provide two possible scenarios which would result in these observed differences. Complete the table below to describe each of your two scenarios.






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Example: Root Mean Squared Speed

Gas particles in an unknown sample are measured to have a root mean square speed of 324.2 m/s at 80 °C. If this sample is known to be a pure element, what is the most likely identity of this gas?

We can generally identify an unknown gas based on its molar mass.
The root mean square speed and temperature are all we need to find this information:

urms=3RTMu_{rms}=\sqrt{\frac{3RT}{M}}

urms=324.2m/s=3(8.314J/mol.K)(353.15K)Mu_{rms}=324.2m/s=\sqrt{\cfrac{3(8.314J/mol.K)(353.15K)}{M}}

M=3(8.314J/mol.K)(353.15K)(324.2m/s)2=0.08380kg/mol=83.80g/molM=\cfrac{3(8.314J/mol.K)(353.15K)}{(324.2m/s)^2}=0.08380kg/mol=83.80g/mol
Notice that our molar mass is initially found with units of kg/mol due to the units included in the Joule [kg m2/s2)! Look over the unit analysis to prove it to yourself.
Given the molar mass is 83.80 g/mol, a quick review of the periodic table tells us that this sample is most likely Krypton.

Practice: Root Mean Squared Speed

Which of the following would have the lowest urms?


Practice: Kinetic Energy of Gases

Which of the following have the same average kinetic energy?

I He at 297 K
II He at 351 K
III H2 at 297 K
IV O2 at 212 K
V SO2 at 297 K
Extra Practice
Molecular Theory of Gases: Kinetic Energy
Which of the following have the same kinetic energy?
He at 297 K He at 351 K H2 at 297 K O2 at 212 K SO2 at 297 K
Collision theory: Root Mean Square Speed
Which of the following would have the lowest root mean square speed?
a) NO2 at 273 K b) NO2 at 453 K c) O2 at 273 K
Consider the following two curves showing distributions of number of molecules vs. their kinetic energies for two different gas samples. Which of the following statements is not consistent with the two curves?
If the temperature of a sample of an unknown gas is increased from 0 °C to 100 °C, what will be the new root mean square speed of the particles if the original value was 330 m/s?
If a mixture of He and Xe are present in a vessel, and the root mean square speed of He is 412 m/s, what is the root mean square speed of Xe? What is the temperature of this vessel?
Gas Laws
Choose the correct statement(s):
Kinetic Molecular Theory
1. Compare the average kinetic energies of SiH4 and CH4 at a specific temperature?
2. Place these gases in order of increasing average molecular speed at 25 °C: Kr, CH4, N2, CH2Cl2.
Kinetic Molecular Theory
Calculate the ratio of the rms speed of N2 molecules 100 ⁰C to the rms speed of the same at 0 ⁰C.
Graham's Law: Kinetic Energy
1. If a mixture of He and Xe are present in a vessel, and the root mean square speed of He is 412 m/s, what is the root mean square speed of Xe? What is the temperature of this vessel?
Kinetic Energy and Temperature
3. If the temperature of a sample of an unknown gas is increased from 0 °C to 100 °C, what will be the new root mean square speed of the particles if the original value was 330 m/s?
How does the root mean square speed of a gas vary with molar mass?
Which of the following have the same kinetic energy?
I He at 297 K
II He at 351 K
Collision theory: Root Mean Square Speed
Which of the following would have the lowest rms?
a) NO2 at 273 K b) NO2 at 453 K c) O2 at 273 K
Molecular Theory of Gases: Kinetic Energy
Which of the following have the same kinetic energy?
He at 297 K He at 351 K H2 at 297 K O2 at 212 K SO2 at 297 K
Molecular theory of gases: Kinetic energy
Which one of the following gases has the greatest average molecular velocity at 300 K?
Atom: Wave-Particle
An proton is confined in a 1-D box with a length of 40 pm.
What is the wavelength of the proton when n = 3?
Atom: Kinetic Energy
An proton is confined in a 1-D box with a length of 40 pm.
What is the kinetic energy of the proton when n = 3?
Atom: Kinetic Energy
A proton is confined in a 1-D box with a length of 40 pm.
Would moving to a larger box result in higher or lower energy?
Kinetic Molecular Theory
Suppose you have a sample of gas at a fixed temperature, and you increase the volume of the container. The pressure changes, since there are...
Gas Laws
Suppose you have a sample of gas at a fixed temperature, and you increase the volume of the container. The pressure changes, since there are...
Kinetic Molecular Theory
Which statement is false?
Consider the following two curves showing distributions of number of molecules vs. their kinetic energies for two different gas samples. Which of the following statements is not consistent with the two curves?
The following curves show the distribution of molecules vs. speed for two different samples of gas, System 1 and System 2. The area under both curves is equal. Which of these descriptions is not consistent with the differences between these two graphs?