Increasing the **temperature** allows a gas to absorb **thermal energy** and convert into **kinetic energy**. Kinetic energy allows the gas to move and the speed at which it moves gives us the **root mean square speed**.

**Concept:** Understanding kinetic energy & Root Mean Square Speed

Welcome back guys. In this new video, we're going to take a look at the kinetic energy of gases. We've talked about kinetic energy really quickly in previous videos. We say that when we increase the temperature of any type of container, the gas particles in there will absorb that thermal energy and convert it to kinetic energy. This is the energy that they're going to use in order to push themselves around inside of this container.

We're going to say in order to measure the average kinetic energy of a gas molecule or particle, we must employ the root mean square equation. The root mean square equation is U, which stands for velocity, equals square root of 3RT over M. When I say velocity, I mean speed. The units for velocity or speed are meters per second.

Here this R is not the same R we're used to seeing. Because we're talking about speed or energy, R is now 8.314 joules over moles times K. Remember when do I use this R? I use this R anytime we're talking about speed, velocity, kinetic energy. Energy is the key word here. The three keywords we look out for are energy speed or velocity. That's when we use this R. The other R is associated with the Ideal Gas Law. It's different.

Here, T represents temperature in Kelvin. Here, capital M means molar mass or molecular mass. But here it's different. We're used to seeing molecular mass or molar mass in grams per mole, but in this case, in this equation, it's going to be in kilograms per mole. You have to remember that. It's in kilograms per mole.

Now the R we're going to look at it a little bit closer. We're going to say here joules is just a form of energy. We're going to say joules are kilograms times meters squared over seconds squared. That's what joules mean. That means that our R is really 8.314 kilograms times meters squared over moles times K times seconds squared. Because the second squared is on the bottom here, we would have to put it on the bottom here. Just remember the units that are involved with joules. This is going to help us in our calculations to see what we isolate when we find our answer.

**Example:** A 1.56 x 10** ^{13}** pg gaseous particle travels at 6.21 m/s. Determine its kinetic energy.

The kinetic energy (in J or kJ) of a gas molecule is directly proportional to its absolute temperature in Kelvins.

**Problem:** Calculate the molar mass, in g/mol, of a gaseous compound with an average root mean velocity of 652 m/s at a temperature of 30^{o }C.

Remember that using the root mean square speed equation deals with molar mass in g/mol, so further conversion may sometimes be needed.

Calculate the root-mean-square speed of methane, CH_{4} (g), at 78 °C.

(a) 23 m/s

(b) 350 m/s

(c) 550 m/s

(d) 667 m/s

(e) 739 m/s

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The average kinetic energy of the molecules in a gas sample depends only on the temperature, T. But given the same kinetic energies, a lighter molecule will move faster than a heavier molecule.

rms speed = √3RT/M where R=8.314 J/(mol • K) and M is molar mass in kilograms per mole. Note that a joule is the same as a kg • m^{2}/s^{2}.

What is the rms speed of N_{2} molecules at 287 K?

What is the rms speed of He atoms at 287 K?

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Find the rms speed of the molecules of a sample of N_{2} (diatomic nitrogen) gas at a temperature of 33.3°C

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Calculate the root mean square velocity of nitrogen molecules at 25 ^{o}C

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Of these gases, which has the fastest-moving molecules (on average) at a given temperature?

a. HBr

b. NO_{2}

c. C_{2}H_{6}

d. they all have the same average speed

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A 0.81 mole sample of CO_{2} is confined in a 20 L container. The volume of the gas sample is decreased to 10L while holding temperature constant. The average molecular speed will.

a. increase

b. decrease

c. remain the same

d. insufficient information to answer

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Which of the following gases will have the largest root mean square speed at 100^{◦}C?

1. water

2. argon

3. oxygen

4. methane

5. nitrogen

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Calculate the root mean square velocity of nitrogen molecules at 25.0°C. (1kg = 1000g)

A) 729 m/s

B) 515 m/s

C) 149 m/s

D) 16.29 m/s

E) 51.2 m/s

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Which gas will have the highest root mean square velocity at the same temperature?

- NH
_{3 }**c)**CO_{2 }d) SO_{2 }e) NO_{2}

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The root mean square speed of nitrogen molecules in air at 20°C is 511 m/s in a certain container. If the gas is allowed to expand to twice its original volume, the root mean square velocity of nitrogen molecules drops to 325 m/s. Calculate the temperature after the gas has expanded.

1. 45.1°C

2. −45.1°C

3. 261°C

4. 347◦C

5. −347°C

6. −261°C

7. 154°C

8. −154°C

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Identify the gas particle that would travel the fastest at a temperature of 293 K.

A. SO_{2}

B. CO_{2}

C. N_{2}

D. Ne

E. Ar

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What is the root mean square speed (in m/s) of hydrogen molecules at 25.0°C?

( R= 8.3145 kg m^{2} s^{-2} mol^{ -1} k^{-1})

a) no given answer is close

b) 3.72 x 10^{6}

c) 6.10 x 10^{1}

d) 1.93 x 10^{3}

e) 3.72 x 10^{3}

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Helium is the lightest noble gas in the air. Calculate its root-mean-square speed (in m/s) in the winter when it is 0^{o}C outside. (R = 8.3145 kg m ^{2} s ^{-2} mol ^{-1} K ^{-1})

a. 1845

b. 1305

c. 1.71 x 10^{3}

d. 1.70 x 10^{6}

e. 41.3

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What is the root mean square speed of carbon dioxide molecules at 98°C?

1. 45.6 m · s ^{-1}

2. 574 m · s ^{-1}

3. 236 m · s ^{-1}

4. 459 m · s ^{-1}

5. 153 m · s ^{-1}

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What is the root mean square velocity (m/s) of H_{2}O steam at 373 K?

A) 2.26

B) 22.7

C) 71.4

D) 372

E) 719

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The root-mean-square speed of gas molecules is 256.0 m/s at a given T. The gas has a molar mass of 32.00 g/mol. What would be the root-mean-square speed for a gas with a molar mass of 131.0 g/mol?

a. 126.5 m/s

b. 62.53 m/s

c. 518.0 m/s

d. 1048 m/s

e. 189.8 m/s

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