You are watching: Which gas molecules have the highest average kinetic energy at this temperature?

Temperature and also pressure space macroscopic nature of gases. This properties are regarded molecular motion, i m sorry is a microscopic phenomenon. **The kinetic theory of gases correlates between macroscopic properties and microscopic phenomena**. Kinetics way the study of motion, and in this case motions the gas molecules.

At the same temperature and also volume, the very same numbers of mole of all gases exert the same pressure on the walls of your containers. This is recognized as **Avogadro"s principle**. His theory means that the same numbers of moles of gas have the same variety of molecules.

Common sense tells united state that the pressure is proportional come the average kinetic power of all the gas molecules. Avogadro"s principle likewise implies the the kinetic energies of various gases room the exact same at the very same temperature. The molecular masses are different from gas to gas, and if all gases have the same median kinetic energy, the mean speed of a gas is unique.

Based on the over assumption or theory, Boltzmann (1844-1906) and also Maxwell (1831-1879) extended the concept to indicate that the mean kinetic power of a gas relies on that temperature.

They let* u* be the average or **root-mean-square speed** the a gas who molar fixed is *M*. Since *N* is the Avogadro"s number, the average kinetic energy is (1/2) (*M/N) u*2 or

(mathrmK.E. = dfracM2 Nu^2 =dfrac3R T2 N=dfrac32,k T)

Note that *M / N* is the fixed of a single molecule. Thus,

(eginalignu &= left(dfrac3k N TM ight)^1/2\&= left(dfrac3 R TM ight)^1/2endalign)

where *k* (= *R/N*) is the **Boltzmann constant**. Keep in mind that *u* so evaluated is based on the average energy of gas molecules being the same, and it is called the root-mean-square speed; *u* is no the median speed of gas molecules.

These formulas correlate temperature, pressure and kinetic power of molecules. The circulation of gas speed has actually been studied by Boltzmann and Maxwell as well, yet this is beyond the scope of this course. However, friend may an alert that at the same temperature, the mean speed of hydrogen gas, (ceH2), is 4 times more than the of oxygen, (ceO2) in order to have actually the same mean kinetic energy.

## Calculation the Effusion rate By Comparison

For 2 gases, in ~ the exact same temperature, with molecular masses *M*1 and also *M*2, and also average speeds* u*1 and *u*2, Boltzmann and also Maxwell theory suggests the adhering to relationship:

Thus,

The repercussion of the over property is that the effusion rate, the root mean square speed, and also the most probable speed, room all inversely proportional to the square root (SQRT) that the molar mass. Simply formulated, the Graham"s law of effusion is

< extrmrate of effusion = dfrackmathrm d^1/2 = dfrack"mathrm M^1/2>

Have you noticed the helium balloons were typically deflated the following day while sizes of common air balloons will save at the very least for a couple of days? tiny helium molecule not only effuse with the tiny holes that the balloons, they likewise effuse much much faster through them.

The theories extended here enable you to make numerous predictions. Apply these theory to settle the complying with problems.

Example 1

Calculate the kinetic energy of 1 mole the nitrogen molecule at 300 K.

**Solution**

Assume nitrogen behaves as suitable gas, then

(eginalign*E_ extrm k &= dfrac32 R T\&= mathrmdfrac32 imes 8.3145: dfracJmol: K imes 300: K\&= mathrm3742: J / mol: (or: 3.74: kJ/mol)endalign*)

**Discussion**

At 300 K, any gas that behaves like perfect gas has actually the same energy per mol.

Example 2

Evaluate the root-mean-square rate of (ceH2), (ceHe), (ceN2), (ceO2) and (ceCO2) in ~ 310 K (the human body temperature).

**Solution**

Recall that

(eginalign*u &= left(dfrac3k N TM ight)^1/2\&= left(dfrac3 R TM ight)^1/2\&= left(dfrac3 imes 8.3145 imes 3100.002 ight)^1/2\&= 1966: mathrmm/sendalign*)

Note that the molecule mass that hydrogen is 0.002 kg/mol. This units room used since the constant *R* has actually been calculated making use of the SI units. The calculation for various other gases is achieved using their molar fixed in kg.

(eginalign*u &= left(dfrac3k N TM ight)^1/2\&= left(dfrac3 R TM ight)^1/2\&= dfrac(3 imes 8.3145 imes 310)^1/2M^1/2\&= dfrac87.9345M^1/2: mathrmm/sendalign*)

The root-mean-square speeds are:

Gas Molar mass (u) (root-mean-square rate in m/s)(ceH) | 2 | 1966 |

(ceHe) | 4 | 1390 |

(ceH2) | 28 | 525 |

(ceO2) | 32 | 492 |

(ceCO2) | 44 | 419 |

**Discussion**

Molar masses space 349 and also 352 because that (ce^235UF6) and (ce^238UF6) respectively. Making use of the an approach above, their root-mean-square speeds room 149 and also 148 m/s respectively.

The separation that these two isotopes of uranium to be a necessity during the time of battle for the united state scientists. Gas diffusion was among the techniques employed for your separation.

Example 3

Assume air and helium molecules pass with the undetected feet in balloons with equal opportunities. If a helium balloon takes 10.0 hrs to mitigate its size by 5.0 %, how long will certainly it take it a nitrogen balloon to reduce its size by 5.0 %?

**Solution**

The effusion rates are

< extitrate of effusion =dfrackmathrm d^1/2=dfrack "mathrm M^1/2 onumber>

Let"s assume an mean rate that effusion that helium to it is in 5/10 = 0.5, climate the effusion rate of nitrogen is 0.5 * (4/28)1/2 = 0.189. The time required to effuse the exact same amount is hence 10*0.5/0.189 = 26.5 hr.

**Discussion**

The time compelled can be evaluate by

<eginalign*time &= mathrm10 imes(28/4)^1/2: hr\&= mathrm26.5: hrendalign*>

## Confidence building Questions

**Calculate the root mean square rate of **(ceN2)** (molar massive = 28) in ~ 37 °C (310 K, body temperature). R = 8.314 kg m2/(s2 mol K). **

Hint: 525 m/s

**Skill:** calculation the root typical square speed *u* that gas molecules.

**Which gas has a greatest root mean square speed:**(ceH2)

**,**(ceN2)

**,**(ceO2)

**,**(ceCH4)

**, or**(ceCO2)

**?**

Hint: hydrogen gas

**Skill:** discovering that (ceH2) has actually the lowest molecular mass renders your decision easy.

**Which gas has a lowest root mean square speed:**(ceH2)

**,**(ceN2)

**,**(ceO2)

**,**(ceCH4)

**, or**(ceCO2)

**?**

Hint: carbon dioxide

**Skill:** know that (ceCO2) has the greatest molecular mass.

**Assuming appropriate gas behavior, calculate the kinetic energy of 1.00 mole of **(ceN2)** at 37 °C (= 310 K). R = 8.314 J/(mol K); **(ceN2) molar massive = 28.0

Hint: 3.87e3 J/mol

**Skill:** calculate the kinetic energy of any type of amount of any gas at any temperature.

**Which of the complying with gases has the greatest effusion rate through a pinhole opening of that container (T = 300K): **(ceHe)** (molar mass 4), **(ceN2)** (28), **(ceO2)** (32), **(ceCO2)** (44), **(ceSO2)** (64), or **(ceAr)** (40)? **

Hint: helium

**Skill:** associate effusion rate with root mean square speed, and also determine the effusion rates.

**For i m sorry gas in the accompanying list is the effusion price the the smallest (T = 400 K): **(ceNH3)** (17), **(ceCO2)** (44), **(ceCl2)** (71), **(ceCH4)** (16)? **

Hint: chlorine gas

**Skill:** calculate the root typical speed because that these gases. The median speed is 324 m/s for (ceCl2), 681 m/s because that (ceCH4).

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**The most probable rate of **(ceCH4)** (molar massive 16.0) at a given temperature is 411 m/s. What is the many probable rate of **(ceHe)** (molar fixed 4.00) at the very same temperature?**

Hint: 822 m/s

**Discussion:** at the exact same T, the many probable speed of a gas is inversely proportional to the square-root the its molar mass.

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