Molecular Speeds

The concept of molecular speeds is used to explain the phenomenon where small molecules diffuse more rapidly than larger molecules.

Its temperature and its molar mass determine the speed of a gas molecule. The molecular speed of a gas is directly proportional to its speed and inversely proportional to its molar mass. Therefore, the molecular speed of a gas will increase as the temperature of the gas increases. 

Since the gas helium has the lowest molar mass, it has the highest molecular speeds. However, xenon, which has the highest molar mass, has the lowest molecular speeds. If we observe the gas molecules of two gases at the same temperature, we will ascertain that the gas with a heavier mass is slower than the gas with a lighter mass. In equation form, molecular speed can be expressed as v = 3 R Tm. The molecular speed of oxygen is 500 m/s.

In this article, we will discuss the different types of molecular speeds. There are three different types of molecular speeds: Root mean square speed, average molecular speed, and most probable speed. We will also look at their formulas and solved examples. 

Types of Molecular Speed 

There are three types of molecular speed. These are average molecular speed, root mean square speed, and most probable speed. 

• Average molecular speed is the average speed of a group of molecules. The average molecular speed formula is u av = 8 R Tπm. 

• Root mean square speed is the measure of the speed of particles in a particular gas. The root mean square speed formula is urms = 3 R Tm . Root mean square speed is expressed as urms = u2

• The speed which is acquired by most of the molecules in a gas is known as the most probable speed. The most probable speed formula is ump = 2 R Tm. 

• The relationship between average molecular speed, root mean square speed, and most probable speed is- urms> uav > ump. Where urms = root mean square speed, uav = average molecular speed, and ump = most probable speed. 

• The ratio between average molecular speed, root mean square speed, and most probable speed is- urms: uav: ump:: 1:1.128:1.224. 

Relationship between Kinetic Energy and Molecular Speed 

• In a gas, individual molecules exhibit a distribution of kinetic energies because speeds are distributed due to the molecules colliding in the gaseous phase. The speed of each individual molecule is subject to change due to collision. Overall, in a sample of gas, all of the molecules share average kinetic energy. 

• The kinetic molecular theory states that the average kinetic energy of gas particles is proportional to the absolute temperature of the gas. 

• To determine the value of average translational kinetic energy for the movement of a gas particle in a straight line, we require the mean of the square of speeds of all molecules. This concept is represented as – 

u2= u12+u22……un2n. 

Maxwell Distribution of Molecular Speeds 

• James Maxwell and Ludwig Boltzmann derived the equation for the distribution of molecular speeds in gas in the mid-19th century. 

• The Maxwell distribution of speeds is graphically represented with the help of this equation. 

• The most probable speed is the speed that corresponds with the peak of the curve. The average molecular speed is the speed that is just a little higher than the most probable speed. And the speed which corresponds with the average kinetic energy of molecules is the root mean square speed.   

• By studying the shape of the speed distribution curve, a lot of information about the gas can be determined. The Maxwell distribution of speed curve shape will be dependent upon the molar mass and temperature of the gas. 

• The Maxwell distribution of the speed curve will spread and flatten out when we observe the gas at increasing temperatures. 

• At increasing temperature, as the most probable speed increases, we will observe the shift of the peak to the right. It is also observed that the particles of gas tend to move faster. 

• The Maxwell distribution of speed curves gets taller and flatter when we consider gases of increasing molar mass. In this case, the peak shifts towards the left as the most probable speed decreases.  

• The distribution of speeds is constant and fixed in a state of equilibrium. 

• Through their experimentation and derivation, Maxwell and Boltzmann were able to show that the distribution of molecular speeds depends upon the molecular mass and temperature of the gas. 

Molecular Speeds Solved Examples 

Example 1: Determine the speed of particles of m = 1 gr/mol and temperature 1000 k if the particles do not interact with each other. 

Solution: From the aforementioned statement we can gather the following information- 

Mass (m) = 1 gr/mol 

Temperature = 1000k 

We know that molecular speed =√ 3 R T / m  

Where, R = ideal gas constant (8.314 kg m2/s2 mol K) 

Therefore, molecular speed = v = √ 3 ×8.314 kgm2s2mol K ×1000k / 0.001 kg/mol  

v = 2883.4 m/s. 

Example 2: Determine the speed of molecules whose mass is 2 gr/mol and whose temperature is 900 k if the particles are not reacting with each other. 

Solution: From the aforementioned statement we can gather the following information-

Mass (m) = 2 gr/mol 

Temperature = 900 K 

We know that molecular speed = 3 R T /m  

Where, R = ideal gas constant (8.314 kg m2/s2 mol K) 

Therefore, molecular speed = v = 3 ×8.314 kgm2s2mol √ K ×900k / 2 gr/mol

v = √11223.9

v = 105.9 m/s

Therefore, if the particles are not reacting with each other, then the speed of the molecules is 105.9 m/s. 

Example 3: Determine which of the following quantities is larger, the rms velocity of oxygen at 75 °C or the rms velocity of nitrogen at 25 °C. 

Solution: The root mean square speed of Oxygen at 75 °C is 521 m/s, and the root means the square speed of nitrogen at 25 °C is 515 m/s. Therefore, oxygen at 75 °C is larger than nitrogen at 25 °C.

Conclusion 

In this article, we looked in depth at kinetic energy and molecular speeds. We also explored concepts such as Maxwell Distribution of molecular speeds. Determining the molecular speed of gases helps ascertain how they will react under different temperatures and conditions in an experimental setup.