Kinetic Energy and Molecular Speeds

To study the action of molecules scientists have thought to study a theoretical model and that model is the Kinetic theory of gases and it assumes that molecules are very small relative to the distance between molecules. Typically, the actual properties of solids and fluids can be depicted by their size, shape, mass, volume, and so on, when talking about gases, they have no shape, size while mass and volume are not directly measurable.

Kinetic theory of gases

The Kinetic theory of gases is helpful and can be applied to this situation, with the assistance of the kinetic theory of gases, the actual properties of any gas can be characterized commonly as far as three measurable properties. The pressure, volume, and temperature of the compartment where the gas is put away or present.
The kinetic theory of gases explains the random movement of molecules in a gas. The kinetic theory of gases depicts how gases act by accepting that the gas is comprised of rapidly moving particles or atoms.

General terms related to the kinetic theory of gases

• Pressure: Pressure is defined to be the amount of force exerted per area.
• Volume: Volume is the amount of 3D space a substance or object occupies
• Temperature: Temperature is the property of matter which reflects the quantity of energy of motion of the component particles. It is a comparative measure of how hot or cold a material is. 
  The gas constant: A  gas constant in the equation of state of gases that are equal in the case of an ideal gas to the product of the pressure and volume of one mole divided by the absolute temperature. R = 8.314Jmol^-1 K^-1 = 2calmol^-1K^-1 = 0.0821L-atm-mol^-1K^-1

Introduction

The kinetic theory of gases is a model of the thermodynamical behavior of gases. This model describes a gas that has a large number of submicroscopic particles which are in rapid, random motion, and frequently collide with each other and with the walls of any container. The higher the temperature, the greater they collide.

Kinetic energy

Kinetic energy is a form of energy that an object by reason of its motion. Kinetic energy is proportional to the speed of the molecules. As the speed of the colliding molecules increases, so does the total kinetic energy of all the gas molecules.
Their size is assumed to be smaller than the average distance between the particles. The kinetic theory of gases explains the macroscopic properties of gases such as volume, pressure, and temperature, as well as properties such as viscosity and thermal conductivity. This model also tells about Brownian motion.

Postulates of the kinetic theory of gases

1. Gases are made up of rigid molecules that are spherical in shape.
2. The volume of the molecule is negligible as compared to the volume of gas [volume of container] as compared. 
3. There are no intermolecular forces.
4. Molecules are constantly in random motion going for perfectly plastic collisions.
5. These molecules when colliding with the walls of the container they exert pressure.
6. There is no impact contain on gravity on the molecule of gas

Average kinetic energy

Average Kinetic energies is directly proportional to temperature

Average kinetic energy = 3/2RT for 1 mole

For n moles, Average kinetic energy = 3/2nRT

Average kinetic energy = 3/2KT for 1 molecule 

Here, K is called Boltzmann constant and this is equal to1.38 × 10^-23 J/K 

PV = 1/3mn(v_rms)²

Here, m is the mass of one molecule

n is the number of moles

V_rms is RMS velocity

Example: Find K.E of 5 moles of O₂ in 370 in Joule?

Answer:

KE = 3/2 × 5 × 8.314 × 300 = 19330J  

Molecular Speeds

The speed related to a gathering of atoms is normal. In an ideal gas, the particles don't interface with one another. There are 3 types of molecular speeds, they are RMS velocity, Average velocity, and Most probable velocity. Here are the respective formulae for different speeds.

• R.M.S velocity: The root-mean-square (RMS) speed is the worth of the square foundation of the amount of the squares of the stacking speed esteems partitioned by the quantity of qualities.

V_rms = √(3RT)/(M) or √(3P)/(d)

• Average velocity: Average speed is the arithmetic mean of the speeds of the molecules.

V_avg = √(8RT)/(πM) or (√8P)/(πd)

• The most probable velocity:-The speed that corresponds to the peak of the curve is called the most probable velocity. 

V_mp = √(2RT)/(M) or √(2P)/(d )

Ratio of V_mp: V_avg: V_rms = 1:1.128:1.224

Sample Problems

Question 1: IF V_rms is 6.12m/s find, V_mp

Solution:

We know that V_mp:V_rms = 1:1.224 
Given that, V_rms = 6.12 so V_mp = V_rms/1.224 
V_mp = V_rms/1.224 = 6.12/1.224 = 5 
So, V_mp = 5m/s 

Question 2: Find K.E of 1mole of O₂ in cal/mole at 27^°C.

Solution:

Avg. KE = 3/2nRT 
Given number of mole(n) = 1, T = 27 + 273 = 300K 
And as asked answer in cal/mol so, R = 2
Substituting the given values in formula,  

Avg KE = 3/2 × 1 × 2 × 300 = 450.
So average kinetic energy = 450cal/mole.       

Question 3: A gas has three molecules with velocities 100m/s,200m/s,500m/s find the rms velocity.

Solution:

V_rms = √[(100)² + (200)² + (500)²]/3
= 100√[1 + 4 + 25]/3
= 100√10
= 100 × 3.3
= 330m/s

Question 4: Find the ratio of He, CH₄, SO₂ at a certain temperature is?

Answer:

Note that average kinetic energy depends only on temperature it doesn't depend on type of molecules, molecular weight of compound, etc. 
So answer is 1:1:1

Question 5: For Helium gas, the RMS velocity at 800K is?

Answer:

V_rms = √(3RT)/(M) = √(3 × 8.314 × 800)/4 × 10^-3) = 500√20 = 2236.06 m/s
Note: R should be in J and weight should be in Kg for S.I units

Question 6: Find the average kinetic energy of the ideal gas per molecule at 25°C?

Solution:

Average kinetic energy per molecule = 3/2KT
Boltzmann constant, K = 1.38 × 10^-23 and temperature (T) = 298K
Average kinetic energy per molecule = 3/2 × 1.38 × 10^-23 × 298 = 6.17 × 10^-23J

Question 7: V_rms, V_avg, V_mp are root mean square, average, and most probable speeds of molecules of a gas obeying Maxwellian velocity distribution arrange them in descending order.

Answer:

V_mp: V_avg: V_rms = 1: 1.128: 1.224 

V_avg = 1.128V and V_rms = 1.224 and V_rms/V_avg = 1.224/1.128

= 1.085

So from above observations we can say that V_rms>V_avg>V_mp

Question 8: V_rms of CO₂ at temperature T is X cm/sec  at what temperature it would be 4X

Solution:

V_rms = √(3RT)/(M), So V_rms is directly proportional to √T. 

Let's assume V_rms at Xcm/sec be V₁ and  V_rms at 4Xcm/sec be V₂  

V₁/V₂ = √T₁/T₂ 

(V₁/V₂)² = T/T₂  (As T₁ = T) 

(X/4X)² =  T₁/T₂  

So ,T₂ = 16T