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Average Kinetic Energy

We know that all objects have some energy. It can be in different forms like kinetic, vibration, rotation; potential etc. Energy is associated with certain kind of movement such as kinetic energy is associated with the motion of object whereas potential energy is associated with the stationary state of the object. According to First Law of Thermodynamics; energy cannot created or destroyed but can convert from one form to another. Therefore gaseous particles can convert their potential energy to kinetic energy and vice versa. 

The energy associated with translational movement is called as translation energy whereas rotational energy is associated with rotation of constituent atoms around their chemical bonds in molecule.

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Average Kinetic Energy Definition

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Kinetic energy is the energy of motion of particles of a system. Here particles could be atoms or molecules. In an object, kinetic energy depends on mass (m) and velocity (v). So the kinetic energy of an object can be written as

KE = $\frac{1}{2}$ x Mass (kg) x $\left [ Velocity\left (\frac{m}{s} \right ) \right ]^{2}$

The SI unit of energy called joule (J).

Any of the system consists of many atoms or molecules which are in random motion especially for gaseous system. The average of kinetic energy of all the molecules gives the average kinetic energy of the system. 

Average Kinetic Energy of Molecules

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We know that in a chemical reaction, reactant molecules converted to product molecules only when there are some effective collisions between reactant molecules. For the formation of products, there must be effective collisions between molecules and for effective collisions, molecules require kinetic energy. The average kinetic energy of molecules depends on the temperature. 

At high temperature, there are more molecules with greater kinetic energy than molecules at low temperature. So we can say that average kinetic energy of molecules depends on temperature and it determines the rate of a chemical reaction. 

We know that there are mainly two types of chemical reactions; exothermic and endothermic reactions. In an exothermic reaction, when energy releases during the reaction, the potential energy is converted to kinetic energy whereas in an endothermic reaction, the kinetic energy is converted to potential energy during absorption of energy and it increases effective collisions between reactant molecules. 

Average Kinetic Energy Equation

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The average kinetic energy for one molecule is $\frac{1}{2}$ mv2. For one mole of molecules it can be written as

$\frac{1}{2}$ M(vrms)2 = $(\frac{3}{2})$ RT

For ideal gases, the average molecular kinetic energy depends only on temperature.

Average Kinetic Energy of a Gas

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Gaseous molecules show random motion as they have greater kinetic energy and can move easily.

The average kinetic energy of each molecule is

KEavg  = $\frac{3}{2}$ kT

We can calculate total kinetic energy by multiplying the average kinetic energy with the number of molecules, nNA, here n is the number of moles and k is Boltzmann constant;

KEtotal$\frac{3}{2}$ nNAkT

Since

NAk  =  R(universal gas constant)

So KEtotal = $\frac{3}{2}$ nRT

Measure of Average Kinetic Energy

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Find out the kinetic energy of an object whose mass is 55 kg and it is travelling with 23 m/s.

The average kinetic energy can be written as

KE = $\frac{1}{2}$  x Mass (kg) x [ Velocity $(\frac{1}{2})$]2

Here;
Mass = 55 kg
Velocity = 23 m/s

Plug in these numbers into the kinetic energy equation

KE = $\frac{1}{2}$ x 55 kg x [23 $\frac{m}{s}$]2

= 14547.5 Joules

Average Kinetic Energy Formula

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The formula for average kinetic energy of ideal gases is;

KE  =
$\frac{3}{2}$ nRT

Here
n = Number of moles of the gas
R = Gas constant (8.3145 J mole-1 K-1)
T =Temperature (Kelvin)

When we apply Boltzmann distribution to average kinetic energy, it becomes
 
$(v_{z}^{2})$$\sqrt{\frac{m}{2\pi kT}} \int_{-\infty}^{\infty} v_{z}^{2} e^{(\frac{mv_{z}^{2}}{2kT})} dv_{z}$
    
The average value of the square of the velocity would be;

$(v_{z}^{2})$$\sqrt{\frac{m}{2\pi kT}} \int_{-\infty}^{\infty} v_{z}^{2} e^{(\frac{mv_{z}^{2}}{2kT})} dv_{z}$
 
The average kinetic energy for one dimension can be written as;

$\frac{1}{2}$$(v_{z}^{2})$ = $\frac{kT}{2}$
 
If the motion in three dimensions, then the average kinetic energy is:

$\frac{1}{2}$ mv2 = $\frac{3}{2}$ kT

The average kinetic energy of molecules is postulated by the kinetic molecular theory. According to kinetic molecular theory, gases are composed of a large number of hard, spherical particles which are in random motion.

Gaseous particles move in a straight line and collide with another particle or the walls of the container. Gaseous particles are too small therefore the most of the volume of gas is empty space.

The intermolecular force of attraction between gaseous particles is negligible. Collisions between gaseous particles are perfectly elastic and there is no loss of energy during collisions.

The average kinetic energy of particles entirely depends on the temperature. The average kinetic energy depends on temperature and can be expressed as
KE = $\frac{1}{2}$ mv2

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