Degree of freedom

 Degree of Freedom

Introduction: We know that every molecule can move in space in three dimension i.e x , y , z axis. This coordinates are enough to specify it's location in a space. If a molecule is restricted to move in a plane then two coordinates are enough to specify it's location. Let us  consider a body which can move in only two direction in a plane i,e  x and y direction of cartesian coordinate system, and if it is in motion then its velocity component in x and y direction  ( v_x and v_y)is enough to specify its position and motion. Thus the number of direction in which molecule can move and number of motion it can perform relates the degree of freedom. In the above fig, a molecule can move in translatory direction and can perform rotational motion. This parameter is enough to describe its position and motion

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Degree of Freedom : It is defined as total numbers of coordinates  or independent quantities required to describe the position and configuration of a system completely.

For monoatomic gases,

Let us consider Helium (He).It can move in three direction in a space i.e  in x , y, z direction. Thus it has only 3 degree of freedom.

For diatomic gases,

Let us consider a diatomic molecule like Oxygen O₂  or Nitrogen N₂ ,

It is free to move in 3 translation direction i.e in x , y, z direction. In additional it is free to perform rotational motion around Y and Z direction, if molecule is situated on X axis. As it is situated on X axis it cannot rotate around X axis in the sense there will be no change in position of molecule while rotating on X axis.

Transalation degree of freedom: 3

Rotational degree of freedom :2

Degree of freedom : 5

If ω₁  and ω₂  are the angular velocities along Y and Z axis and I₁  and I₂  are corresponding moment of inertia then total energy is given by,

Total Energy = E_tr+ E_ro 

E_tr+ E_ro =  1/2 mv²_x  + 1/2 mv²_y  +  1/2 mv²_z+  1/2 I₁ω²₁ +  1/2 I₁ω²₂

_{Molecules like, CO have additional vibrational motion,Thus}

Total Energy = E_tr+ E_ro + E_vr 

_{Evr= 1/2 (dy/dt)²  + 1/2(ky)}² 

_{where k is force constant and y is vibrational coordinate.}

_{Important note:E ach translation and rotational motion constitute 1/2 Kb} T of energy  as it have only one squared term but vibrational frequency contributes 

 E_vr = 2 x _{1/2 Kb} T  = _Kb T 

<<<Kinetic Energy of gas>>>│<<<Mayers equation>>>

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