DERIVE THE RELATION BETWEEN GAMMA AND DEGREE OF FREEDOM (f).
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Suppose, the number of degrees of freedom in a polyatomic gas molecule = n
Internal energy of one gram mole of the gas,
U = n * (1/2)kT * N
Where k = Boltzmann constant, T = temerature of system and N = Avogadro number
Or U = n/2 RT
Now we know,
Cv =dU/dT = > Cv = d/dT (U)
= (n / 2) R
Also we have a relation,
Cp = Cv + R
= ( n / 2) R + R
= (n/ 2 + 1) R
Again, Cp /Cv = Y
= ( n / 2 + 1) R / ( n / 2) R = ( 1 + 2 / n)
Therefore, Y = ( 1 + 2/ n)
Internal energy of one gram mole of the gas,
U = n * (1/2)kT * N
Where k = Boltzmann constant, T = temerature of system and N = Avogadro number
Or U = n/2 RT
Now we know,
Cv =dU/dT = > Cv = d/dT (U)
= (n / 2) R
Also we have a relation,
Cp = Cv + R
= ( n / 2) R + R
= (n/ 2 + 1) R
Again, Cp /Cv = Y
= ( n / 2 + 1) R / ( n / 2) R = ( 1 + 2 / n)
Therefore, Y = ( 1 + 2/ n)
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