Heat, First Law of Thermodyna

Assume the mixture is given a qunatity of heat energy Q which causes a rise of temperature T in the mixture.

Let Q1, Q2 and Q3 be the heat abosrbed by the 3 gases respectively.
Hence, Q1 = (n1).(C1).T
Q2 = (n2).(C2).T
Q3 = (n3).(C3).T

By conservation of energy,
Q = Q1 + Q2 + Q3
i.e. Q = (n1).(C1).T +(n2).(C2).T + (n3).(C3).T
Q = [(n1).(C1) +(n2).(C2) + (n3).(C3)].T

But when consider the mixture as a whole, we have,
Q = M.C.T
where M is the mass (in noles) of the mixture
C is the molar specific heat at constant volume of the mixture

Therefore, M.C.T, = [(n1).(C1) +(n2).(C2) + (n3).(C3)].T
C = [(n1).(C1) +(n2).(C2) + (n3).(C3)]/M
but M = (n1 + n2 + n3)
Hence, C = [(n1).(C1) +(n2).(C2) + (n3).(C3)]/(n1 + n2 + n3)

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