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Calculate the specific heat capacity C(r...

Calculate the specific heat capacity `C_(r)` of a gaseous mixture consisting of `v_(1)` moles of a gas of adiabatic exponent `gamma_(1)` and `v_(2)` moles of another gas of adiaqbatic exponent `gamma_(2)`.

Text Solution

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The internal energy of an ideal gas of mass m is given by
`U = (pV)/(gamma - 1)`
Internal energy is an extensive property.
`:. U_(mix) = U_(1) + U_(2) implies (p_(mix))/(gamma - 1) = (P_(1))/(gamma_(1) - 1) + (P_(2))/(gamma_(2) - 1)`
(as volume is the same for all)
From the formula `PV = nRT`
`._(P_(mix)) V = (v_(1) + v_(2)) RT` , `._(P_(1))V = ._(v_(1)) RT` , `._(P_(2)) V = ._(v_(2)) RT`
`:. P_(1) = (v_(1))/(v_(1) + v_(2)) P_(mix)` and `P_(2) = (v_(2))/(v_(1) + v_(2)) P_(mix)`
`:. (1)/(gamma - 1) = (1)/(v_(1) + v_(2)) ((v_(1))/(gamma_(1) - 1) + (v_(2))/(gamma_(2) - 1))`
`C_(v) = (R )/(gamma - 1) = (R )/(v_(1) + v_(2)) ((v_(1))/(gamma_(1) - 1) + (v_(2))/(gamma_(2) - 1))`
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