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Question
Two ideal gases have the same value of Cp / Cv = γ. What will be the value of this ratio for a mixture of the two gases in the ratio 1 : 2?
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Solution
For the first ideal gas,
Cp1 = specific heat at constant pressure
Cv1 = specific heat at constant volume
n1 = number of moles of the gas
`"C"_("p"1)/"C"_("v"1) = gamma and "C"_("p"1) -"C"_("v"1) = "R"`
`=> gamma "C"_("v"1) - "C"_("v"1) ="R"`
`=> "C"_("v"1)(gamma-1) ="R"`
`=> "C"_("v"1) = "R"/(gamma -1)`
`"C"_("p"1) = gamma "R"/((gamma-1))`
For the second ideal gas,
Cp2 = specific heat at constant pressure
Cv2 = specific heat at constant volume
n2 = number of moles of the gas
`"C"_("p"2)/"C"_("v"2) = gamma and "C"_("p"2) -"C"_("v"2) = "R"`
`=> gamma "C"_("v"2) - "C"_("v"2) ="R"`
`=> "C"_("v"2)(gamma-1) ="R"`
`=> "C"_("v"2) = "R"/(gamma -1)`
`"C"_("p"2) = gamma "R"/((gamma-1))`
Given:
n1 = n2 = 1 : 2
dU1 = nCv1dt
dU2= 2nCv2dT
When the gases are mixed,
nCv1dT + 2nCv2dT = 3nCvdT
`"C"_"v" = ("C"_("v"1) +2"C"_("v"2))/3`
`= (" R"/(gamma-1) +(2"R")/(gamma-1))/3`
`= (3"R")/((gamma-1)3) = "R"/(gamma-1)`
Hence, Cp / Cv in the mixture is γ.
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