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Question
An ideal gas is taken through a process in which the pressure and the volume are changed according to the equation p = kV. Show that the molar heat capacity of the gas for the process is given by `"C" ="C"_"v" +"R"/2.`
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Solution
Relation between pressure and volume of a gas is P = kV.
Ideal gas equation is PV = nRT.
`=> ("n""R""T")/"V" = "k""V"`
`=> "n""R""T" = "k""V"^2`
For simplicity, take the number of moles of a gas, n = 1.
⇒ RdT = 2 kVdV
`=> ("R""d""T")/ (2"k""V") = "d""V"`
From the first law of thermodynamics,
dQ = dU + dW
⇒ nCPdT = CVdT + PdV
`=> "n""C"_"p""d""T" = "C"_"v""d""T" +("P""R""d""T")/(2"k""V")`
`=> 1 xx"C"_"P" ="C"_"v" +("P""R")/(2"k""v")`
`therefore "C"_"p" = "C"_"v"+"R"/2`
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