During an adiabatic process, the the pressure of a mixture of monatomic and diatomic gases is found to be proportional to the cube of the temperature . Find the value of `gamma=(C_P//C_V)`

Explanation

Meyer's relation is, `C_P-C_V=R`
`:. C_P = C_V+R`
`gamma=C_P/C_V=(C_V+R)/(C_V)`
`gamma=1+R/(C_V)`
`:. C_V=R/(r-1)`
For monoatomic gas,
`C_V=R/(5/3-1)=R/(((5-3))/3)`
`=R/(2//3)=3/2R`
For diatomic gas, `C_V'=R/((7)/(5)-1)=R/(((7-5))/5)`
`=R/(2//5)=(5R)/2`
`:.C_(V("mixture"))=(nC_V+n'C_V')/(n+n')`
`n= 1 , n' = 1`
`:.C_(V("mixture"))=(1xx(3/2R)+1(5/2R))/(1+1)`
`=(3/2R+5/2R)/2`
`=(8R)/(2xx2)=2R`
`gamma_("mixture")=1+R/(C_(V("mixture")))`
`1+R/(2R)`
`=1+1/2=3/2`
`gamma=C_P/C_V=3/2`
`:. C_P/C_V=3/2`