The ratio of the molar heat capacities of an ideal gas is `(C_p / C_v = 7/6)`. Calculate the change in internal energy of 1.0mole of the gas when its temperature is raised by `50 K `, (a) keeping the pressure constant , (b) keeping the volume constant and (c ) adiabatically.

Explanation

`(C_p/C_V)=7/6`, `n=1 mol.`, `DeltaT=50 K`
(a) Keeping the pressure constant,
`dQ = dU + dW`,
`DeltaT = 50 K, gamma =7/6`, `n=1 mol.`
`dQ=dU=dW`
`implies nC_pdT=dU+RdT`
`dU=nC_pdT-RdT`
`=1xx(Rgamma)/(gamma-1)xxdT-RdT`
`=7Rdt - RdT`
`=7Rdt-RdT=6RdT`
`=6xx8.3xx50 = 2490J.`
(b) Keeping volume constant,
`dU=nC_vdT`
`=1 xx R/(gamma-1) xxdT`
`=1((8.3)/(7//6)-1)xx50 `
`=8.3 xx 50 xx 6 = 2590 J.`
(c) Adiabatically, `dQ=0`,
`dU=-dW`
`=[(nxxR)/(gamma-1)(T_1-T_2)]`
`=(1xx83)/(7//6-1)= (T_2-T_1)`
`=8.3 xx 6 xx 50 = 2490 J`.