A sample of 2 kg of monatomic helium (assumed ideal) is taken through the process ABC another sample of 2 kg of the same gas is taken through the process ADC as shown in Fig. Given molecular mass of helium = 4.
a. What is the temperature of helium in each of the states A, B, C and D ?
b. Is there any way of telling afterwards which sample of helium went through the process ABC and which went through the process ADC ? Write yes or no.
How much is the heat involved in each of the process ABC and ADC ?

Explanation

Number of moles `n=(m)/(M_0)=(2000)/(4)=500`
`C_V=(3R)/(2)`
(a) `T_A=(P_(A)V_A)/(nR)=(5xx10^4xx10)/(500R)=(1000)/(R)`
`T_B=(P_(B)V_(B))/(nR)=(10xx10^4xx20)/(500R)=(2000)/(R)`
`T_C=(P_(C)V_C)/(nR)=(10xx10^4xx20)/(500R)=(4000)/(R)`
`T_D=(P_(D)V_D)/(nR)=(5xx10^4xx20)/(500R)=(2000)/(R)`
(b) No, change in internal energy is independent path
(c) Process ABC:
`DeltaW_(ABC)=DeltaW_(AB)+DeltaW_(BC)`
`=0+10xx10^4xx10^4xx(20-10)`
`=10^6J`
`DeltaU=nC_VDeltaT=nC_V(T_C-T_A)`
`=500R.(3R)/(2)((4000)/(R)-(1000)/(R))`
`=750xx3000=2.25xx10^6j`
`DeltaQ=DeltaU+DeltaW=3.25xx10^6J`
Process ADC:
`DeltaW_(ADC)=DeltaW_(AD)+DeltaW_(DC)`
`=5xx10^4xx(20-10)+0`
`=10^6J`
`DeltaU=2.25xx10^6J`
`DeltaQ=DeltaU+DeltaW=2.75xx10^6J`