An ideal monoatomic gas occupies volume `10^(-3)m^(3)` at temperature `3K` and pressure `10^(3)`Pa. The internal energy of the gas is taken to be zero at this point. It undergoes the following cycle: The temperature is raised to `300K` at constant volume, the gas is then expanded adiabatically till the temperature is `3K` , followed by isothermal compression to the original volume . Plot the process on a `PV` diagram. Calculate (i) The work done and the heat transferred in each process and the internal energy at the end of each process, (ii) The thermal efficiency of the cycle.

Answer

(i) For process `A rarr W =0 , Q = 148.5J`
For process `B rarr W =148.5J , Q =0 , U = 148.5J`
For process `C rarr W= 6.9 J ,Q=-6.9J , U = 0` (ii) `eta= 0.954`
`(##ALN_PHY_C07(I)_E01_282_A01##)`

Explanation

`PV = nRT implies n =(PV)/(RT) = ((10^(3))(10^(-3)))/((25//3)(3)) = (1)/(25)`
For process `AB W = 0 , Q = DeltaU = nC_(V)DeltaT`
=`((1)/(25))((3)/(2)R) (300-3) = (297)/(2) = 148.5J`
For process `BC Q= 0 , W = (nR)/(gamma-1)(T_(1)-T_(2)) = (((1)/(25))((25)/(3)))/(((5)/(3)-1))[300-3]`
=`148.5 J "and" DeltaU = -W = -148.5J`
For process `CA DeltaU=0`
`W= nRTln((V_(A))/(V_(C))) = -6.9J"and" Q = W = -6.9J`
Thermal efficiency `(Q_("net"))/(Q_("supplied")) = (148.5 -6.9)/(148.5) = 0.954`