Three moles of an ideal gas `(C_p=7/2R)` at pressure, `P_A` and temperature `T_A` is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally gas is compressed at constant volume to its original pressure `P_A`.
(a) Sketch P-V and P-T diagrams for the complete process.
(b) Calculate the net work done by the gas, and net heat supplied to the gas during the complete process.

Answer

(a)`(##BSL_PHY_KT_E01_094_A01##)`
(b)`triangleW=3rT_ain2-(3)/(2)rT_A+0=3RT_a((In2-(1)/(2));`
`triangleQ=3RT_aIn2-(21rT_A)/(4)+(15RT)/(4)=3RT_a(In2-(1)/(2))`

Explanation

(b)In the process 1 to 2 the state chages from `(p_(A)V,T_(A))` to`(p_(A)V,T_(A))`.
Hence `p_(2)=p_(A)/(2)`
Here `triangleU =0triangleW=overset(v)underset(2v)intpdV=3RT_(A)`In 2, `triangleQ=triangleU+triangleW=triangleW`
In the process`2 rarr 3` the state change from `((P_(A))/(2),2V,T_(A)) `"to"` (p_(A)//2,V,T_(3))` so that
`(P_(A))/(2)xx(2V)/(T_(A))=(p_(0)//2xxV)/(T_(3))` "or" `T_(3)=T_A//2`
Here `triangle U = 3C_(v)triangleT_(A)=3(R)/(gamma-1)((T_(A))/(2)-T_(A))=(3RT_(A))/(2(gamma-1))`
`gamma=(C_(p))/(C_(v))=(C_(p))/(C_(p)-R)=((7)/(2)R)/((7)/(2)R-R)=(7)/(5)`
`therefore triangleU= -(3RT_(A))/(((7)/(5)-1))xx2=-(15RT_(A))/(4)`
`triangleW=overset(v)underset(2v)intpdV=(p_(A))/(2)(V-2V)=(p_(A)V)/(2)=-(2RT_(A))/(2)`
`therefore triangleQ+=triangleU+triangleW=-(15)/(4)RT_(A)-(3)/(2)RT_(A)=-(21RT_(A))/(4)`
In the process 3 `rarr` 1, the state changesf from`(p_(A)/(2),V,T_(A)/(2))` to `(p_(A),V,T)`that
`(p_(A)//2xxV)/(T_(A)//2)=(p_(A)V)/(T)` or `T=T_(A)`
`triangleU=3C_(v)(T_(A)`-`T_(A)/(2))=(3R)/(7/5-1)xx(T_A)/(2)=(15)/(4)RT_(A)`
`triangleW =0`
`therefore `triangleQ=triangleU=(15)/(4)RT_(A)`
`therefore` Net `triangleW=3RT_(A)in2-(3)/(2)RT_(A)+0=3RT(In2-1/2) `therefore`"Net"`triangle`Q=3RT_(A)In2-(21RT)_(A)/(4)+(15RT)/(4)=3RT_(A)(In2-(1)/(2))`