Problem on Second Law of Thermodynamics

[adipta_datta]

A reversible engine works between three thermal reservoirs-A,B and C.The engine absorbs an equal amount of heat from the thermal reservoirs A & B kept at temperatures Ta and Tb respectively,and rejects heat to the thermal reservoir C kept at temperature Tc.The efficiency of the engine is\alpha times the efficiency of the reversible engine,which works between the two reservoirs A & C.Prove that:Ta/Tb=(2\alpha -1)+2(1-\alpha )Ta/Tc

 

[tiny-tim]

welcome to pf!

hi adipta_datta! welcome to pf!

(have an alpha: α )

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[adipta_datta]

Below is the thing which I have attempted:

eta of Heat engine 2(between the reservoirs A&C)=(Ta-Tc)/Ta
therefore,eta of heat engine 1(between reservoirs A,B&C)=\alpha[(Ta-Tc)/Ta]
where eta is the efficiency of the heat engine.
Now heat absorbed by heat engine 1=2Q1
Heat rejected by heat engine 2=Q2.
therefore,eta of heat engine1=1-(Q2/2Q1)=1-(Tc/2Tb)

therefore,
1-(Tc/2Tb)=\alpha[(Ta-Tc)/Ta]-1,
which gives to me Ta/Tb=2\alpha+2(1-alpha)(Ta/Tc)

I am not getting the -1 term after 2 alpha.

Please help and rectify the problem.The diagram is attached.
Thanks.