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Mechanical Properties Of Fluids

Question
CBSEENPH11019819

A Carnot engine P works between 827°C and T and another engine Q is working between T and 227°C. Find the value of T at which the efficiency of both the engines is same.

Solution

The efficiency of engine P is, 
            space space space straight eta subscript straight P equals 1 minus straight T subscript 2 straight P end subscript over straight T subscript 1 straight P end subscript 
                 equals 1 minus fraction numerator straight T over denominator 827 plus 273 end fraction
equals 1 minus straight T over 1100 
The efficiency of engine Q is, 
           straight eta subscript straight Q equals 1 minus straight T subscript 2 straight Q end subscript over straight T subscript 1 straight Q end subscript equals 1 minus fraction numerator 227 plus 273 over denominator straight T end fraction
space space space space space equals space 1 minus 500 over straight T 
Here,         straight eta subscript straight P equals straight eta subscript straight Q
∴   1 minus straight T over 1100 equals 1 minus 500 over straight T
rightwards double arrow    space space space straight T over 1100 equals 500 over straight T
rightwards double arrow              straight T equals square root of 550000 space

space space equals space 741.62 straight K 
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This is the required temperature at which the efficiency of both engines is same.