Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $T_{1}$ and $T_{2}$ . The temperature of the hot reservoir of the first engine is $T_{1}$ and the temperature of the cold reservoir of the second engine is $T_{2}$ . $T$ is temperature of the sink of first engine which is also the source for the second engine. How is $T$ related to $T_{1}$ and $T_{2}$ , if both the engines perform equal amount of work?

Option 1 - <math> <mi mathvariant="normal">T</mi> <mo>=</mo> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> </math>

Option 2 - <math> <mi mathvariant="normal">T</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mrow> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math>

Option 3 - <math> <mi mathvariant="normal">T</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>2</mn> <msub> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mrow> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math>

Option 4 - <math> <mi mathvariant="normal">T</mi> <mo>=</mo> <mn>0</mn> </math>

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Answered by

Alok Kumar Singh | Contributor-Level 10

6 months ago

Correct Option - 2

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