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, <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </math> and <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </math> . The temperature of the hot reservoir of the first engine is <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </math> and the temperature of the cold reservoir of the second engine is <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </math> . <math> <mi>T</mi> </math> is temperature of the sink of first engine which is also the source for the second engine. How is <math> <mi>T</mi> </math> related to <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </math> and <math> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </math> , if both the engines perform equal amount of work?

W = Q 1 - Q 2 = Q 2 - Q 3 <!- [endif]-><!- [endif]-> Q 2 = Q 1 + Q 3 2 <!- [endif]-><!- [endif]-> 2 = Q 1 Q 2 + Q 3 Q 2 <!- [endif]-><!- [endif]-> 2 = T 1 T + T 2 T <!- [endif]-><!- [endif]-> T = T 1 + T 2 2 <!- [endif]-><!- [endif]->

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Physics Thermodynamics Physics Class 11th Thermodynamics

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 -

$T = \sqrt[]{T_{1} T_{2}}$

Option 2 -

$T = \frac{T_{1} + T_{2}}{2}$

Option 3 -

$T = \frac{2 T_{1} T_{2}}{T_{1} + T_{2}}$

Option 4 -

$T = 0$

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Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 2

Detailed Solution:

$W = Q_{1} - Q_{2} = Q_{2} - Q_{3}$

$Q_{2} = \frac{Q_{1} + Q_{3}}{2}$

$2 = \frac{Q_{1}}{Q_{2}} + \frac{Q_{3}}{Q_{2}}$

$2 = \frac{T_{1}}{T} + \frac{T_{2}}{T}$

$T = \frac{T_{1} + T_{2}}{2}$

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