Question

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 ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$. The temperature of the hot reservoir of the first engine is ${\mathrm{T}}_1$ and the temperature of the cold reservoir of the second engine is ${\mathrm{T}}_2$. $\mathrm{T}$ is the temperature of the sink of first engine which is also the source for the second engine. How is $\mathrm{T}$ related to ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ if both the engines perform equal amount of work?

• A. $\mathrm{T}=\frac{2{\mathrm{T}}_1{\mathrm{T}}_2}{({\mathrm{T}}_1+{\mathrm{T}}_2)}$
• B. $\mathrm{T}=\frac{{\mathrm{T}}_1+{\mathrm{T}}_2}{2}$
• C. $\mathrm{T}=0$
• D. $\mathrm{T}=\sqrt{{\mathrm{T}}_1{\mathrm{T}}_2}$

Answer 1

The correct option is B

$\mathrm{T}=\frac{{\mathrm{T}}_1+{\mathrm{T}}_2}{2}$

Step 1: Given data

Temperature of the hot reservoir of the first engine is ${\mathrm{T}}_1$

Temperature of the cold reservoir of the second engine is ${\mathrm{T}}_2$

Temperature of the sink of the first engine and the temperature of the source for the second engine are equal, i.e., $\mathrm{T}$

Step 2: Formula used

To find the relation between $\mathrm{T}$ , ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$, we will use our understanding of the Carnot cycle.

According to the question, the situation can be described as:

So, for the source: $\frac{{\mathrm{Q}}_{\mathrm{H}}}{{\mathrm{Q}}_{\mathrm{L}}}=\frac{{\mathrm{T}}_1}{\mathrm{T}}$ ………………(1)

and $\mathrm{W}={\mathrm{Q}}_{\mathrm{H}}-{\mathrm{Q}}_{\mathrm{L}}$ ………………(2)
And for the sink: $\frac{{\mathrm{Q}}_{\mathrm{L}}}{{\mathrm{Q}}_{\mathrm{L}}^{\text{'}}}=\frac{\mathrm{T}}{{\mathrm{T}}_2}$ ………………(3)

and $\mathrm{W}={\mathrm{Q}}_{\mathrm{L}}-{\mathrm{Q}}_{\mathrm{L}}^{\text{'}}$ ………………(4)

Where, ${\mathrm{Q}}_{\mathrm{H}}$ : Heat input to$1^{\text{st}}$ engine
${\mathrm{Q}}_{\mathrm{L}}$ : Heat rejected from $1^{\text{st}}$ engine
${\mathrm{Q}}_{{\mathrm{L}}^{\text{'}}}$ : Heat rejected from $2^{\text{nd}}$ engine

Step 3: Find the relation between $\mathrm{T}$ , ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$,

According to the question:

Work done by $1^{\text{st}}$ engine is equal to work done by $2^{\text{nd}}$ engine

Or, From (2) and (4): ${\mathrm{Q}}_{\mathrm{H}}-{\mathrm{Q}}_{\mathrm{L}}={\mathrm{Q}}_{\mathrm{L}}-{\mathrm{Q}}_{{\mathrm{L}}^{\text{'}}}$

$2{\mathrm{Q}}_{\mathrm{L}}={\mathrm{Q}}_{\mathrm{H}}+{\mathrm{Q}}_{{\mathrm{L}}^{\text{'}}}$
Now using the equation (1) and (3) in the above expression, we get:

$2=\frac{{\mathrm{T}}_1}{\mathrm{T}}+\frac{{\mathrm{T}}_2}{\mathrm{T}}$
$\mathrm{T}=\frac{{\mathrm{T}}_1+{\mathrm{T}}_2}{\mathrm{T}}$

So, option (B) is correct.