Question

a carrnot engine performs 2000 j of work and rejects 4000 j of heat of the sink if the difference of temp b/w the source and sink is 85 degreefind the temp of the source and sink?

Answer 1

Solve. A Carnot engine performs 2000 j of work and rejects 4000 j of heat of the sink if the difference of temp b/w the source and sink is 85 degree find the temp of the source and sink?

Solution.

Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.

This maximum efficiency $\eta$ is defined to be:

$$\eta = \frac{W}{Q_h} = 1 - \frac{T_c}{T_h}$$

where

- $W$ is the work done by the system (energy exiting the system as work),

- $Q_h$ is the heat put into the system (heat energy entering the system),

- $T_c$ is the absolute temperature of the cold reservoir, and

- $T_h$ is the absolute temperature of the hot reservoir.

In our case,

$$W = 2000J$$ $$Q_c = 4000J$$ $$T_h - T_c = 85{}^\circ \text{C}$$

The work done by the system:

$$W = Q_h - Q_c$$

So

$$Q _ {h} = W + Q _ {c} = 2 0 0 0 + 4 0 0 0 = 6 0 0 0 J$$ $$\eta = \frac {W}{Q _ {h}} = \frac {2 0 0 0}{6 0 0 0} = \frac {1}{3}$$

From the other side:

$$\eta = 1 - \frac {T _ {c}}{T _ {h}} = \frac {T _ {h} - T _ {c}}{T _ {h}} = \frac {1}{3}$$ $$\frac {8 5}{T _ {h}} = \frac {1}{3}$$

So

$$T _ {h} = 8 5 \cdot 3 = 2 5 5 {}^ {\circ} \mathrm {C}$$ $$T _ {c} = T _ {h} - 8 5 = 2 5 5 - 8 5 = 1 7 0 {}^ {\circ} \mathrm {C}$$

Answer:

$$T _ {h} = 2 5 5 {}^ {\circ} \mathrm {C}$$ $$T _ {c} = 8 5 {}^ {\circ} \mathrm {C}$$