Chapter 21: Q. 25 (page 595)

A heat engine does $10 J$ of work and exhausts $15 J$ of waste heat during each cycle.
a. What is the engine's thermal efficiency?
b. If the cold-reservoir temperature is $20 ° C$ , what is the minimum possible temperature in ${}^{\circ}C$ of the hot reservoir?

Short Answer

Expert verified

a.Engine thermal efficiency is $40 %$ .

b.The minimum temperature of the heat reservoir $T_{h, \min}$ is $215 ° C$ .

Step by step solution

Calculation for engine thermal efficiency (part a)

The efficiency of a heat engine,

$η = \frac{W}{Q_{h}}$

W is work done , $η$ is efficiency and $Q_{h}$ is heat used up by machine.

$Q_{h} = W + Q_{c}$

$Q_{c}$ is heat dumped to cold reservoir.

So, we can write our expression for the efficiency as.

$η = \frac{W}{W + Q_{c}}$

$η = \frac{10}{10 + 15} = 40 %$

Calculation for the minimum temperature of the heat reservoir (part b)

The maximum efficiency of a Carnot engine is calculated as a function of the minimum and maximum temperatures as follows:

$η_{m} = 1 - \frac{T_{c}}{T_{h}}$

The actual efficiency is calculated as a function of the amount of work done and the amount of heat lost.

$η = \frac{W}{W + Q_{c}}$

We can write since our engine's efficiency cannot be higher than Carnot's,

$η \leq η_{m}$

$\frac{W}{W + Q_{c}} \leq 1 - \frac{T_{c}}{T_{h}}$

$\frac{T_{c}}{T_{h}} \leq 1 - \frac{W}{W + Q_{c}} = \frac{Q_{c}}{W + Q_{c}}$

The temperature of the heat reservoir,

$T_{h} \geq \frac{W + Q_{c}}{Q_{c}} T_{c}$

$T_{h} \geq \frac{10 + 15}{15} \cdot (20 + 273) = 488 K$

$T_{h, \min} = 215 ° C$

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A heat engine does $10 J$ of work and exhausts $15 J$ of waste heat during each cycle.
a. What is the engine's thermal efficiency?
b. If the cold-reservoir temperature is $20 ° C$ , what is the minimum possible temperature in ${}^{\circ}C$ of the hot reservoir?

Short Answer

Step by step solution

Calculation for engine thermal efficiency (part a)

Calculation for the minimum temperature of the heat reservoir (part b)

One App. One Place for Learning.

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A heat engine does10Jof work and exhausts15Jof waste heat during each cycle.a. What is the engine's thermal efficiency?b. If the cold-reservoir temperature is 20°C, what is the minimum possible temperature in C∘of the hot reservoir?

Short Answer

Step by step solution

Calculation for engine thermal efficiency (part a)

Calculation for the minimum temperature of the heat reservoir (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

A heat engine does $10 J$ of work and exhausts $15 J$ of waste heat during each cycle.
a. What is the engine's thermal efficiency?
b. If the cold-reservoir temperature is $20 ° C$ , what is the minimum possible temperature in ${}^{\circ}C$ of the hot reservoir?