Problem 18 - University Physics with Modern Physics

Question

20.18. A Carnot device extracts 5.00 \(\mathrm{kJ}\) of heat from a body at \(-10.0^{\circ} \mathrm{C} .\) How much work is done if the device exhausts heat into the environment at \((a) 25.0^{\circ} \mathrm{C} ;(\mathrm{b}) 0.0^{\circ} \mathrm{C} ;(\mathrm{c})-25.0^{\circ} \mathrm{C} ;\) In each case, is the device acting as an engine or as a refrigerator?

Step-by-Step Solution

Answer

(a) Work: 585 J, Refrigerator; (b) Work: 185 J, Refrigerator; (c) No work, Non-functional.

Step 1: Convert Temperatures to Kelvin

The first step is to convert all given temperatures from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \). \(-10.0^{\circ} \text{C} = 263.15 \text{K}\), \(25.0^{\circ} \text{C} = 298.15 \text{K}\), \(0.0^{\circ} \text{C} = 273.15 \text{K}\), \(-25.0^{\circ} \text{C} = 248.15 \text{K}\).

Step 2: Calculate Efficiency of Carnot Device

For a Carnot device, the efficiency can be calculated using the formula: \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_h \) is the temperature of the hot reservoir and \( T_c \) is the temperature of the cold reservoir.

Step 3: Case a: Efficiency and Work at 25.0°C

Here, \( T_h = 298.15 \text{K} \) and \( T_c = 263.15 \text{K} \). Calculate efficiency: \( \eta = 1 - \frac{263.15}{298.15} \approx 0.117 \). Work done: \( W = \eta \times Q_h = 0.117 \times 5000 \text{ J} \approx 585 \text{ J} \). The device acts as a refrigerator because heat is absorbed at a lower temperature and released at a higher temperature.

Step 4: Case b: Efficiency and Work at 0.0°C

Here, \( T_h = 273.15 \text{K} \) and \( T_c = 263.15 \text{K} \). Efficiency: \( \eta = 1 - \frac{263.15}{273.15} \approx 0.037 \). Work done: \( W = \eta \times Q_h = 0.037 \times 5000 \text{ J} \approx 185 \text{ J} \). The device acts as a refrigerator because it extracts heat from a cold reservoir and exhausts it to a warmer environment.

Step 5: Case c: Efficiency and Work at -25.0°C

Here, \( T_h = 248.15 \text{K} \) and \( T_c = 263.15 \text{K} \). Since the hot temperature is less than the cold temperature, the efficiency is negative: \( \eta = 1 - \frac{263.15}{248.15} \). This scenario isn't physically valid for a Carnot cycle acting as a refrigerator or engine, and thus no work is done and the device does not function.

Step 6: Conclusion

In cases a and b, the device acts as a refrigerator due to the natural direction of heat flow (hot to cold). In case c, the device cannot function because the cold reservoir is warmer than the hot reservoir.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and the interrelation of various forms of energy. It helps us understand how energy moves and changes form in various systems. In thermodynamics, we often talk about systems, which can be as small as a single gas container or as large as a power plant.
Some important concepts in thermodynamics include:

• The First Law of Thermodynamics: Energy cannot be created or destroyed, only transferred or transformed.
• The Second Law of Thermodynamics: Heat energy will naturally flow from a hot body to a cold one.

The Carnot cycle, which you encountered in this exercise, is an idealized sequence of processes that provides a standard by which we can measure real engines' efficiency. It's important because it defines the maximum possible efficiency a heat engine can achieve.
Understanding thermodynamics is crucial for designing efficient engines, refrigerators, and other systems that rely on heat transfer.

Heat Engine

A heat engine is a device that converts thermal energy into mechanical work. The engine absorbs heat from a high-temperature reservoir, performs work, and then releases some heat to a lower-temperature reservoir.
The Carnot engine is an idealized example of a heat engine where all processes are reversible and there are no energy losses. In your exercise, when the Carnot device operates as a heat engine, it would absorb heat (like the 5.00 kJ here), convert a portion into work, and exhaust the remainder to a cold reservoir.
Key points about a heat engine include:

• It follows a cycle, continuously converting heat into work.
• Its efficiency depends on the temperature difference between the hot and cold reservoirs.

The effectiveness of the conversion process is encapsulated by its efficiency, which is the ratio of work output to heat input.

Refrigerator

A refrigerator is essentially the reverse of a heat engine. Instead of converting heat into work, it uses work to move heat from a cold reservoir to a hot one.
In the context of the Carnot cycle, when the given device acts as a refrigerator, as in cases (a) and (b) of your exercise, it extracts heat from the colder environment and expels it to a warmer one. This is crucial in cooling applications, like domestic fridges and air conditioners.
Key characteristics of a refrigerator include:

• It requires an external work input to move heat against the natural flow (from cold to hot).
• Its performance is often measured by the Coefficient of Performance (CoP), which is different from efficiency.

A CoP quantifies how effectively a refrigerator performs its function compared to the energy input required.

Efficiency

Efficiency is a measure of how well a system transforms input energy into useful work. In a perfect Carnot cycle, efficiency is determined by the temperatures of the reservoirs it operates between. The formula for Carnot efficiency is given by:\[\eta = 1 - \frac{T_c}{T_h}\]Where:

• \(T_c\) is the absolute temperature of the cold reservoir.
• \(T_h\) is the absolute temperature of the hot reservoir.

In your exercise, this formula helped calculate the efficiency of the Carnot device in different scenarios. Efficiency is crucial because it tells us what fraction of heat intake can be converted into useful work.
Real-world systems always have efficiencies less than a Carnot engine due to losses and irreversibilities. However, understanding this theoretical limit helps engineers strive for designs that approach it, optimizing the balance of work done and energy used.