Problem 21 - College Physics

Question

A Carnot engine is operated between two heat reservoirs at temperatures of 520 \(\mathrm{K}\) and 300 \(\mathrm{K}\) . (a) If the engine receives 6.45 \(\mathrm{kJ}\) of heat energy from the reservoir at 520 \(\mathrm{K}\) in each cycle, how many joules per cycle does it reject to the reservoir at 300 \(\mathrm{K} ?\) (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

Step-by-Step Solution

Answer

(a) The engine rejects approximately 3723.8 J per cycle. (b) It performs about 2726.2 J of work per cycle. (c) The thermal efficiency is 42.31%.

Step 1: Calculate the Carnot Efficiency

The efficiency of a Carnot engine depends only on the temperatures of the two heat reservoirs. The efficiency \( \eta \) is given by: \[ \eta = 1 - \frac{T_C}{T_H} \]Where \( T_C = 300 \, \text{K} \) and \( T_H = 520 \, \text{K} \).Substituting the given values, we have:\[ \eta = 1 - \frac{300}{520} = 1 - 0.5769 \approx 0.4231 \]So, the efficiency \( \eta \) is approximately 0.4231 or 42.31%.

Step 2: Determine the Heat Rejected

To find the heat rejected \( Q_C \), use the relationship between the efficiency, heat absorbed \( Q_H \), and heat rejected:\[ \eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} \]Rearrange the equation to find:\[ Q_C = Q_H (1 - \eta) \]Substitute \( Q_H = 6.45 \, \text{kJ} = 6450 \, \text{J} \) and \( \eta = 0.4231 \):\[ Q_C = 6450 \, \text{J} \times (1 - 0.4231) = 6450 \, \text{J} \times 0.5769 \approx 3723.8 \, \text{J} \]

Step 3: Compute the Mechanical Work Done

Using the efficiency \( \eta \) we can also calculate the work done \( W \):\[ W = Q_H \times \eta \]Substitute the known values:\[ W = 6450 \, \text{J} \times 0.4231 \approx 2726.2 \, \text{J} \]

Step 4: Verify Thermal Efficiency

Finally, verify the calculations for thermal efficiency by comparing with the work done and heat absorbed:\[ \eta = \frac{W}{Q_H} = \frac{2726.2}{6450} \approx 0.4231 \text{ or } 42.31\% \]This confirms the previous calculation for efficiency.

Key Concepts

Thermal Efficiency

Thermal efficiency is a measure of how well a heat engine converts the thermal energy it receives into useful work. For a Carnot engine, thermal efficiency is determined solely by the temperatures of the hot and cold reservoirs, following the equation: \[ \eta = 1 - \frac{T_C}{T_H} \]where \( T_C \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir.

• The efficiency indicates the fraction of heat energy that is converted into mechanical work.
• A higher efficiency means less energy wasted as heat.
• Carnot engines, being idealized, represent the maximum possible efficiency that any heat engine can achieve.

Understanding thermal efficiency helps in grasping how energy is utilized in practical systems and the limits imposed by temperature differences.

Mechanical Work

Mechanical work in a Carnot engine is the amount of energy it produces to perform tasks. The work done per cycle is directly linked to the efficiency of the engine. In the Carnot cycle, the mechanical work \( W \) is given by: \[ W = Q_H \times \eta \]where \( Q_H \) is the heat absorbed from the hot reservoir. This relationship highlights:

• How work is derived from the absorbed heat and the thermal efficiency.
• That improvements in thermal efficiency increase the work output for the same heat input.
• The importance of optimizing temperatures to maximize work.

Mechanical work helps in understanding the practical applications of energy conversion in engines and machinery.

Heat Reservoirs

Heat reservoirs are large bodies capable of absorbing or supplying infinite amounts of heat without changing their temperatures significantly. In the Carnot cycle, two reservoirs are involved:

• The hot reservoir at a higher temperature \( T_H \), from which the engine absorbs heat \( Q_H \).
• The cold reservoir at a lower temperature \( T_C \), to which the engine rejects heat \( Q_C \).

These reservoirs enable the cyclic process of the Carnot engine:

• The engine absorbs heat from the hot reservoir to perform work.
• It then expels some residual heat to the cold reservoir, completing the cycle.

The concept of heat reservoirs is key to understanding the flow of energy in engines and the limits of efficiency.

Carnot Cycle

The Carnot cycle is a theoretical model that describes an idealized heat engine cycle, achieving maximum efficiency due to its reversible processes. The cycle consists of four distinct stages:

• Isothermal Expansion: The gas absorbs heat \( Q_H \) from the hot reservoir, doing work on its surroundings while maintaining a constant temperature.
• Adiabatic Expansion: The gas continues to expand without heat exchange, doing work and lowering its temperature.
• Isothermal Compression: Heat \( Q_C \) is expelled to the cold reservoir as the gas is compressed at a constant, lower temperature.
• Adiabatic Compression: The gas is compressed further, raising its temperature back to the initial state without heat exchange.

This cycle helps visualize how the theoretical maximum efficiency of heat engines can be achieved and why real engines fall short of this ideal due to irreversibilities.