Problem 19 - College Physics

Question

A Carnot engine whose high-temperature reservoir is at 620 \(\mathrm{K}\) takes in 550 \(\mathrm{J}\) of heat at this temperature in each cycle and gives up 335 \(\mathrm{J}\) to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir' (c) What is the thermal efficiency of the cycle"

Step-by-Step Solution

Answer

(a) 215 J; (b) 378 K; (c) 39.1\% efficiency.

Step 1: Calculate Work Done (Part a)

The work done by the engine in each cycle is the difference between the heat absorbed and the heat released. We use the formula: \( W = Q_h - Q_c \) where \( Q_h = 550 \, \text{J} \) (heat absorbed) and \( Q_c = 335 \, \text{J} \) (heat released). Thus, \( W = 550 \, \text{J} - 335 \, \text{J} = 215 \, \text{J} \). So, the work done is 215 J.

Step 2: Calculate the Low-Temperature Reservoir (Part b)

Using the fact that for a Carnot engine the efficiency \( \, e \, \) is given by: \( e = 1 - \frac{T_c}{T_h} \) and that \( e = \frac{W}{Q_h} \), we can equate those efficiencies: \( \frac{W}{Q_h} = 1 - \frac{T_c}{T_h} \). We have \( W = 215 \), \( Q_h = 550 \), \( T_h = 620 \) K. Thus, solve for \( T_c \): \[ \frac{215}{550} = 1 - \frac{T_c}{620} \]Where, \( \frac{215}{550} = 0.3909 \) hence, \[ 0.3909 = 1 - \frac{T_c}{620} \] Solving, we obtain:\[ \frac{T_c}{620} = 1 - 0.3909 = 0.6091 \] \[ T_c = 620 \times 0.6091 = 377.6 \] K.Thus, \( T_c \approx 378 \) K.

Step 3: Calculate Thermal Efficiency (Part c)

The thermal efficiency of the Carnot engine is calculated using \( e = \frac{W}{Q_h} \). We found \( W = 215 \) J and \( Q_h = 550 \) J in the previous steps. So:\[ e = \frac{215}{550} = 0.3909 \] Converting to percentage gives:\[ e = 39.09\% \]Thus, the thermal efficiency is approximately 39.1\%.

Key Concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy, especially mechanical work. The laws of thermodynamics describe how energy moves and changes in a system. The Carnot engine is an idealized model used to understand these principles. It operates in a cycle, taking in heat from a high-temperature reservoir, performing mechanical work, and releasing some of the heat to a low-temperature reservoir.

• The first law of thermodynamics, also known as the law of energy conservation, states that the energy that goes into a system is equal to the energy that comes out of it.
• The second law of thermodynamics states that heat cannot spontaneously flow from a colder location to a hotter location.

These laws are fundamental to understanding how a Carnot engine operates.

Thermal Efficiency

The thermal efficiency of a heat engine, like a Carnot engine, is a measure of how well it converts heat into mechanical work. It's expressed as a percentage, representing the fraction of heat absorbed from the high-temperature reservoir that is converted into useful work.
The formula for thermal efficiency, \( e \), is given by:
\[ e = \frac{W}{Q_h} \] where \( W \) is the work done by the engine, and \( Q_h \) is the heat absorbed from the high-temperature reservoir.
A higher efficiency means more of the absorbed heat is used for work and less is wasted, usually as heat to the low-temperature reservoir. For a Carnot engine, the maximum possible efficiency is determined by the temperatures of the reservoirs, making it crucial to know these values when calculating efficiency.

Temperature Reservoirs

Temperature reservoirs are large systems that maintain a constant temperature and are crucial in thermodynamic cycles like those performed by Carnot engines. There are two types of temperature reservoirs involved: high-temperature and low-temperature reservoirs.
The high-temperature reservoir provides the heat energy necessary for the engine to do work. In the given exercise, the high-temperature reservoir is at 620 K.
On the other hand, the low-temperature reservoir absorbs the excess heat that is not converted into work. In our exercise, calculating the temperature of this reservoir is key to understanding the engine's overall cycle and efficiency. For accurate calculations, it's important to apply formulas like: \( \frac{T_c}{T_h} \), where \( T_c \) is the low-temperature reservoir and \( T_h \) is the high-temperature reservoir.

Mechanical Work

Mechanical work is the energy transferred by the engine due to its motion. In a Carnot cycle, mechanical work is the difference between the heat absorbed from the high-temperature reservoir and the heat expelled to the low-temperature reservoir.
To calculate the mechanical work \( W \) done by the Carnot engine during each cycle, use the formula:
\[ W = Q_h - Q_c \] where \( Q_h \) is the heat absorbed, and \( Q_c \) is the heat released.
In the given exercise, this calculation gives us a clear understanding of how much work is done per cycle. Mechanical work is a tangible output of the heat input and temperature management in a thermodynamic system, illustrating the core functions of the Carnot engine.