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As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 mol of a monatomic ideal gas as its working substance and operates from a hightemperature reservoir at \(500^{\circ} \mathrm{C}\) . The engine is to lift a 15.0 -kg weight 2.00 \(\mathrm{m}\) per cycle, using 500 \(\mathrm{J}\) of heat input. The gas in the engine chamber can have a minimum volume of 5.00 \(\mathrm{L}\) during the cycle. (a) Draw a \(p V\) -diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

Step by step solution

01

Convert Temperatures to Kelvin

To begin, we need to convert the given temperature of the high-temperature reservoir from Celsius to Kelvin. The formula for this is: \[ T (K) = T (^{\circ} C) + 273.15 \] Given, the high-temperature reservoir is at 500°C. Thus, \[ T_h = 500 + 273.15 = 773.15 \text{ K} \]

02

Draw the PV Diagram

For a Carnot cycle involving an ideal gas, the pv diagram consists of two isothermal processes and two adiabatic processes, forming a rectangle-like loop. 1. Isothermal expansion where the gas absorbs heat \( Q_h \).2. Adiabatic expansion where the gas performs work without heat exchange.3. Isothermal compression where the gas rejects heat \( Q_c \).4. Adiabatic compression returning to the initial state.Heat enters during the isothermal expansion and leaves during the isothermal compression.

03

Calculate Temperature of Cold Reservoir

The thermal efficiency \( \eta \) of a Carnot engine is given by \[ \eta = 1 - \frac{T_c}{T_h} \] where \( \eta \) is the efficiency, \( T_c \) is the cold reservoir temperature, and \( T_h \) is the hot reservoir temperature. Given the engine lifts a 15.0 kg weight by 2.0 m using 500 J of heat input:- Work done \( W \) = \( mgh = 15 \times 9.81 \times 2.0 = 294 \) J - Efficiency \( \eta = \frac{W}{Q_h} = \frac{294}{500} = 0.588 \)Solving \[ 0.588 = 1 - \frac{T_c}{773.15} \] for \( T_c \):\[ T_c = (1 - 0.588) \times 773.15 = 316.48 \text{ K} \]

04

Calculate Thermal Efficiency

We've already calculated the thermal efficiency in the previous step as \( 0.588 \). Thus, the thermal efficiency is \[ 58.8\% \]

05

Calculate Heat Energy Wasted

The heat energy wasted per cycle, \( Q_c \), can be found using:\[ Q_c = Q_h - W = 500 - 294 = 206 \text{ J} \]

06

Calculate Maximum Pressure

Using the ideal gas law \( PV = nRT \), where \( n = 2.00 \text{ mol} \), \( R = 8.314 \text{ J/mol K} \):For maximum pressure at minimum volume (5.00 L = 0.005 m³) at \( T_h \) (773.15 K):\[ P_{max} = \frac{nRT_h}{V_{min}} = \frac{2.00 \times 8.314 \times 773.15}{0.005} = 2562.5 \text{ Pa} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics

At its core, thermodynamics is the branch of physics that studies heat, work, and the energy involved in physical and chemical processes. It provides insights into how energy is transferred and transformed. In a Carnot engine, which is a theoretical model of idealized heat engine, thermodynamics plays a vital role. This engine operates by cyclically converting heat into work using two reservoirs: one hot and one cold.
The operation of a Carnot cycle involves four main processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. In simple terms:

• Isothermal processes occur at constant temperatures where heat is transferred.
• Adiabatic processes occur without heat exchange, only work is done or gained.

Understanding these processes allows engineers to predict the engine's performance under certain conditions.

Ideal Gas Law

The ideal gas law is a fundamental principle that describes the behavior of an idealized gas. It's encapsulated by the equation: \[ PV = nRT \]Where:

• P is the pressure of the gas.
• V is the volume of the gas.
• n is the number of moles of the gas.
• R is the ideal gas constant (approximately 8.314 J/mol·K).
• T is the temperature in Kelvin.

In the context of a Carnot engine, this law helps calculate maximum pressure. For instance, using the given inputs, maximum pressure is derived during isothermal expansion when the gas is at a high temperature, providing insights into the needed robustness of the engine’s components.

Thermal Efficiency

Thermal efficiency is a measure of how effectively a heat engine converts heat into work. For a Carnot engine, the thermal efficiency \( \eta \) is given by:\[ \eta = 1 - \frac{T_c}{T_h} \]Here, \( T_h \) and \( T_c \) are the temperatures of the hot and cold reservoirs, respectively. The concept highlights that no heat engine operating between two heat reservoirs can be more efficient than a Carnot engine.

• A higher efficiency means less heat energy wasted.
• The maximum theoretical efficiency is achieved if the cold reservoir is at absolute zero, which is practically impossible.

Hence, any real-world engine has an efficiency lower than that of a Carnot engine.

Heat Exchange

Heat exchange is the process by which heat energy is transferred from one body or system to another. In a Carnot cycle, this occurs during the isothermal processes:

• During isothermal expansion, heat \( Q_h \) is absorbed from the hot reservoir.
• During isothermal compression, heat \( Q_c \) is rejected to the cold reservoir.

The heat absorbed and rejected can be quantitatively scrutinized to determine how much work is performed and what energy is wastefully expelled, respectively. Such quantification is critical for analyzing the efficiency and for designing more effective heat engines in the real world.

PV Diagram

A PV diagram is a graphical representation of pressure (P) versus volume (V) for a thermodynamic process or cycle. For a Carnot cycle, it forms a loop comprising four segments:

• Two horizontal lines representing isothermal processes where pressure changes with constant temperature.
• Two curves representing adiabatic processes where the pressure-volume relation follows a particular curve without heat transfer.

Analyzing this loop helps visualize how work and energy transformations occur in the cycle. It's an effective tool for engineers to ensure engines operate efficiently and inform necessary adjustments to engine parameters for optimal performance.