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The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), and temperature (T) of a gas with the equation:

• \(PV = nRT\)

Here, \(n\) represents the number of moles, and \(R\) is the universal gas constant. In the context of our exercise with one kilogram of air, we use the specific gas constant for air, \(R = 287 \frac{J}{kg \, K}\). It's crucial to convert all units correctly. For example, pressure should be in Pascals and volume in cubic meters.
In step 3, we used this law to calculate the high temperature state during the isothermal expansion. By rearranging the formula, we found:

• \(T_H = \frac{PV}{R}\)

This equation helped us determine the temperature \(T_H\) given the pressure and volume at the end of the isothermal expansion. Understanding how to apply the ideal gas law is essential in solving many thermodynamics problems.

Thermal efficiency represents how effectively a heat engine converts heat into work. For the Carnot cycle, thermal efficiency \(\eta\) is given by:

• \(\eta = 1 - \frac{T_L}{T_H}\)

Where \(T_L\) and \(T_H\) are the lower and higher temperatures of the cycle, respectively. This formula indicates that the efficiency depends on the temperature difference. In our problem, the thermal efficiency is 50%, which means:

• \(\frac{T_L}{T_H} = 0.5\)

This relationship helped us establish a connection between \(T_L\) and \(T_H\). By knowing \(T_H\) from the isothermal process, we calculated \(T_L\). The concept of thermal efficiency is fundamental, as it dictates the maximum possible efficiency any heat engine can achieve, setting a benchmark for engine performance.

An isothermal process is a thermodynamic process where the temperature remains constant. In an isothermal expansion or compression, performed in the Carnot cycle, the internal energy of an ideal gas remains unchanged. This implies that the work done by or on the gas is exactly equal to the heat absorbed or released, described as:

• \(Q_{in} = W_{out}\)

In our problem, during the isothermal expansion, the heat transfer to the system is \(Q_{in} = 50 \mathrm{kJ}\). Using the equation for work done during an isothermal process, we solved for the initial volume \(V_1\):

• \(Q_{in} = nRT_H \ln\frac{V_2}{V_1}\)

Where \(\ln\) denotes the natural logarithm, \(V_2\) is the final volume, and \(V_1\) is the initial volume. This relationship is critical for understanding how changes in volume affect heat transfer during isothermal processes.

An adiabatic process is one in which no heat is transferred to or from the system. During adiabatic compression or expansion, the temperature of the gas changes as work is done, but the total heat transfer \((Q)\) is zero. This process for an ideal gas follows the equation:

• \(PV^\gamma = \textrm{constant}\)

Where \( \gamma \) is the adiabatic index, typically \( \frac{C_p}{C_v} \) where \( C_p \) is the specific heat at constant pressure and \( C_v \) at constant volume. In the Carnot cycle, two adiabatic processes occur: an adiabatic compression and an adiabatic expansion. During these, the pressure and volume change, but no heat is exchanged. Understanding adiabatic processes helps in analyzing parts of the Carnot cycle without involving heat transfer, which simplifies calculations for these sections.

A PV diagram, or Pressure-Volume diagram, visualizes the relationship between the pressure of the gas and its volume during different thermodynamic processes. For the Carnot cycle, the PV diagram typically has four distinct paths: two isothermal and two adiabatic.
In our exercise, sketching the cycle on PV coordinates involves plotting these processes:

• An isothermal expansion where volume increases at constant temperature.
• An adiabatic expansion where temperature and pressure drop without heat transfer.
• An isothermal compression at a lower temperature where volume decreases.
• An adiabatic compression bringing the system back to the initial state.

The diagram helps visualize how the gas transitions through different states, showing the areas under the curves as representative of the work done during each process. A clear understanding of how to read and sketch PV diagrams is essential for grasping the entirety of thermodynamic cycles like the Carnot cycle.