At point A in a Carnot cycle, 2.34 mol of a monatomic ideal...

At point $A$ in a Carnot cycle, $2.34$ mol of a monatomic ideal gas has a pressure of $1400 \mathrm{kPa}$, a volume of $10.0 \mathrm{~L}$, and a temperature of $720 \mathrm{~K}$. It expands isothermally to point $B$, and then expands adiabatically to point $C$, where its volume is $24.0 \mathrm{~L}$. An isothermal compression brings it to point $D$, where its new volume is 15.0 L. An adiabatic process returns the gas to point $A$. (a) Determine all the unknown pressures, volumes, and temperatures as you fill in the following table: $$ \begin{array}{lccc} & \boldsymbol{P} & \boldsymbol{V} & \boldsymbol{T} \\ \hline A & 1400 \mathrm{kPa} & 10.0 \mathrm{~L} & 720 \mathrm{~K} \\ B & & & \\ C & & 24.0 \mathrm{~L} & \\ D & & 15.0 \mathrm{~L} & \\ & & & \\ \hline \end{array} $$ (b) Find the energy added by heat, the work done, and the change in internal energy for each of the following steps: $A \rightarrow B, B \rightarrow C, C \rightarrow D$, and $D \rightarrow A$. (c) Show that $W_{\text {net }} / Q_{\text {in }}=1-T_{C} / T_{A}$, the Carnot efficiency.

At point $A$ in a Carnot cycle, $2.34$ mol of a monatomic ideal gas has a pressure of $1400 \mathrm{kPa}$, a volume of $10.0 \mathrm{~L}$, and a temperature of $720 \mathrm{~K}$. It expands isothermally to point $B$, and then expands adiabatically to point $C$, where its volume is $24.0 \mathrm{~L}$. An isothermal compression brings it to point $D$, where its new volume is 15.0 L. An adiabatic process returns the gas to point $A$.
(a) Determine all the unknown pressures, volumes, and temperatures as you fill in the following table:
$$
\begin{array}{lccc}
& \boldsymbol{P} & \boldsymbol{V} & \boldsymbol{T} \\
\hline A & 1400 \mathrm{kPa} & 10.0 \mathrm{~L} & 720 \mathrm{~K} \\
B & & & \\
C & & 24.0 \mathrm{~L} & \\
D & & 15.0 \mathrm{~L} & \\
& & & \\
\hline
\end{array}
$$
(b) Find the energy added by heat, the work done, and the change in internal energy for each of the following steps: $A \rightarrow B, B \rightarrow C, C \rightarrow D$, and $D \rightarrow A$. (c) Show that $W_{\text {net }} / Q_{\text {in }}=1-T_{C} / T_{A}$, the Carnot efficiency.

Key Concepts

Thermal Efficiency

Thermal efficiency in thermodynamics measures the fraction of heat input that is converted to work by a heat engine. For the Carnot cycle, the efficiency is determined by the temperatures of the hot and cold reservoirs and is expressed as 1 - (T_C/T_H). This concept illustrates the theoretical maximum efficiency an engine can achieve, serving as a benchmark for real engines.

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that represents the maximum possible efficiency any heat engine can achieve operating between two temperature reservoirs. It consists of two reversible isothermal processes, during which heat is exchanged with the surroundings, and two reversible adiabatic processes, during which no heat is exchanged. The cycle serves as a standard to measure the efficiency of real engines and to illustrate the second law of thermodynamics.

Isothermal Process

An isothermal process is one during which the temperature of the system remains constant. For an ideal gas, this implies that any change in volume is accompanied by a corresponding change in pressure in such a way that the product of pressure and volume remains constant. In the context of the Carnot cycle, the isothermal expansion and compression phases involve heat transfer to or from the gas while the internal energy remains constant.

Adiabatic Process

An adiabatic process is one in which no heat exchange occurs between the system and its surroundings. For an ideal gas undergoing an adiabatic process, the relationship between pressure and volume is governed by the adiabatic condition, and the process leads to changes in temperature as work is done on or by the system. In the Carnot cycle, the adiabatic expansion and compression connect the isothermal processes and result in changes in temperature without heat transfer.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, and temperature of an ideal gas with the number of moles present. Represented by PV = nRT, it provides the basis for calculating state variables during various thermodynamic processes, including those in the Carnot cycle. Understanding this law is essential to analyze and compute the changes in state properties during isothermal and adiabatic processes.

First Law of Thermodynamics

The first law of thermodynamics is the principle of energy conservation, stating that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. This law is crucial for analyzing how energy is transferred and transformed during the different steps of the Carnot cycle, allowing calculations of work, heat transfer, and changes in internal energy during each process.

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