A heat engine takes 0.350 mol of an ideal diatomic gas...

A heat engine takes 0.350 mol of an ideal diatomic gas around the cycle shown in the $\underline{p V}$ diagram of Figure 16.17 . Process $1 \rightarrow 2$ is at constant volume, process $2 \rightarrow 3$ is adiabatic, and process $3 \rightarrow 1$ is at a constant pressure of 1.00 atm. The value of $\gamma$ for this gas is $1.40 .$ The magnitude of the change in internal energy for each process is $\left|\Delta U_{1 \rightarrow 2}\right|=2180 \mathrm{~J},\left|\Delta U_{2 \rightarrow 3}\right|=785 \mathrm{~J},$ and $\left|\Delta U_{3 \rightarrow 1}\right|=1396 \mathrm{~J}$ (a) Find the pressure and volume at points $1,2,$ and $3 .$ (b) Calculate $Q$ and $W$ for each of the three processes. (c) Find the net work done by the gas in the cycle. (d) Find the net heat flow into the engine in one cycle. (e) What is the thermal efficiency of the engine? How does this efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximum temperatures $T_{1}$ and $T_{2} ?$

Key Concepts

Thermal Efficiency and Carnot Cycle

Thermal efficiency measures a heat engine’s ability to convert heat into work, defined as the ratio of net work output to the heat input. The Carnot cycle represents an idealized engine with maximum possible efficiency operating between two temperatures. Comparing a real engine’s efficiency with that of a Carnot engine helps gauge the performance limitations dictated by thermodynamic laws, including the inevitable losses in practical systems.

Thermodynamic Cycles

A thermodynamic cycle is a series of processes that return a system to its initial state, allowing the system to perform work by transferring heat. These cycles form the basis of heat engines, where the net work done over one complete cycle is the difference between the heat added and the heat removed. Understanding the cycle’s stages and their interconnections is crucial for analyzing engine performance and energy efficiency.

Isothermal, Isobaric, and Isochoric Processes

In thermodynamic systems, processes often occur under constant temperature (isothermal), constant pressure (isobaric), or constant volume (isochoric) conditions. Each process type has unique implications for heat transfer and work done on or by the gas. For example, during an isochoric process the volume remains constant so all heat transfer results in a change in internal energy, while during an isobaric process, heat may do work as the volume changes.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. Changes in internal energy during an adiabatic process are solely due to work done by or on the system. The relationship between pressure, volume, and temperature during adiabatic changes is governed by specific equations, using the adiabatic index (?), which reflects the ratio of specific heats.

Ideal Gas Law

The ideal gas law (PV = nRT) is a fundamental equation relating the pressure, volume, temperature, and amount of an ideal gas. It serves as a cornerstone for understanding and predicting the behavior of gases under different conditions and is essential for determining the state variables at various points in thermodynamic cycles.

First Law of Thermodynamics

The first law of thermodynamics, which is a statement of energy conservation, posits that the change in internal energy of a system equals the net heat added to the system minus the work done by the system. This principle is crucial in analyzing energy changes during individual processes within a cycle, as it helps balance heat transfer and work done, thereby describing the overall energy flow in the system.

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