One mole of a monatomic ideal gas has an initial pressure,...

One mole of a monatomic ideal gas has an initial pressure, volume, and temperature of $P_{0}, V_{0},$ and $438 \mathrm{~K},$ respectively. It undergoes an isothermal expansion that triples the volume of the gas. Then, the gas undergoes an isobaric compression back to its original volume. Finally, the gas undergoes an isochoric increase in pressure, so that the final pressure, volume, and temperature are $P_{0}, V_{0},$ and $438 \mathrm{~K},$ respectively. Find the total heat for this three-step process, and state whether it is absorbed by or given off by the gas.

Key Concepts

Monatomic Ideal Gas

A monatomic ideal gas consists of single-atom molecules and exhibits simple behavior with respect to energy distribution and degrees of freedom. Its internal energy is directly proportional to the temperature, typically described as (3/2)nRT for one mole of gas. This property simplifies calculations in thermodynamic processes and is important in understanding how energy is stored and transferred in ideal gas systems.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. This fundamental equation is essential for predicting a gas’s behavior under different conditions and is widely used in thermodynamics to analyze processes such as expansion, compression, and heat transfer in gases.

Isothermal Process

An isothermal process is one in which the temperature remains constant throughout. For an ideal gas undergoing an isothermal process, the internal energy does not change, which means that any work done by or on the system is exactly balanced by heat transfer. This concept is critical for understanding energy exchange during gas expansions or compressions at constant temperature.

Isobaric Process

An isobaric process is a thermodynamic process that occurs at constant pressure. In such processes, work is done by the system when the volume changes, and heat transfer occurs to accommodate the changes in internal energy. This type of process is common in practical applications where pressure constraints exist, and it helps illustrate the relationship between heat, work, and temperature changes in a gas.

Isochoric Process

An isochoric process is one in which the volume remains constant. Since there is no volume change, no work is done by or on the system, and any heat transferred results solely in a change in the internal energy of the gas. This concept is fundamental when considering processes where volume constraints are significant, allowing for a direct analysis of energy changes in the system.

First Law of Thermodynamics

The First Law of Thermodynamics is a statement of energy conservation, asserting 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 (?U = Q - W). This principle provides the framework for analyzing energy exchanges in any thermodynamic process and is crucial for determining the heat flow and work interactions in a variety of gas processes.

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