A sample of an ideal gas expands isothermally, doubling in...

A sample of an ideal gas expands isothermally, doubling in volume. (a) Show that the work done on the gas in expanding is $W=-n R T \ln 2 .$ (b) Because the internal energy $E_{\text {int }}$ of an ideal gas depends solely on its temperature, the change in internal energy is zero during the expansion. It follows from the first law that the energy input to the gas by heat during the expansion is equal to the energy output by work. Does this process have $100 \%$ efficiency in converting energy input by heat into work output? (c) Does this conversion violate the second law? Explain.

Key Concepts

Thermodynamic Efficiency

Thermodynamic efficiency often refers to the ratio of work output to heat input in a process. In the context of an isothermal expansion, even though all the heat input is converted to work, this process does not equate to having 100% efficiency in the broader sense used in engines, as it is not a complete cycle. Efficiency must be evaluated over a full cycle where energy conversion and losses are considered.

First Law of Thermodynamics

The first law of thermodynamics states 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. In an isothermal process for an ideal gas, because the internal energy is dependent solely on temperature (which remains constant), any heat input into the system is exactly balanced by the work output. This concept underscores the conservation of energy in thermodynamic processes.

Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and that processes occur in the direction of increasing entropy. Even when a process like an isothermal expansion appears to convert all the incoming heat to work, it does not violate this law because the process is carried out reversibly, meaning there is no overall increase in net work output beyond what is allowed by the thermodynamic constraints. Entropy considerations ensure that perpetual or 100% efficient engines over entire cycles remain impossible.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas. This law is crucial in determining how the state variables change in a thermodynamic process. In an isothermal process, the product of pressure and volume remains constant because temperature is held fixed, which simplifies the integration needed to compute work done during expansion or compression.

Work in Isothermal Expansion

The work done during an isothermal expansion of an ideal gas can be derived by integrating the pressure with respect to volume. Using the ideal gas law to substitute for pressure, the work is given by W = ?(nRT/V) dV, which leads to W = nRT ln(V_final/V_initial). This fundamental concept shows how energy is transferred as work during volume changes, with the logarithmic dependence reflecting the exponential nature of the volume change.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout. For an ideal gas, this implies that any change in volume directly influences pressure according to the ideal gas law, and the work done during the process can be calculated by integrating the pressure with respect to the change in volume. In such processes, the internal energy of an ideal gas stays constant, making the heat added to the system equal to the work done by the system.

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