Adiabatic expansion of an ideal gas

The first consists of two steps (1) an isothermal reversible expansion at the temperature Ta until the volume V is reached, and (2) an adiabatic reversible expansion from V to Vj,. The entropy change for the gas is given by the sum of the entropy changes for the two steps  

As V Vfl, the entropy change for the gas is clearly positive for the reversible path and, therefore, also for the irreversible change. 

A reversible adiabatic expansion of an ideal gas has a zero entropy change, and an irreversible adiabatic expansion of the same gas from the same initial state to the same final volume has a positive entropy change. This statement may seem to be inconsistent with the statement that 5 is a thermodynamic property. The resolution of the discrepancy is that the two changes do not constitute the same change of state the final temperature of the reversible adiabatic expansion is lower than the final temperature of the irreversible adiabatic expansion (as in path 2 in Fig. 6.7). 

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Orifice Discharge for Gas Flow The analytic solution for discharge through an orifice of an ideal gas is derived by invoking the equation of state for adiabatic expansion of an ideal gas ... 

TABLE 5.2. Thermodynamic Changes in Adiabatic Expansions of an Ideal Gas... 

Points a and b in Figure 6.7 represent the initial and final states of an irreversible adiabatic expansion of an ideal gas. The path between is not represented because the temperature has no well-defined value in such a change different parts of the system may have different temperatures. The inhomogeneities in the system that develop during the irreversible change do not disappear until a new equilibrium is reached at b. 

A reversible adiabatic expansion of an ideal gas is infinitely slow, so the system maintains internal equilibrium (mechanical, thermal, and material) and equilibrium with its surroundings. Mechanical equilibrium with the surroundings requires that the external pressure be only infinitesimally less than the internal pressure. We can therefore set P = Pext. Thermal and material equilibria with the surroundings are not at issue, because the system is closed with adiabatic walls. A reversible adiabatic expansion is a highly idealized process Nevertheless, it will serve as a cornerstone in our discussions of thermodynamics. Applying the first law to such a process,... 

Here we begin with the relation dE = Cy dT = dW = —PdV, which holds for reversible adiabatic expansion of an ideal gas. By definition Cv is a constant. We immediately find that... 

By combining equations (10.5) and (10.8) derive an expression for the work of reversible, adiabatic expansion of an ideal gas in terms of the initial and final volumes. Determine the work done in liter-atm. when 1 mole of a diatomic gas at 0 C expands from 10 ml. to 1 liter. 

The assumption that the contraction process is ideally adiabatic, while perhaps not entirely permissible practically, seems indicated by modern theory of the behavior of molecular chains, which pictures these as undergoing, when freed of restraints, a sort of segmental diffusion, much like the adiabatic expansion of an ideal gas into a vacuum (155). In the case of the molecular chain, it diffuses to the most probable, randomly coiled configuration, which is much less asymmetric, hence shorter, than an initially extended chain. Because rubber most nearly presents this ideal behavior, those fibers which develop increased tension (a measure of the tendency toward assumption of the contracted form) when held isometrically under conditions of increasing temperature (favoring the diffusion ) are said to be rubber-like. Most normal elastic solids upon stress are strained from some stable structure and relax as the temperature is raised. 

Derive an expression for the work done in the reversible adiabatic expansion of an ideal gas. 

EXAMPLE 8.8 The quasi-static adiabatic expansion of an ideal gas. Let s start with an idealization, a gas expanding slowly in a cylinder with no heat flow, Sq = 0. (Nearly adiabatic processes are common in real piston engines because the heat transfer processes are much slower than the volume changes within the cylinders.) What is the temperature change inside the cylinder as the gas expands ... 

Figure 2.7 Final Temperature as a Function of Final Volume for the Adiabatic Expansion of an Ideal Gas.

Both the processes in this section can be considered polytropic. The isothermal expansion of an ideal gas follows Equation (2.49) with 7 = 1 while the reversible, adiabatic expansion of an ideal gas with constant heat capacity has y = k = c lc . Can you think of another example of a polytropic process ... 

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