Reversible adiabatic expansion of an ideal gas

This section derives temperature-volume and pressure-volume relations when a fixed amount of an ideal gas is expanded or compressed without heat. 

First we need a relation between internal energy and temperature. Since the value of the internal energy of a fixed amount of an ideal gas depends only on its temperature (Sec. 3.5.1), an infinitesimal change dT will cause a change dU that depends only on T and dT  

The relation given by Eq. 3.5.3 is valid for any process of a closed system of an ideal gas of uniform temperature, even if the volume is not constant or if the process is adiabatic, because it is a general relation between state functions. 

In a reversible adiabatic expansion with expansion work only, the heat is zero and the first law becomes 

It is convenient to make the approximation that over a small temperature range, Cv is constant. When we divide both sides of the preceding equation by T in order to separate the variables T and F, and then integrate between the initial and final states, we obtain 

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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). 

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. 

Derive an expression for the work done in the reversible 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 ... 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

A hypothetical cycle for achieving reversible work, typically consisting of a sequence of operations (1) isothermal expansion of an ideal gas at a temperature T2 (2) adiabatic expansion from T2 to Ti (3) isothermal compression at temperature Ti and (4) adiabatic compression from Ti to T2. This cycle represents the action of an ideal heat engine, one exhibiting maximum thermal efficiency. Inferences drawn from thermodynamic consideration of Carnot cycles have advanced our understanding about the thermodynamics of chemical systems. See Carnot s Theorem Efficiency Thermodynamics... 

We divide this equation by T and integrate between the limits T and T for the temperature and V and V for the volume, because the heat capacity of an ideal gas is a function of the temperature alone. Thus, for an adiabatic reversible expansion of an ideal gas... 

The expansion of an ideal gas in the Joule experiment will be used as a simple example. Consider a quantity of an ideal gas confined in a flask at a given temperature and pressure. This flask is connected through a valve to another flask, which is evacuated. The two flasks are surrounded by an adiabatic envelope and, because the walls of the flasks are rigid, the system is isolated. We now allow the gas to expand irreversibly into the evacuated flask. For an ideal gas the temperature remains the same. Thus, the expansion is isothermal as well as adiabatic. We can return the system to its original state by carrying out an isothermal reversible compression. Here we use a work reservoir to compress the gas and a heat reservoir to remove heat from the gas. As we have seen before, a quantity of heat equal to the work done on the gas must be transferred from the gas to the heat reservoir. In so doing, the value of the entropy function of the heat reservoir is increased. Consequently, the value of the entropy function of the gas increased during the adiabatic irreversible expansion of gas. 

It is well known that the change in entropy for an adiabatic reversible expansion of an ideal gas is equal to zero. Using the equation given in Problem 5, find the final temperature when an ideal gas at 300K expands adiabatically from 1.00 liter to 5.00 liters. Take Cv = / and /f= 8.314 J/mol-K. 

The adiabatic expansion of a gas is an example of (b). In the reversible adiabatic expansion of one mole of an ideal monatomic gas, initially at 298.15 K, from a volume of 25 dm3 to a final volume of 50 dm3, 2343 J of energy are added into the surroundings from the work done in the expansion. Since no heat can be exchanged (in an adiabatic process, q = 0), the internal energy of the gas must decrease by 2343 J. As a result, the temperature of the gas falls to 188 K. 

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

We can use these equations to calculate the changes in various properties of an ideal gas undergoing a reversible, adiabatic expansion or compression. This is illustrated in Example 10.16. 

A 1.50-mol sample of an ideal gas is allowed to expand adiabatically and reversibly to twice its original volume. In the expansion the temperature dropped from 296 K to 239 K. Calculate A and AH for the gas expansion. 

Problem Calculate the work of expansion in ergs when the pressure of 1 mole of an ideal gas at 25 C is changed adiabatically and reversibly from 1.0 atm. to 5.0 atm. The molar heat capacities may be taken as equal to those of air. (Compare the problem in 8b, which is for an isothermal expansion between the same pressure limits.)... 

The solid curve in Fig. 3.6 on page 77 shows the path of a reversible adiabatic expansion or compression of a fixed amount of an ideal gas. Information about the gas is given in the figure... 

In an adiabatic expansion or compression, the system is thermally isolated from the surroundings so that q = 0. If the change is reversible, we can derive a general relationship between p, V, and T, that can then be applied to a fluid (such as an ideal gas) by knowing the equation of state relating p, V, and T. 

So far we have not specified whether the adiabatic expansion under consideration is reversible. Equations (5.40), (5.42), and (5.44) for the calculation of the thermodynamic changes in this process apply to the reversible expansion, the free expansion, or the intermediate expansion, so long as we are dealing with an ideal gas. However, the niunerical values of W, AU, and AH will not be the same for each of the three types of adiabatic expansion because T2, the final temperature of the gas, will depend on the type of expansion, even though the initial temperature is identical in aU cases. 

A reversible Joule cycle consists of the following stages (i) an expansion at the constant pressure P2, (ii) an adiabatic expansion to a lower pressure Pi, (iii) a compression at the constant pressure Pi, (iv) an adiabatic compression which restores the system to its initial state. Draw the indicator (P-F) diagram for the cycle, and prove that with an ideal gas as the working substance the efficiency is given by... 

It is desired to quench the coagulation of an aerosol composed of very small panicles dp (p)- If the rate of coagulation is to be reduced to 1% of its original value by isoihemial. constant pressure dilution with particIc-frce gas, dctcnninc the dilution ratio. The rate is to be reduced by the same factor by a reversible adiabatic expansion. Determine the volume e.xpansion ratio assuming the gas is ideal. 

Adiabatic expansion may be carried out as a butch process in a cloud chamber or as a steady-flow process in the diverging section of the nozzle of a steam turbine or supersonic wind tunnel. If the process is carried out reversibly (this is often a good approximation), the conditions along the path for an ideal gas are related by the expression... 

In an adiabatic expansion of a gas, mechanical work is done by the gas as its volume increases and the gas temperature falls. For an ideal gas undergoing a reversible adiabatic change it can be shown that pvy=Ki V p -r=K2... 

In the remaining part of this AppendixA.1, we obtain the important result (A.9) using an ideal cyclic process from subset C of Sect. 1.2, namely the Carnot cycle [1, 2, 4, 5]. Carnot cycle is a cyclic process with (fixed number of mols, n, of) uniform ideal gas composed from isothermal and adiabatic (no heat exchange) expansions followed by isothermal (at lower temperature) and adiabatic compressions back to the starting state. All these processes pass the equilibrium (stable) states and they are reversible (cf. definition in Sect. 1.2), see also Rem. 48 in Chap. 3. 

Equation (2.4-21) can be used for a reversible adiabatic compression as well as for an expansion. It is an example of an important fact that holds for any system, not just an ideal gas For a reversible adiabatic process in a simple system the final temperature is a function of the final volume for a given initial state. All of the possible final state points for reversible adiabatic processes starting at a given initial state lie on a single curve in the state space, called a reversible adiabat. This fact will be important in our discussion of the second law of thermodynamics in Chapter 3. 

Figure 2.16 An ideal gas in a piston-cylinder assembly undergoing a reversible, adiabatic expansion. In this example, is constant. See if you can predict the signs of At/, (), and W for this process in the table.

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