Reversible, Isothermal Expansion Compression

Since the internal energy of an ideal gas is only a function of temperature, 

For a reversible process, we can integrate over the system pressure (see Section 2.3)  

Substituting Equation (2.37) into Equation (2.36) and integrating gives  

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Figure 4.3 Reversible Camot cycle, showing steps (1) reversible isothermal expansion at th (2) reversible adiabatic expansion and cooling from th to tc (3) reversible isothermal compression at tc (4) reversible adiabatic compression and heating back to the original starting point. The total area of the Camot cycle, P dV, is the net useful work w performed in the cyclic process (see text).

What might a reversible isothermal expansion of an ideal gas be This process will occur only if initially, when the gas is confined to half the qdinder, the external pressure acting on the piston exactly balances the pressure exerted by the gas on the piston. If the external pressure is reduced infinitely slowly, the piston will move outward, allowing the pressure of the confined gas to readjust to maintain the pressure balance. This infinitely slow process in which the external pressure and internal pressure are always in equilibrium is reversible. If we reverse the process and compress the gas in the same infinitely slow manner, we can return the gas to its original volume. The complete cy cle of expansion and compression in this hypothetical process, moreover, is accomplished without any net change to the surroundings. 

FIGURE 3.2 A representation of the Carnot cycle performed on a gaseous system. The steps are (1) Reversible isothermal expansion. (2) Reversible adiabatic expansion. (3) Reversible isothermal compression. (4) Reversible adiabatic compression. The system ends up at the same conditions it started at the area inside the four-sided figure is representative of the p — V work performed by the cycle. 

Figure 2J Schematic of infinitesimal-step, reversible isothermal expansion (process E) and compression (process F) processes. The corresponding plots of the processes on a Peo diagram is shown at the bottom.

Figure 3.2 Illustration of irreversible and reversible processes, (a) Mechanical process of isothermal expansion/compression. (b) Thermal process in which work can be obtained from a Carnot engine.

A reversible cycle also can be completed in three steps, such as isothermal expansion (at from V to V2, cooling (at constant V2) from 2 to Ti, and adiabatic compression back to the initial state. 

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... 

Derive an equation for the work of mechanically reversible, isothermal compression of 1 mol of a gas from an initial pressure P, to a final pressure P2 when the equation of state is the virial expansion [Eq. (3.10)] truncated to... 

The Carnot65 cycle, using a perfect gas as the working fluid and reversible steps, will maximize t/. The full cycle consists of four steps (i) an adiabat (S = constant) followed by (ii) an isothermal expansion (constant TH), then (iii) one more adiabat (S = constant), then (iv) a final isothermal compression (constant Tc). Other cycles are given in Table 4.3. 

To integrate this function, the relationship between pressure and volume must be known. In process design, an estimate of the work done in compressing or expanding a gas is often required. A rough estimate can be made by assuming either reversible adiabatic (isentropic) or isothermal expansion, depending on the nature of the process. For isothermal expansion (expansion at constant temperature) ... 

That V2 is greater than Vi implies that w < 0 and q> 0-, m an isothermal expansion, the system does work against the surroundings and heat must be transferred into it to maintain T constant. In an isothermal compression, the reverse is true The surroundings do work on the system, and the system must then lose heat to the bath to maintain T constant. 

It will be recalled from the statements in 9d that in an isothermal, reversible expansion of an ideal gas the work done is exactly equal to the heat absorbed by the system. In other words, in this process the heat is completely converted into work. However, it is important to observe that this conversion is accompanied by an increase in the volume of the gas, so that the system has undergone a change. If the gas is to be restored to its original volume by reversible compression, work will have to be done on the system, and an equivalent amount of heat will be liberated. The work and heat quantities involved in the process are exactly the same as those concerned in the original expansion. Hence, the net result of the isothermal expansion and compression is that the system is restored to its original state, but there is no net absorption of heat and no work is done. The foregoing is an illustration of the universal experience, that it is not possible to convert... 

Let ns next consider a particular type of a non-isothermal reversible cycle consisting of an isothermal expansion of a system (solid, liquid, or gas), followed by an adiabatic expansion, this in turn being followed by an isothermal compression, and this by an adiabatic compression, thereby bringing the system back to its original state Such a cycle, consisting of two isothermal volume changes and two adiabatic volume T.,. changes, is called a Car-... 

Isothermal reversible Reversible adiabatic Isothermal reversible expansion at 7 expansion compression at T ... 

We have regretfully eliminated at this point a discussion of Carnot cycles (a combination of reversible adiabatic and isothermal expansions and compressions of a gas or any other working substance arranged in cycles and producing work), which leads to the result that... 

The most efficient cycle of operation for a reversible heat engine. It consists of four operations, as in the four-stroke internal combustion engine, namely isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression to the initial state. 

Carnot cycle /kar-noh/ The idealized reversible cycle of four operations occurring in a perfect heat engine. These are the successive adiabatic compression, isothermal expansion, adiabatic expansion, and isothermal compression of the working substance. The cycle returns to its initial... 

Since a reversible change provides the maximum (minimum) amount of work for a given expansion (compression), the change in Helmholtz energy provides a bound on the work associated with an isothermal process. 

Carnot cycle The idealized reversible cycle of four operations occurring in a perfect heat engine. These are the successive adiabatic compression, isothermal expansion, adiabatic expansion, and isothermal compression of the working substance. The cycle returns to its initial pressure, volume, and temperature, and transfers energy to or from mechanical work. The efficiency of the Carnot cycle is the maximum attainable in a heat engine. It was published in 1824 by the French physicist Nicolas L. S. Carnot (1796-1832). See Carnot s principle. 

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