Reversible adiabatic expansion

Figure A2.1.4. Adiabatic reversible (isentropic) paths that do not intersect. (The curves have been calculated for the isentropic expansion of a monatomic ideal gas.)...

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

E3.13 A gas obeys the equation of state PVm RT + Bp and has a heat capacity Cy m that is independent of temperature. Derive an expression relating T and Vm in an adiabatic reversible expansion. 

For the reversible adiabatic expansion, we can see from Equation (5.42) that the final temperature T2 must be less than Ti, because W is negative and Cy is always positive. Thus, the adiabatic reversible expansion is accompanied by a temperature drop, and W, AU, and AH can be calculated from the measured initial and final temperatures using Equations (5.42) and (5.43). 

For a monatomic ideal gas, C i =3/2 R. Calculate the work performed in an adiabatic reversible expansion of 1 mole of this gas by integrating Equation (5.41). 

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 ... 

In Step II, a drop in temperature occurs in the adiabatic reversible expansion, but no change in entropy occurs. The isentropic nature of II is emphasized by the vertical line. Step III is an isothermal reversible compression, with a heat numerically equal to Qi being evolved. As this step is reversible and isothermal, we have from Equation (6.53)... 

Reversible Adiabatic Expansion Under reversible adiabatic conditions (q = 0), the first law and (3.74a) establish that... 

We next carry out an adiabatic reversible expansion from B to C in which the additional quantity of work, W2, represented by the area BCV3V2 is done on the system (a negative value). During this expansion the cylinder is enclosed by an adiabatic envelope. No heat is transferred to the fluid, and its temperature decreases to 0X. 

We complete the proof by considering the adiabatic reversible expansion of an ideal gas. In this case we know that... 

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 idealization of the gas-turbine cycle (based on air, and called the Bray cycle) is shown on a PV diagram in Fig. 8.12. The compression step AB represented by an adiabatic, reversible (isentropic) path in which the press increases from PA (atmospheric pressure) to PB. The combustion process replaced by the constant-pressure addition of an amount of heat QBC. Work produced in the turbine as the result of isentropic expansion of the air to press... 

A certain gas obeys the equation of state P(V -nb) - nRT and has a constant volume heat capacity, Cv, which is independent of temperature. The parameter b is a constant. For 1 mol, find W, AE, Q, and AH for the following processes (a) Isothermal reversible expansion. (b) Isobaric reversible expansion. (c) Isochoric reversible process, (d) Adiabatic reversible expansion in terms of Tlf Vlt V2, Cp, and Cv subscripts of 1 and 2 denote initial and final states, respectively. (c) Adiabatic irreversible expansion against a constant external pressure P2, in terms of Plf P2, Tj, and 7 = (Cp/Cy). 

Perfect (reversible) one-time-expansion adiabatic ERR of our gas, or even imperfect (irreversible) adiabatic or even polytropic (intermediate between adiabatic and isothermal)... 

When no friction is present and the expansion is reversible and adiabatic, entropy is preserved. For this isentropic expansion, we may set ds = 0 in (14.30) to give ... 

Show that the work done by an ideal gas in an adiabatic reversible expansion is P2V2 — P V )/ y — 1). Verify the statement in the text that an ideal monatomic gas with 7=1.667 would do more than twice the work between points A and B (Table 8.1). 

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. 

We can also understand the phenomena in the adiabatic reversible expansion of a gas in such an expansion, 4Qrev = so that dS = 0. Since the volume goes up, the distribution over space broadens, and this part of the entropy increases. If the total entropy change is to be zero, the distribution over energies must get narrower this corresponds to a decrease in energy that is reflected in a decrease in the temperature of the gas. The... 

All the foregoing concerned zero-clearance compressors, ones in which no gas is left in the cylinder at the end of the discharge stroke. For mechanical reasons it is impractical to build a compressor with zero clearance. So in real compressors there is always a small amount of gas in the top of the cylinder, which is repeatedly compressed and expanded. If the compression and expansion are reversible, either adiabatic or isothermal, then they contribute as much work on the expansion step as they require on the compression step, and thus they contribute nothing to the net work requirement of the compressor. For real compressors the compression and the expansion of the gas in the clearance volume contribute to the inefficiency of the compressor compressor designers make the clearance volume as small as practical. 

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... 

From state 2, the adiabatic reversible expansion (i.e., with zero heat exchange) to state 3 follows, resulting in volume V3 and temperature da- Internal energy is U2 = t/(i9a).Using the molar heat capacity Cv at constant volume (which is positive and may be a function of d, see Rem. 5) as nCv = AU/Ad, the differential form of energy balance (1.13) gives with state equation (A.3) for this reversible adiabatic expansion... 

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. 

The expansion of the combustion product gases in the gas turbine again is represented as an adiabatic, reversible (constant S) process. Next, heat is removed from these gases at nearly constant pressure in the heat recovery steam generator and they pass out through the stack. 

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. 

The Carnot engine operates on a two-stroke cycle that is called the Carnot cycle. We begin the cycle with the piston at top dead center and with the hot reservoir in contact with the cylinder. We break the expansion stroke into two steps. The first step is an isothermal reversible expansion of the system at the temperature of the hot reservoir. The final volume of the first step is chosen so that the second step, which is an adiabatic reversible expansion, ends with the system at the temperature of the cold reservoir and with the piston at bottom dead center. The compression stroke is also broken into two steps. The third step of the cyclic process is a reversible isothermal compression with... 

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. 

Consider when the same ideal gas undergoes an adiabatic, reversible expansion (as opposed to isothermal). We will assume that the heat capacity of this gas does not change with temperature, that is, constant beat capaci. This process is illustrated in Figure 2.16. 

An ideal gas undergoes an adiabatic, reversible expansion process in a closed system (a) If Cj, is constant, show that ... 

---