Ideal gas expansion

Figure 3.1 Comparison of reversible isothermal and adiabatic (C =jR) ideal gas expansions.

Reversible Isothermal Expansion Let us consider the heat and work of ideal gas expansion from V to V2 under isothermal conditions (AT = 0). We recognize from (3.74a) that... 

Several Type HI studies were performed at very high injectant-to-chamber pressure ratios to imitate SCRAMJET conditions. Wu and Chen [24, 25] and Lin and Cox-Stouffer [26] studied the location of shock structures resulting fi"om this type of injection. Far from the critical point, jet behavior resembled ideal-gas expansion. In contrast, homogeneous droplet nucleation was observed near the critical point. Locations of the observed shock structures, i.e. Mach disks, matched well with those from under-expanded ideal-gas jet predictions. However, the Mach disks disappeared as the injectant-to-chamber pressure ratio decreased. 

Figure 8.11 shows a container in which moles of ideal gas A at temperature T, pressure P, and volume are separated by a partition from % moles of ideal gas B at the same temperature and pressure as gas A, but with volume Vb- When the partition is removed, the gases mix spontaneously and the entropy of the system increases. To calculate the entropy of mixing, AS jx, we can treat the process as two separate isothermal ideal gas expansions using Equation 8.10. For gas A ... 

Equation (8.49) shows how the temperature decreases as the volume increases in an adiabatic ideal gas 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. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, l/t)(3 0/3T)p for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (3p/3T),, should be calculated either from the equation of state or from tabulated PVT data. 

The thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

The coiTclation of AH° for the ideal gas at low temperature is based on a series expansion in temperature and is expressed as ... 

For an ideal gas, pV is constant for isentropic expansion (that is, without energy addition or energy loss). Therefore, V2 is ... 

This equation is well known and often used to calculate initial fragment velocity, but its application can result in gross overestimation. Assuming adiabatic expansion of the ideal gas, it can be derived that ... 

A particular case is a cyclic process an example of a non-cyclic aschistic change is afforded by the expansion of an ideal gas at constant temperature ( 71). 

Illustrative Example.—Consider the isothermal expansion of an ideal gas between fixed limits of volume ... 

The isothermal expansion of an ideal gas is an aschistic process.— If a mass of gas expands isothermally, the heat absorbed is equal to the external work done. 

Ideal Gases.—The state of unit mass of an ideal gas, undergoing adiabatic compression or expansion, is completely defined by the equations... 

If a relationship is known between the pressure and volume of the fluid, the work can be calculated. For example, if the fluid is the ideal gas, then pV = nRT and equation (2.14) for the isothermal reversible expansion of ideal gas becomes... 

Figure 2.6 Comparison of the work obtained from the two isothermal expansions at 300 K of ideal gas following path (i) in (a) and path (ii) in (b). In each instance, the initial pressure is 0.200 MPa and the final pressure is 0.100 MPa.

Example 2.1 Compare the work obtained from the two isothermal expansions at 300 K of one mole of ideal gas following the paths shown in Figure 2.6. 

Example 2.3 Calculate q for the isothermal reversible expansion of the ideal gas under the conditions given in Example 2.1. 

Solution AC = 0 for the isothermal expansion of the ideal gas. Hence, from equation (2.33)... 

In summary, in the isothermal expansion of ideal gas, work flowing out of a system is balanced by heat flowing into the system so that AC = 0. 

Calculation of AS for the Reversible Isothermal Expansion of an Ideal Gas Integration of equation (2.38) gives... 

From example 2.3 we saw that for the reversible isothermal expansion of ideal gas... 

---