Ideal gas reversible

Assuming the heat capacity of an ideal gas is constant with temperature, calculate the entropy change associated with lowering the temperature of 1.47 mol of monatomic ideal gas reversibly from 99.32°C to — 78.54°C at (a) constant pressure and (b) constant volume. 

TiVi = T2V2 (ideal gas, reversible adiabatic process) (Section 12.6)... 

Ideal-Gas Law. Here Eq. XVII-67 applies, and on reversing the procedure that led to it, one finds... 

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. 

For an ideal gas and a diathemiic piston, the condition of constant energy means constant temperature. The reverse change can then be carried out simply by relaxing the adiabatic constraint on the external walls and innnersing the system in a themiostatic bath. More generally tlie initial state and the final state may be at different temperatures so that one may have to have a series of temperature baths to ensure that the entire series of steps is reversible. 

The total energy of condensation from the ideal gas to the liquid state (the reverse process of vaporization) as a consequence of 1-1 contacts (i.e., intermolecular interactions of component 1 with like molecules) is the product of the energy of condensation per unit volume, the volume of liquid, and the volume fraction of component 1 in the liquid, or... 

Corollary 1.—If an ideal gas changes its volume reversibly without alteration of temperature, the quantities of heat absorbed or emitted form an arithmetical progression whilst the volumes form a geometrical progression (Sadi Carnot, 1824). 

Example.—An ideal gas of constant specific heat is taken round a reversible Carnot s cycle, represented by four curves (Fig. 22) ... 

Let unit mass of an ideal gas pass reversibly from the state... 

Example.—If a mol. of an ideal gas changes reversibly from a state of 10 litres at 15° C. to 100 litres at 50° C, show that the increase of entropy is... 

We shall suppose the solute to be a mol of an ideal gas, occupying a volume v at the pressure o and the solvent a volume Y of Iig. 56. liquid just sufficient to dissolve all the gas under the pressure j)o- If the gas is brought directly into contact with the liquid, an irreversible process of solution occurs, but if it is first of all expanded to a very large volume, the dissolution may be made reversible, except for the first trace of gas entering the... 

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... 

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

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... 

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

E2.16 For an ideal gas under reversible conditions, a differential element of heat can be expressed in the form... 

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. 

Example 3.8 Show that for the reversible adiabatic expansion of ideal gas with constant heat capacity... 

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... 

Figure 3.2 compares a series of reversible isothermal expansions for the ideal gas starting at different initial conditions. Note that the isotherms are parallel. They cannot intersect since this would give the gas the same pressure and volume at two different temperatures. Figure 3.3 shows a similar comparison for a series of reversible adiabatic expansions. Like the isotherms, the adiabats cannot intersect. To do so would violate the Caratheodory principle and the Second Law of Thermodynamics, since the gas would have two different entropies at the same temperature, pressure, and volume. 

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

Figure 3.2 Reversible isotherms of an ideal gas starting with the same initial and final volumes, but with different pressures.

A reversible isothermal expansion of the ideal gas is made from an initial volume V to a volume Vz at an absolute (ideal gas) temperature 73. The amount of pressure-volume work in done by the system is obtained by substituting into Equation (2.16). The result is... 

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.

The above relations apply for an ideal gas to a reversible adiabatic process which, as already shown, is isentropic. 

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