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Temperature

That temperature can be measured in absolute terms has been discussed in Sect. 1.1.1. Volume

The absolute nature of volume is a consequence of our notions of three-dimensional space. The manner by which volume changes with pressure and temperature given by Eq (1.2) are likewise properties with absolute values.

Entropy

Entropy is at once the most fundamental of the thermodynamic concepts and the least connected to common experience. As reviewed in Sect. 1.1.5, entropy acquired meaning as a measure of the degree of order of a system. Highly ordered systems such as solids have low entropies, and conversely, systems possessing a high degree of disorder, such as gases, exhibit high entropies. The regularity of a liquid is intermediate between that of solids and gases, and as a result, the entropy of a liquid falls between that of its corresponding solid and gaseous states. By virtue of the 3rd law of thermodynamics (S = 0 at 0 K), entropy has an absolute value.

Internal Energy, enthalpy and free energy

Internal energy in various forms is stored by the molecules or atoms of a substance. The energy of crystalline solids is contained principally as the interparticle potential energy that is responsible for the stability of the solid phase. Additional energy is stored as kinetic energy of the atoms or molecules that vibrate about their equilibrium positions in the crystal. The increase in vibrational energy of a solid with increasing temperature is responsible for the specific heat of this state of matter. The intermolecular (or interatomic) potential energy is independent of temperature.

The chief mode of energy storage in gases is the translational kinetic energy of the moving particles. Interparticle potential energy in gases is small; in an ideal gas, this component

6 A turnbuckle is a cast-metal sleeve with a left-handed screw thread at one end and right-handed one at the other end. Twisting the sleeve causes the threaded rods to move outward is by definition zero. When the interparticle interactions are not negligible, the gas behaves nonideally.

Molecular gases also hold energy in the form of vibrations and rotations of the individual molecules (atomic gases such as helium do not have contributions of this type). These motions become significant contributors to the internal energy, and hence to the heat capacity, only at high temperatures. For the most part, the specific heats of gases are due to the increase in molecular speeds with increasing temperature.

The internal energy of the liquid state is due mainly to particle vibrations, as in the solid state. Even though the structure is not as regular as that of crystalline solids, liquids possess none of the translational motion of gases. The proximity of the properties of many liquids and solids (e.g., density, internal energy) makes the term condensed phase a useful description when the distinction between liquid and solid is not important.

The internal energy u and its energy-like cousins h, f, and g do not have absolute values. This lack is not an impediment to thermodynamic calculations, however, which involve only changes in these properties. To facilitate computation, the internal energy (or enthalpy or free energy) is set equal to zero at an arbitrarily chosen temperature and pressure called the reference state. The most common reference state is room temperature (298 K) and one atm pressure. Other choices are possible. For example, the enthalpy rather than the internal energy of a substance may be assigned a value of zero at 298 K at 1 atm pressure. However, the reference state must be unique: u can be set equal to zero at some reference condition denoted by To and po, but h cannot also be zero in the same state. This is because h = u + pv, so in the reference state, the specific volume vo is implicitly fixed by the values of To and po. Thus, if u = 0 at po and To, h must be povo in this state.

By virtue of their definitions as derivatives, the specific heats of Eq (1.1) are independent of the reference state chosen for u or h. CP and CV are absolute properties. Similarly, the volume dependencies on temperature and pressure, a and P, possess absolute values.

1.7 Reversible and Irreversible processes

The old English nursery rhyme provides a good description of an irreversible process:

Humpty Dumpty sat on a wall

Humpty Dumpty had a great fall

All the King's horses and all the King's men

Couldn't put Humpty Dumpty together again

Although not all processes are as dramatic as Humpty-Dumpty's fall, the implication of permanent changes in the system and/or surroundings captures the essence of mechanical irreversibility. In thermodynamic processes, irreversibilities occur in a number of ways:

1. Friction between moving parts, which degrades potentially-useful work into heat

2. Rapid (pressure-unbalanced) expansion or compression of a gas, often contained in a cylinder with a piston. If not done slowly, change in gas volume is accompanied by diminishing oscillations of the piston about its final equilibrium position.

3. Heat transfer between system and surroundings through a finite temperature difference.

These three modes of irreversibility are not independent of each other. Damping of piston oscillation in No. 2 is a consequence of solid-solid friction between piston and cylinder wall (No. 1). Friction of any sort ultimately degrades kinetic energy or work to heat, a process that cannot be reversed.

Irreversibility can occur within the system, in the surroundings, or in both. Within the system, internal irreversibility is associated with friction within a moving fluid caused by turbulence, viscous flow, or rapid changes that imbalance the uniform conditions in a gas or liquid. Expansion or compression work done by a system can be calculated by Eq (1.3) only if internal reversibility prevails. External irreversibilities occur in the surroundings, in the boundary between system and surroundings (e.g., friction between the moving piston and the cylinder in Fig. 1.8a), or jointly between system and surroundings , as in heat exchange over a nonzero temperature difference. When irreversibilities are absent from both system and surroundings, the process is said to be totally reversible.

All real processes contain various degrees of irreversibility, but the concept of the perfectly reversible process is extraordinarily useful in thermodynamic analysis. Irreversible processes can be quantitatively treated by the First law only, while both the First and Second laws can be applied to reversible processes.

A reversible process is one that can be made to go backward without any change in the system or the surroundings. To possess this feature, the reversible process must proceed through a series of infinitesimal stages in which internal equilibrium and external equilibrium are maintained. Such processes are sometimes called quasistatic.

Reversibility can also be identified by comparing the work done by (or on) the system with the work done on (or by) the surroundings. If the two works are identical, the process is totally reversible. The following very popular example illustrates this method of detecting irreversibility by analyzing compression of an ideal gas by adding weights to the top of a piston. The book by Van Wylen and Sonntag (Ref. 3, Sect. 6.3) gives a brief qualitative version of this example. Abbott and Van Ness (Ref. 4, Sect. 1.5) present a slightly modified version. Van Ness (Ref. 2, pp. 19 - 22) gives a more colorful qualitative description of this example using grains of sand as the small masses added to the piston.

Example: Compression of an ideal gas using sliding weights

The two methods of compressing a gas by reversible and irreversible processes with common initial and final states are illustrated in Figs. 1.12 and 1.13. In both cases, a total mass M is added to the top of a piston enclosing a cylinder containing n moles of an ideal gas. The piston is assumed to move freely in the cylinder, so that its motion does not introduce irreversibilities into either process. The only difference between the two methods is in the way that the mass is placed on the piston's pedestal. In Fig. 1.12, the mass M is divided into a large number n of very small masses m such that M = nm. The small masses are slid onto the piston one at a time from shelves at different elevations. In Fig. 1.13, the entire mass M is placed on the piston at once.

In both cases, the cylinders are immersed in a constant-temperature bath, which maintains the gas in Fig. 1.12 at temperature T during the entire process. In Fig. 1.13, only the initial and final states are sure to be at temperature T because these are equilibrium states whereas the middle one is not. The constant-temperature condition is not critical to the argument. The key issue is the manner that the masses are added to the piston.

Reversible compression (Fig. 1.12)

Sliding a small mass onto the pedestal causes the piston to very slightly compress the gas in the cylinder. Each addition of a small mass approximates an infinitesimal equilibrium stage so that the overall process is reversible. At all points in the compression process, the conditions of external equilibrium, namely:

are satisfied. Tsurr and psurr are, respectively, the constant temperature and constant pressure of the surroundings (i.e., the environment in which the cylinder is placed). The number of small masses added to the piston is denoted by j (0<j<n), g is the acceleration of gravity, and A is the cross sectional area of the piston and cylinder. The shelves holding the small masses are not equally spaced.

Because the process is reversible, the work involved in compressing the gas can be calculated in two ways. The first is the (negative) pV work done by the gas as its volume is reduced from Vo to Vf by the weight of the small masses slid onto the pedestal of the piston (work is considered to be positive if done by the system):

The integral can be performed by expressing p in terms of V using the ideal gas law, pV = nRT. Since the process is isothermal, T is constant and:

The second method of calculating the work involved in the process depicted in Fig. 1.12 is from the point of view of the surroundings. pV work is performed by the external pressure as the piston descends; in addition, there is a loss of gravitational potential AEp as the small masses are slid onto the pedestal and sink with the piston. These two contributions yield:

Fig. 1.12 Reversible, isothermal compression of an ideal gas

Detailed calculation shows that Wrev = WSurr. This equality is a hallmark of a reversible process. Problem 1.2 analyzes the analogous problem of loading weights onto a spring rather than a piston.

The process in Fig. 1.12 can be reversed by sequentially sliding small masses off the pedestal at their original elevations. When the initial state is recovered, both system and surroundings will have been restored to their original states.

Irreversible Compression (Fig. 1.13)

When the entire mass M is placed on the piston, as in Fig. 1.13, the system is not in equilibrium with the surroundings. The piston rapidly descends, oscillates as the gas acts as a spring, and eventually settles down to the final elevation. Although Wrev does not apply to this situation (because the process is not internally reversible), the equation for WSurr does. In this case, the potential energy loss AEp is easily determined from the elevation change of the large mass, which is directly related to the volume change of the gas by A(elevation) = (Vo - Vf)/A. The work done by the surroundings in this irreversible process is:

At the final equilibrium state, the force balance on the piston gives:

Combining these two equations and using the ideal gas law yields:

* D. Olander, " Compression of an Ideal Gas as a Classroom Example of Reversible and Irreversible Processes", International Journal of Engineering Education 16 (2000) 524

Comparing this equation with Wrev for Vo/Vf = 3 (as an example) shows that the ratio of the reversible work of compression to the irreversible work is:

That is, the surroundings need to supply only about one half as much work for the reversible process as it does for the irreversible process. Finally, simply sliding the large mass off the piston in the final state in Fig. 1.13 will return the system (gas plus piston) to its initial state but will leave the surroundings with the large mass at a lower elevation than it was initially. This is characteristic of an irreversible process.

In the above example, the work done by the surroundings on the system in the irreversible case is always greater than the work required were the process reversible. As a corollary, the work done by the system on the surroundings is always greater than can be had from the irreversible process. The first stricture says that the effort to do a job is always greater than you think it should be and the second warns that you can never do as much as theoretically possible. This_foreshadows the Second Law of Thermodynamics.

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