PHASE

in thermodynamics, chemically and physically uniform or homogeneous quantity of matter that can be separated mechanically from a nonhomogeneous mixture and that may consist of a single substance or of a mixture of substances. The three fundamental phases of matter are solid, liquid, and gas (vapour), but others are considered to exist, including crystalline, colloidal, glassy, amorphous, and plasma phases. Matter is considered to form one homogeneous phase if its atomic or molecular dispersion is uniform; e.g., a glass of water containing dissolved salt, sugar, bicarbonate of soda, and a dye constitutes only a single liquid phase. If hundreds of grains of sand were added, all the grains together would constitute only a single additional (solid) phase. The different phases of a pure substance bear a fixed relationship to one another in terms of temperature and pressure. Thus, if the pressure on some liquids is raised, they will freeze at a higher temperature. This relationship is extremely important in industrial as well as scientific work (see phase diagram). in thermodynamics, chemically and physically uniform or homogeneous quantity of matter that can be separated mechanically from a nonhomogeneous mixture and that may consist of a single substance or of a mixture of substances. The three fundamental phases of matter are solid, liquid, and gas (vapour), but others are considered to exist, including crystalline, colloid, glassy, amorphous, and plasma phases. When a phase in one form is altered to another form, a phase change is said to have occurred. in astronomy, any of the varying appearances of a celestial body as different amounts of its disk are seen (from the Earth, ordinarily) to be illuminated by the Sun. The Moon displays four main phases: new, first quarter, full, and last quarter. Earth, as seen from the Moon, shows the same phases in opposite order; e.g., Earth is full when the Moon is new. Planets more distant than the Earth from the Sun display only full or gibbous (more than half but not entirely full) phases to an observer on the Earth; i.e., they are always seen with more than half of their apparent disks in sunlight. Mercury and Venus, closer to the Sun than Earth is, show full cycles of phases like the Moon's. in mechanics, the fraction of a period (i.e., the time required to complete a full cycle) that a point completes after last passing through the reference, or zero, position. For example, the reference position for the hands of a clock is at the numeral 12, and the minute hand has a period of one hour. At a quarter past the hour the minute hand has a phase of one-quarter period, having passed through a phase angle of 90, or p/2 radians. In this example the motion of the minute hand is a uniform circular motion, but the concept of phase also applies to simple harmonic motion such as that experienced by waves and vibrating bodies. If the position y of a point or particle changes according to a simple harmonic law, then it will change in time t according to the product of the amplitude, or maximum displacement, r, of the particle and a sine or cosine function composed of its angular speed, symbolized by the Greek letter omega (w), the time t, and what is called the epoch angle, symbolized by the Greek letter epsilon (e): y = r sin (wt + e). The angle (wt + e) is called the phase angle at time t, which at zero time is equal to e. Phase itself is a fractional value-the ratio of elapsed time t to the period T, or t/T-and is equal to the ratio of the phase angle to the angle of the complete cycle, 360, or 2p radians. Thus, phase for uniform circular or harmonic motion has the value (wt + e)/2p. Applying this expression to the example of the moving minute hand cited above, e is zero (zero phase angle at zero time), angular speed is 2p radians per hour, and time t is 1/4 hour, giving a phase of 1/4. When comparing the phases of two or more periodic motions, such as waves, the motions are said to be in phase when corresponding points reach maximum or minimum displacements simultaneously. If the crests of two waves pass the same point or line at the same time, then they are in phase for that position; however, if the crest of one and the trough of the other pass at the same time, the phase angles differ by 180, or p radians, and the waves are said to be of opposite phase. Two periodic motions represented by the equations y1 = r1 sin (w1t + e1) and y2 = r2 sin (w2t + e2) have a phase-angle difference (w2t + e2) - (w1t + e1), or (w2 - w1)t + (e2 - e1). At zero time, or if the angular speeds w1 and w2 are identical, the phase-angle difference is (e2 - e1) and the phase difference is (e2 - e1)/2p. The measurement of phase difference is of central importance in alternating-current technology. In the diagram, two waves represent the voltage (E) and the current (I) in an alternating-current (ac) circuit with pure inductance. The difference in phase angle (eE - eI) is 90, and the phase difference is 90/360 = 1/4; the current is said to lag one-quarter cycle in phase. This lag may also be seen from the diagram: the voltage has already completed one-quarter cycle by the time the current has reached zero. In ac power transmission, the terms multiphase and polyphase are applied to currents that are out of phase with one another. In a two-phase system, there are two currents with a phase-angle difference of 90; in a three-phase system, the currents differ in phase angle by 120. Additional reading Daniel S. Barker, Igneous Rocks (1983), ch. 3, "Phase Relations," pp. 24-57, presents a clear summary of the interpretation of petrologic phase diagrams. Ernest G. Ehlers, The Interpretation of Geological Phase Diagrams (1972, reprinted 1987), provides step-by-step nonmathematical procedures for understanding both simple and complex phase diagrams. Ernest G. Ehlers and Harvey Blatt, Petrology: Igneous, Sedimentary, and Metamorphic (1982), ch. 2, "Experiments with Molten Silicates: Unary and Binary Systems," pp. 43-73, provides an introduction to phase equilibria of petrologic systems. Donald W. Hyndman, Petrology of Igneous and Metamorphic Rocks (1972), in the second half of ch. 1, "Environment and Materials," pp. 15-30, summarizes the interpretation of some important petrologic phase diagrams. W.G. Ernst, Petrologic Phase Equilibria (1976), is a concise introduction to phase equilibria; some knowledge of thermodynamics on the part of the reader would be helpful. Ernest G. Ehlers