Archive:New Draft of the Week

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The New Draft of the Week is a chance to highlight a recently created Citizendium article that has just started down the road of becoming a Citizendium masterpiece.
It is chosen each week by vote in a manner similar to that of its sister project, the Article of the Week.

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Table of Nominees
Nominated article Vote
Score
Supporters Specialist supporters Date created
Developing Article Clean Air Act (U.S.) 1 Milton Beychok; 2009-06-27
Developing Article Euler angles 2 Milton Beychok; Meg Ireland; 2009-07-10

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The next New Draft of the Week will be the article with the most votes at 1 AM UTC on Thursday, 16 July 2009. I did the honors this time. Milton Beychok 01:11, 16 July 2009 (UTC)

Nominated article Supporters Specialist supporters Dates Score
Developing Article Clean Air Act (U.S.): A law enacted by the U.S. Congress that defines the responsibilities of the U.S. Environmental Protection Agency (U.S. EPA) for protecting and improving the nation's air quality and the stratospheric ozone layer. [e]

(PD) Photo: Jon Sullivan, www.pdphoto.org
Poor air quality over Los Angeles (August, 2003)
See also: National Ambient Air Quality Standards

The Clean Air Act is a law enacted by the U.S. Congress that defines the responsibilities of the U.S. Environmental Protection Agency (U.S. EPA) for protecting and improving the nation's air quality and the stratospheric ozone layer. The latest major amendments were enacted as the Clean Air Act Amendments of 1990 (Public Law 101–549).

The Clean Air Act Amendments of 1990 were preceded by various other pieces of legislation enacted by the U.S. Congress dating back to the Air Pollution Control Act of 1955.

Implementation of the Act

In the same year that Congress created the Clean Air Act of 1970 (see History section below), Congress also created the U.S. EPA and gave it the primary role in carrying out the law. Since 1970, the U.S. EPA has been responsible for a variety of programs to reduce air pollution nationwide.

However, the environmental regulatory agencies of the states, Indian tribes and local governments do a lot of the work to meet the Act's requirements. Those agencies work with industrial and commercial companies to reduce air pollution. They also review and approve permit applications for construction and operation of industrial plants and commercial facilities involving sources of air pollution. They are able to develop solutions for pollution problems that require special understanding of local industries, geography, housing, and travel patterns, as well as other factors. State, local, and tribal governments also monitor air quality, inspect facilities under their jurisdictions and enforce Clean Air Act regulations.[1]

States must also develop State Implementation Plans (SIPs) that outline how each state will control air pollution under the Clean Air Act. An SIP is a collection of the regulations, programs and policies that a state will use to clean up polluted areas. In developing their SIPs, the states must involve the public and industries through hearings and opportunities to comment on the development of each state plan.[1]

Contents of the Act

Legislation enacted by the U.S. Congress since 1990 has made several minor changes in the Act. The current version, including amendments through February 24, 2004, is available on the Internet.[2] This is a listing of its major parts:

  • Title I - Air Pollution Prevention and Control
    • Part A - Air Quality and Emission Limitations (CAA § 101-131; USC § 7401-7431 )
    • Part B - Ozone Protection (replaced by Title VI)
    • Part C - Prevention of Significant Deterioration of Air Quality (CAA § 160-169b; USC § 7470-7492)
    • Part D - Plan Requirements for Nonattainment Areas (CAA § 171-193; USC § 7501-7515)
  • Title II - Emission Standards for Moving Sources
    • Part A - Motor Vehicle Emission and Fuel Standards (CAA § 201-219; USC § 7521-7554)
    • Part B - Aircraft Emission Standards (CAA § 231-234; USC § 7571-7574)
    • Part C - Clean Fuel Vehicles (CAA § 241-250; USC § 7581-7590)
  • Title III - General (CAA § 301-328; USC § 7601-7627)
  • Title IV - Acid Deposition Control (CAA § 401-416; USC § 7651-7651o)
  • Title V - Permits (CAA § 501-507; USC § 7661-7661f )
  • Title VI - Stratospheric Ozone Protection (CAA § 601-618; USC § 7671-7671q )

Key elements of the act

The U.S. EPA's mission is to protect human health and the environment. To achieve this mission, EPA implemented a variety of programs under the Clean Air Act that focus on:

  • Reducing outdoor ambient concentrations of air pollutants that cause smog, haze, acid rain, and other problems
  • Reducing emissions of toxic air pollutants that are known to, or are suspected of, causing cancer or other serious health effects
  • Phasing out production and use of chemicals that destroy stratospheric ozone.

The above programs have required the development of specific regulations by the U.S. EPA and the state, tribal and local governments to limit the emissions of air pollutants from mobile sources (like automotive vehicles and airplanes) and stationary sources (like petroleum refineries, petrochemical and chemical manufacturing plants, power plants, and gas or petrol stations).

More specifics concerning each of the above programs are available on the Internet.[3]

What the Clean Air Act has accomplished

The implementation of the Clean Air Act has accomplished very significant reductions in the emission of air pollutants during the period of 1970 – 2008. The reductions were accomplished despite even larger increases in activities, during that same period, that produce air pollutants as measured by the national Gross Domestic Product (GDP), consumption of energy, and usage of automobiles. The details of that accomplishment are presented in the two tables below:[1][4]


Air pollutant emission reductions
(1970 – 2008)
Pollutant Emissions Reductions
Criteria air pollutants More than 50%
Toxic air pollutants Nearly 70%
Automobile emissions More than 90%
           
Activities that produce air pollutants
(1970 – 2008)
Factors Increases
Gross Domestic Product 200 %
Energy consumption 50%
Automobile usage Almost 200%

History

In October 1948, a cloud of air pollution formed above the industrial town of Donora, Pennsylvania and lingered for five days. It caused sickness in 6,000 of the town's 14,000 people and the death of 20 people. Four years later, in 1952, over 3,000 people died in what became known as London's "Killer Fog".[1][4] Events like those led to the enactment of several federal and state laws which established funding for the study and the cleanup of air pollution. But there was no comprehensive federal response to address air pollution until Congress passed the Clean Air Act in 1970 and created the U.S. EPA.

In 1990, Congress amended and greatly expanded the Clean Air Act, providing EPA even broader authority to implement and enforce regulations reducing air pollutant emissions. The 1990 Amendments also placed an increased emphasis on more cost-effective approaches to reduce air pollution.

The principal milestones in the evolution of the Clean Air Act are:[4][5][6]

The Air Pollution Control Act of 1955

The Air Pollution Control Act of 1955 was the first federal legislation that involved air pollution. It funded research of air pollution and state assistance resources.

Clean Air Act of 1963

The Clean Air Act of 1963 was the first federal legislation regarding air pollution control. It established a federal program within the U.S. Public Health Service and authorized research into techniques for monitoring and controlling air pollution. It included grants to the states for developing state and local air pollution control programs.

In 1966, the Clean Air Act was extended to add the authority for grants to maintain state and local programs rather than just develop them.

Air Quality Act of 1967

In 1967, an Air Quality Act was enacted in order to expand federal government activities. In accordance with this law, enforcement proceedings were initiated in areas subject to interstate air pollution transport. As part of these proceedings, the federal government for the first time conducted extensive ambient monitoring studies and stationary source inspections.

The Air Quality Act of 1967 also authorized expanded studies of air pollutant emission inventories, ambient monitoring techniques, and control techniques.

Clean Air Act 1970

The enactment of the Clean Air Act of 1970 constituted a major change of the federal government's role in air pollution control. It authorized the development of comprehensive federal and state regulations to limit emissions from stationary industrial sources and from mobile sources.

Four major regulatory programs involving stationary sources were initiated:

It also authorized requirements for the control of automotive vehicle emissions (i.e., mobile sources). Furthermore, the authority to enforce air quality standards and air pollution emission controls was substantially expanded.

As mentioned above, the EPA was created at about the same time in order to implement the various requirements included in the Clean Air Act of 1970.

1977 Amendments to the Clean Air Act of 1970

The 1977 Amendments established major review requirements to ensure the attainment and maintenance of the National Ambient Air Quality Standards.

It codified provisions for the Prevention of Significant Deterioration (PSD) program. It also codified requirements pertaining to air pollution sources in geographical areas called "non-attainment areas" because they had not attained on or more of the National Ambient Air Quality Standards.

1990 Amendments to the Clean Air Act of 1970

Another set of major amendments to the Clean Air Act were enacted in 1990 that substantially increased the authority and responsibility of the federal government. New regulatory programs were authorized for:

  • Control of acid deposition (acid rain)
  • Established the requirements for stationary source operating permits
  • Established a program to control 189 toxic air pollutants, including those previously regulated by the National Emission Standards for Hazardous Air Pollutants (NESHAPS)

The provisions for attainment and maintenance of the National Ambient Air Quality Standards were significantly modified and expanded. Other revisions involved stratospheric ozone protection, increased enforcement authority, and expanded research programs.

References

  1. 1.0 1.1 1.2 1.3 Understanding the Clean Air Act (From the U.S. EPA website)
  2. The Clean Air Act (As Amended Through P.L. 108–201, February 24, 2004)
  3. Key Elements of the Clean Air Act (From the U.S. EPA website)
  4. 4.0 4.1 4.2 Air Today, Yesterday, and Tomorrow (Part 1, prepared for the U.S. EPA by John Bachmann, 2008)
  5. Air Today, Yesterday, and Tomorrow (Part 2, prepared for the U.S. EPA by John Bachmann, 2008)
  6. History of the Clean Air Act (From the U.S. EPA website)
 (Read more...)
Milton Beychok; 1


Developing Article Euler angles: three rotation angles that describe any rotation of a 3-dimensional object. [e]

PD Image
Figure 1. Euler angles. From left to right: initial configuration, after rotation over angle α, after rotation over angle β, and after rotation over angle γ.

In physics, mathematics, and engineering, Euler angles are three rotation angles, often denoted by 0 ≤ α ≤ 2π, 0 ≤ β ≤ π, and 0 ≤ γ ≤ 2π, although the notation φ, θ, ψ is also common. Any rotation of a 3-dimensional object can be performed by three consecutive rotations over the three Euler angles.

Different conventions are in use: a rotation can be active (the object is rotated, the system of axes is fixed in space), or passive (the object is fixed in space, the axes are rotated).

Also the choice of rotation axes may vary; an active convention common in quantum mechanical applications is the z-y′-z′ convention. Attach a system of Cartesian coordinate axes to the body that is to be rotated (the coordinate frame is fixed to the body and is rotated simultaneously with it); in the figure the body-fixed frame is shown in red and labeled by lowercase letters. First rotate around z, then around the new body-fixed y-axis, y′, and finally around z′. Another convention often used is the z-x′-z′ convention, where instead of over the new y-axis the second rotation is over the new x-axis. Also the z-y-x convention is used (and will be discussed below).

The right-hand screw rule is practically always followed: the rotation axis is a directed line and a positive rotation is as a cork screw driven into the positive direction of the axis. In older literature left-handed Cartesian coordinate frames appear sometimes, but in modern literature right-handed frames are used exclusively.

Euler angles are used in many different branches of physics and engineering. The present article is written from the point of view of molecular physics, where the objects to be rotated are molecules and applications are often of quantum mechanical nature.

The angles are named after the 18th century mathematician Leonhard Euler who introduced in 1765 two of the three for an axially symmetric body where the third angle, γ, does not play a role.[1]

Geometric discussion

In Figure 1 the space-fixed (laboratory) axes are labeled by capital X, Y, and Z and are shown in black. The body to be rotated is not shown, but a system of axes fixed to it is shown in red. One may use any convenient orthonormal frame as a body-fixed frame. Often the body-fixed axes are principal axes, that means that they are eigenvectors of the inertia tensor of the body. Also symmetry axes, when present, may be used. When the body has symmetry axes, the principal axes often coincide with these.

PD Image
Figure 2. Rotation of r to r′. On the left around z-axis over α (φ increases), on the right around y-axis over β (φ decreases). Both rotation axes point toward the reader.

The z-y′-z′ convention will be followed. Initially, the two frames coincide, and the path to a final arbitrary orientation of the body—and its frame—is depicted on Figure 1. The first rotation is around the z-axis, which coincides with the Z-axis. The x- and y-axis move in a plane perpendicular to the z-axis over an angle α. The second rotation is in a plane through the origin perpendicular to the y′-axis. The angle is β. The present convention has the practical advantage that the z′-axis has the usual spherical polar coordinates α ≡ φ (longitude angle) and β ≡ θ (colatitude angle) with respect to the space-fixed frame.[2] The final rotation is in a plane perpendicular to the z′-axis over an angle γ. From geometric considerations follows that any orientation of the body-fixed frame in space may be obtained.

Write for the rotation matrix that describes a rotation around the unit vector over an angle . Clearly the three consecutive Euler rotations correspond to rotations around the unit vectors along the body-fixed axes z, y′, and z′ over angles α β, and γ, respectively. Because a matrix acts on a column vector to its right, the order in the matrix product is as in the leftmost term in the following equation. It will be shown that the corresponding matrix product can be written in reverse order (but around fixed, unprimed, axes z, y, z), that is,

Note that the third column contains the Cartesian coordinates with respect to the space-fixed frame of expressed in sines and cosines of spherical polar angles. The first and second column contain by definition expressions for the Cartesian coordinates of and , respectively, but evidently these are not solely in terms of spherical polar angles, γ also enters.

Before proving the first equality in the above equation (reversal of order), we derive the matrix for a rotation around the z-axis, see the left drawing in Figure 2. The rotated vector has components

We used here the relations well-known from trigonometry for the sine and cosine of a sum angle. The derivation of the matrix for a rotation around the y-axis proceeds along the same lines. Note, however, that the angle of a vector with the x-axis decreases by a rotation around the positive y-axis (see right-hand drawing in Figure 2).

To prove the first equality (reversal of the order in the angles), a property of rotation matrices is used. A rotation (orthogonal 3×3) matrix A, transforming a rotation axis, gives rise to the following similarity equation,

where the superscript T indicates the transpose of the matrix. For rotation matrices the transposed matrix is equal to the inverse of the matrix. From this similarity relation follows that

so that

Also

so that

where it is used that rotations around the same axis commute, that is,

and the required result is proved.

Algebraic treatment

In the proof that any rotation can be written as three consecutive rotations, an appeal was made to the geometric insight of the reader. The same result can be proved more rigorously by algebraic means. To that end the notation is somewhat shortened:

Theorem

A proper rotation matrix R can be factorized thus

which is referred to as the Euler z-y-x parametrization, or also as

the Euler z-y-z parametrization.

Proof

First the Euler z-y-x-parametrization will be proved by an algorithm for the factorization of a given matrix R ≡ (r1, r2, r3). Second the z-y-z parametrization will be proved; this parametrization is—as shown above—equivalent to the z′-y′-z parametrization with angles in reverse order.

:A Fortran subroutine based on the algorithm is given on the code page.

To prove the z-y-x parametrization we consider the matrix product

The columns of the matrix product are for ease of reference designated by a1, a2, and a3. Note that the multiplication by

on the right does not affect the first column, so that a1 = r1 (the first column of R). Solve and from the first column of R (which is known),

This is possible. First solve for from

Then solve for from the two equations:

The angles and determine fully the vectors a2 and a3.

Since a1, a2 and a3 are the columns of a proper rotation matrix they form an orthonormal right-handed system. The plane spanned by a2 and a3 is orthogonal to and hence the plane contains and . Thus the latter two vectors are a linear combination of the first two,

Since are known orthonormal vectors, we can compute

These equations give with .

The angle ω1 gives the matrix with

This gives the required z-y-x factorization of the arbitrary proper orthogonal matrix R.

The proof of the Euler z-y-z parametrization is obtained by a small modification of the previous proof. We start by retrieving the spherical polar coordinates and of the unit vector , the third column [the rightmost multiplication by Rz1) does not affect r3]. Then consider

or, The equation for R can be written as

which proves the Euler z-y-z parametrization. Clearly, this factorization is equal to the one given in the previous section, with

Note

  1. Translation by Ian Bruce of L. Euler, Theoria Motus Corporum Solidorum Seu Rigidorum (Theory of the motion of solid or rigid bodies), Rostock (1765), pdf page 11. Later Euler returned to the angles and gave an alternative derivation, see Translation by Johan Sten of Formulae generales pro translatione quacunque corporum rigidorum (General formulas for the translation of arbitrary rigid bodies), Novi Commentarii academiae scientiarum Petropolitanae, vol. 20, (1776), pp. 189-207
  2. In the z-x′-z′ convention the first two Euler angles are not equal to spherical polar angles, in consequence the (m, m′) Wigner D-matrix-element carries the complex phase exp[iπ(mm′)/2]. This phase is absent in the z-y′-z′ convention
 (Read more...)
Milton Beychok; Meg Ireland 2


Current Winner (to be selected and implemented by an Administrator)

To change, click edit and follow the instructions, or see documentation at {{Featured Article}}.


PD Image
Energy of the hot gas flame flows into the kettle and the liquid water in it.

Heat is a form of energy that is transferred between two bodies that are in thermal contact and have different temperatures. For instance, the bodies may be two compartments of a vessel separated by a heat-conducting wall and containing fluids of different temperatures on either side of the wall. Or one body may consist of hot radiating gas and the other may be a kettle with cold water, as shown in the picture. Heat flows spontaneously from the higher-temperature to the lower-temperature body. The effect of this transfer of energy usually, but not always, is an increase in the temperature of the colder body and a decrease in the temperature of the hotter body.

Change of aggregation state

A vessel containing a fluid may lose or gain energy without a change in temperature when the fluid changes from one aggregation state to another. For instance, a gas condensing to a liquid does this at a certain fixed temperature (the boiling point of the liquid) and releases condensation energy. When a vessel, containing a condensing gas, loses heat to a colder body, then, as long as there is still vapor left in it, its temperature remains constant at the boiling point of the liquid, even while it is losing heat to the colder body. In a similar way, when the colder body is a vessel containing a melting solid, its temperature will remain constant while it is receiving heat from a hotter body, as long as not all solid has been molten. Only after all of the solid has been molten and the heat transport continues, the temperature of the colder body (then containing only liquid) will rise.

For example, the temperature of the tap water in the kettle shown in the figure will rise quickly to the boiling point of water (100 °C). Then, when the flame is not switched off, the temperature inside the kettle remains constant at 100 °C for quite some time, even though heat keeps on flowing from flame to kettle. When all liquid water has evaporated—when the kettle has boiled dry—the temperature of the kettle will quickly rise again until it obtains the temperature of the burning gas, then the heat flow will finally stop. (Most likely, though, the handle and maybe the metal of the kettle, too, will have melted before that).

Units

At present the unit for the amount of heat is the same as for any form of energy. Before the equivalence of mechanical work and heat was clearly recognized, two units were used. The calorie was the amount of heat necessary to raise the temperature of one gram of water from 14.5 to 15.5 °C and the unit of mechanical work was basically defined by force times path length (in the old cgs system of units this is erg). Now there is one unit for all forms of energy, including heat. In the International System of Units (SI) it is the joule, but the British Thermal Unit and calorie are still occasionally used. The unit for the rate of heat transfer is the watt (J/s).

Equivalence of heat and work

Although heat and work are forms of energy that both obey the law of conservation of energy, they are not completely equivalent. Work can be completely converted into heat, but the converse is not true. When converting heat into work, part of the heat is not—and cannot be—converted to work, but flows to the body of lower temperature that is out of necessity present to generate a heat flow.

Heat and temperature

The important distinction between heat and temperature (heat being a form of energy and temperature a measure of the amount of that energy present in a body) was clarified by Count Rumford, James Prescott Joule, Julius Robert Mayer, Rudolf Clausius, and others during the late 18th and 19th centuries. Also it became clear by the work of these men that heat is not an invisible and weightless fluid, named caloric, as was thought by many 18th century scientists, but a form of motion. The molecules of the hotter body are (on the average) in more rapid motion than those of the colder body. The first law of thermodynamics, discovered around the middle of the 19th century, states that the (flow of) heat is a transfer of part of the internal energy of the bodies. In the case of ideal gases, internal energy consists only of kinetic energy and it is indeed only this motional energy that is transferred when heat is exchanged between two containers with ideal gases. In the case of non-ideal gases, liquids and solids, internal energy also contains the averaged inter-particle potential energy (attraction and repulsion between molecules), which depends on temperature. So, for non-ideal gases, liquids and solids, also potential energy is transferred when heat transfer occurs.

Forms of heat

The actual transport of heat may proceed by electromagnetic radiation (as an example one may think of an electric heater where usually heat is transferred to its surroundings by infrared radiation, or of a microwave oven where heat is given off to food by microwaves), conduction (for instance through a metal wall; metals conduct heat by the aid of their almost free electrons), and convection (for instance by air flow or water circulation).

Entropy

If two systems, 1 (cold) and 2 (hot), are isolated from the rest of the universe (i.e., no other heat flows than from 2 to 1 and no work is performed on the two systems) then the entropy Stot = S1 + S2 of the total system 1 + 2 increases upon the spontaneous flow of heat. This is in accordance with the second law of thermodynamics that states that spontaneous thermodynamic processes are associated with entropy increase. In general, the entropy S of a system at absolute temperature T increases with

when it receives an amount of heat Q > 0. Entropy is an additive (size-extensive) property.

The hotter system 2 loses an amount of heat to the colder system 1. In absolute value the exchanged amounts of heat are the same by the law of conservation of energy (no energy escapes to the rest of the universe), hence

Here it is assumed that the amount of heat Q is so small that the temperatures of the two systems are constant. One can achieve this by considering a small time interval of heat exchange and/or very large systems.

Remark: the expression ΔS = Q/T is only strictly valid for a reversible (also known as quasistatic) flow of energy. It is possible[1] to define:

It is assumed that ΔSint is much smaller than ΔSext, so that it can be neglected.

Semantic caveats

It is strictly speaking not correct to say that a hot object "possesses much heat"—it is correct to say, however, that it possesses high internal energy. The word "heat" is reserved to describe the process of transfer of energy from a high temperature object to a lower temperature one (in short called "heating of the cold object"). The reason that the word "heat" is to be avoided for the internal energy of an object is that the latter can have been acquired either by heating or by work done on it (or by both). When we measure internal energy, there is no way of deciding how the object acquired it—by work or by heat. In the same way as one does not say that a hot object "possesses much work", one does not say that it "possesses much heat". Yet, terms as "heat reservoir" (a system of temperature higher than its environment that for all practical purposes is infinite) and "heat content" (a synonym for enthalpy) are commonly used and are incorrect by the same reasoning.

The molecules of a hot body are in agitated motion and, as said, it cannot be measured how they became agitated, by work or by heat. Often, especially outside physics, the random molecular motion is referred to as "thermal energy". In classical (phenomenological) thermodynamics this is an intuitive, but undefined, concept. In statistical thermodynamics, thermal energy could be defined (but rarely ever is) as the average kinetic energy of the molecules constituting the body. Kinetic and potential energy of molecules are concepts that are foreign to classical thermodynamics, which predates the general acceptance of the existence of molecules.

Quotation

As a result Carathéodory was able to obtain the laws of thermodynamics without recourse to fictitious machines or objectionable concepts as the flow of heat.[2]

Reference

  1. E. A. Guggenheim, Thermodynamics, 5th edition, North Holland (1967). p. 17
  2. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, 2nd edition, Van Nostrand Company, New York (1956) p. 29

(Read more...)

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