Conservation of energy

From Wikipedia, the free encyclopedia.

This article is about the physics principle. For information on using energy resources sustainably, see Energy Conservation.

Laws of thermodynamics
Zeroth law of thermodynamics
First law of thermodynamics
Second law of thermodynamics
Third law of thermodynamics
Philosophy of thermal and statistical physics
Edit

Conservation of energy, also known as the first law of thermodynamics, is possibly the most important, and certainly the most practically useful of several conservation laws in physics.

The law states that the total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. In other words, energy can be converted from one form to another, but it cannot be created or destroyed.

The law of conservation of energy excludes the possibility of perpetuum mobile of the first kind.

1 Historical development
2 Modern physics
3 Mathematical formulations
4 Reference
5 Bibliography
6 External links

[edit]

Historical development

To understand the significance of the conservation of energy in the context of the development of thermodynamics, see Thermodynamics timeline Edit

Although ancient philosophers as far back as Thales of Miletus had inklings of the first law, it was the German Gottfried Wilhelm Leibniz during 1676-1689 who first attempted a mathematical formulation. Leibniz noticed that in many mechanical systems (of several masses, m_i each with velocity v_i) the quantity:

$\sum_{i} m_i v_i^2$

was conserved. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in many situations. However, many physicists were influenced by the prestige of Sir Isaac Newton in England and of René Descartes in France, both of whom had set great store by the conservation of momentum as a guiding principle. Thus the momentum:

$\,\!\sum_{i} m_i v_i$

was held by the rival camp to be the conserved vis viva. It was largely engineers such as John Smeaton, Peter Ewart, Karl Hotzmann, Gustave-Adolphe Hirn and Marc Séguin who objected that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston.

Members of the academic establishment such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics but in the 18th and 19th centuries, the fate of the lost energy was still unknown. Gradually it came to be suspected that the heat inevitably generated by motion was another form of vis viva. In 1783, Antoine Lavoisier and Pierre-Simon Laplace reviewed the two competing theories of vis viva and caloric^[1]. Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat. Vis viva now started to be known as energy, after the term was first used in that sense by Thomas Young in 1807.

The recalibration of vis visa to

$\frac {1} {2}\sum_{i} m_i v_i^2$

was largely the result of the work of Gaspard-Gustave Coriolis and Jean-Victor Poncelet over the period 1819-1839. The former called the quantity quantité de travail and the latter, travail mécanique and both championed its use in engineering calculation.

In a paper Uber die Natur der Warme, published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."

A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. The caloric theory maintained that heat could neither be created nor destroyed but conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.

The mechanical equivalence principle was first stated in its modern form by the German surgeon Julius Robert von Mayer.^[2] Mayer reached his conclusion on a voyage to the Dutch East Indies, where he found that his patients' blood was a deeper red because they were consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He had discovered that heat and mechanical work were both forms of energy, and later, after improving his knowledge of physics, he calculated a quantitative relationship between them.

Joule's apparatus for measuring the mechanical equivalent of heat

Meanwhile, in 1843 James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the thermal energy (heat) gained by the water by friction with the paddle.

Over the period 1840-1843, similar work was carried out by engineer Ludwig A. Colding though it was little-known outside his native Denmark.

Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that, perhaps unjustly, eventually drew the wider recognition.

For the dispute between Joule and Mayer over priority, see Mechanical equivalent of heat: Priority

Drawing on the earlier work of Joule, Sadi Carnot and Émile Clapeyron, in 1847, Hermann von Helmholtz postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single force (energy in modern terms). He published his theories in his book Über die Erhaltung der Kraft (On the Conservation of Force, 1847). The general modern acceptance of the principle stems from this publication.

In 1877, Peter Guthrie Tait claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the Philosophiae Naturalis Principia Mathematica. This is now generally regarded as nothing more than an example of Whig history.

[edit]

Modern physics

With the discovery of special relativity by Albert Einstein, mass and energy were shown to be interchangeable by the famous equation E = mc². As a result, the rule of conservation of energy was shown to be a special case of a more general rule, the conservation of mass and energy, which is now usually just referred to as conservation of energy.

Conservation of energy can be shown through Noether's theorem to be the result of the time-invariance of the laws of physics (=time has no effect on any physical process).

Within the realm of quantum mechanics, conservation of energy is not applicable when energy can not be defined (say, for time scales shorter than the uncertainty principle defines).

[edit]

Mathematical formulations

One formulation for the first law of thermodynamics is

(1)

Q = Δ U + W

where Q is heat transferred into the system from the surroundings, W is work done by the system, and U is the internal energy of the system. This energy is mostly kinetic energy: the potential energy can be assumed to be negligible. Pressure-volume work (e.g. done by a gas on a piston) is defined to be

(2)

W = P Δ V .

Equation (1) can be interpreted as follows: Q is heat energy being input into the system. The system then can use this incoming energy to do two things: (1) do work, or (2) increase its own internal energy. Here is an analogy: Q is income, which can then be spent to buy things (W), or it can be saved in a bank account ( $Δ U$ ).

If all the heat is used to do work ( $Q = W$ and $Δ U = 0$ ) then the system is undergoing an isothermal process, which means that its temperature remains constant. This is because the system's internal energy is proportional to its temperature.

If all the heat is used to increase internal energy, ( $Q = Δ U$ and $W = 0$ ) then the system is undergoing an isochoric process, also called isometric process. This is a process in which the system's volume is constant: $Δ V = 0$ so that, according to equation (2), W = 0.

It is also possible for the heat energy to be used up partially by doing work and partially by increasing internal energy. An example of such processes is the isobaric process.

Equation (1) is the one preferred by engineers. Another form preferred by chemists is

(3)

Δ U = Q + W

where W is work done on the system by the surroundings. In this case pressure-volume work is defined to be

(4)

W = - P Δ V .

Equation (3) can be interpreted to mean that heat Q and work W are energies being transferred into or out of the system. The system then responds by increasing its internal energy accordingly. In Equation (3) neither Q nor W are state functions. A state function does not depend on which particular thermodynamic process is chosen to connect the initial and final thermostatic states.

[edit]

Reference

^ Lavoisier, A.L. & Laplace, P.S. (1780) "Memoir on Heat", Académie Royal des Sciences pp4-355
^ von Mayer, J.R. (1842) "Remarks on the forces of inorganic nature" in Annalen der Chemie und Pharmacie, 43, 233

[edit]

Bibliography

[edit]

Modern accounts

Kroemer, Herbert; Kittle, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889
Nolan, Peter J. (1996). Fundamentals of College Physics, 2nd ed., William C. Brown Publishers.
Oxtoby & Nachtrieb (1996). Principles of Modern Chemistry, 3rd ed., Saunders College Publishing.
Papineau, D. (2002). Thinking about Consciousness, Oxford University Press: Oxford.
Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.), Brooks/Cole. ISBN 0534408427
Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.), W. H. Freeman. ISBN 0716708094

[edit]

History of ideas

Cardwell, D.S.L. (1971). From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Heinemann: London. ISBN 0435541501
Guillen, M. (1999). Five Equations That Changed the World. ISBN 0349110646
Hiebert, E.N. (1981). Historical Roots of the Principle of Conservation of Energy, Ayer Co Pub. ISBN 0405138806
Kuhn, T.S. (1957) “Energy conservation as an example of simultaneous discovery”, in M. Clagett (ed.) Critical Problems in the History of Science pp.321–56
Smith, C. (1998). The Science of Energy: Cultural History of Energy Physics in Victorian Britain, Heinemann: London. ISBN 0485114313

[edit]

Classic accounts

Mach, E. (1872). History and Root of the Principles of the Conservation of Energy, Open Court Pub. Co., IL.
Poincaré, H. (1905). Science and Hypothesis, Walter Scott Publishing Co. Ltd; Dover reprint, 1952. ISBN 0486602214, Chapter 8, "Energy and Thermo-dynamics"

[edit]