CONSERVATION OF ENERGY

Conservation of energy has two very different meanings. In the popular sense, "conserve" means to "save" or "preserve." Electric energy for lights likely had its origin in burning coal, so turning off lights tends to preserve coal, a valuable natural resource. In the scientific sense, conservation alludes to constancy. Succinctly stated, the energy of the universe is constant. Energy can be converted from one form to another, but ultimately there are as many joules of energy after the conversion as before. Every second, an operating 100-watt lightbulb converts 100 joules of energy. If 10 joules are in the form of light, then 90 joules are in some other form, notably heat to the room. Even though the total amount of energy following the conversion is unchanged, the energy may not be available for some desired purpose. For example, the heat produced by a lightbulb is in the surroundings and is no longer available for other uses. In fact, when energy runs through all possible conversions, the energy ends up as thermal energy in the environment. Addition of thermal energy to the environment can produce local increases in temperature, leading to what is called thermal pollution. This effect is not to be confused with a possible global increase in temperature due to accumulation of carbon dioxide and other gases in the atmosphere.

CONSERVATIVE FORCES

The illustration in Figure 1 depicts a person pushing a box up a ramp. In the process, the person works against the gravitational force on the box and a frictional force between the box and the ramp. The person, the gravitational force, and the frictional force all do work on the box. The same would be said if the ramp were made longer. But interestingly, from the way work is defined, the work done by the gravitational force depends only on the vertical height through which it moves. The work is the same no matter how long or how short the ramp, as long as the vertical height is the same. If the work done by a force depends only on where it started and where it ended up, the force is said to be conservative. Unlike the gravitational force, the frictional force is nonconservative because the work done by it does depend on the path of the movement, that is, the length of the ramp. Potential energy is associated with the work done by a conservative force. For a mass (m) a height (h) above its lowest level the potential energy is U = mgh where g is the acceleration due to the gravitational force (9.80 m/s²).

MECHANICAL ENERGY

Kinetic energy (the energy of motion) and potential energy (the energy based on position) added together are called mechanical energy. Mechanical energy equals kinetic energy plus potential energy.

In an isolated system—one devoid of friction—the mechanical energy does not change. Although friction can never be totally eliminated, there are situations where it is small enough to be ignored. For example, when you hold a book in your outstretched hands, it has potential energy but no kinetic energy because its speed is zero. When dropped, the book acquires speed and kinetic energy. As its height above the floor decreases, its potential energy decreases. The gain in kinetic energy is balanced by a loss in potential energy, and the sum of kinetic energy and potential energy does not change. This is the idea of the conservation principle. Of course, this assumes that there is no friction (a nonconservative force), which is rarely the case in the real world. Frictional forces will always extract energy from a system and produce heat that ends up in the environment. The falling book will come to rest on the floor and have neither kinetic energy nor potential energy. All the energy it had before being dropped will have been converted to heat (and a very small amount of sound energy from its impact with the floor).

CONSERVATION OF ENERGY WITH A SIMPLE PENDULUM

A simple pendulum with ignorable friction illustrates the conservation of mechanical energy. Pulling the bob (the mass) from its lowest position and holding it, the pendulum has only potential energy; the mechanical energy is all potential. When released, the pendulum bob gains kinetic energy and loses potential energy, but at any instant the sum never differs from the sum at the beginning. At the lowest point of the movement the potential energy is zero and the mechanical energy is all kinetic. As the pendulum bob moves to higher levels, the potential energy increases, and the kinetic energy decreases. Throughout the motion, kinetic energy and potential energy change continually. But at any moment the sum, the mechanical energy, stays constant.

A simple pendulum isolated from nonconservative forces would oscillate forever. Complete isolation can never be achieved, and the pendulum will eventually stop because nonconservative forces such as air resistance and surface friction always remove mechanical energy from a system. Unless there is a mechanism for putting the energy back, the mechanical energy eventually drains and the motion stops. A child's swing is a pendulum of sorts. If you release a swing from some elevated position it will oscillate for a while but eventually will stop. You can push the swing regularly and keep it going, but in doing so you do work and put energy back into the system.

CONSERVATION OF ENERGY IN A SPRING-MASS SYSTEM

When a spring is stretched by pulling on one end, the spring pulls back on whatever is pulling it. Like the gravitational force, the spring force is conservative. Accordingly, there is potential energy associated with the spring that is given by U = kx², where k, the spring constant, reflects the strength of the spring and x is the amount of stretching from the relaxed position.

A horizontal spring with a mass attached to one end is a form of oscillator—that is, something that periodically returns to the starting position. To the extent that the spring-mass system can be isolated, the mechanical energy is conserved. When stretched but not yet in motion, the system has only potential energy. When released, the mass gains kinetic energy, the spring loses potential energy, but the sum does not change; it is conserved. Much like the oscillating pendulum, both kinetic energy and potential energy change continually, but the sum is constant. Most people do not think of atoms in a molecule as being connected by springs, yet the forces that bind them together behave like springs, and the atoms vibrate. The mechanical energy of the spring-like atomic system is an important energy attribute. A better understanding of the spring-like characteristics of atoms in materials has made possible many advances in sporting goods equipment, from graphite composite vaulting poles to titanium drivers for golfers.

Even though mechanical energy is rigorously conserved only when a system is isolated, the principle is elegant and useful. Water flowing over the top of a dam has both kinetic energy and potential energy. As it plummets toward the bottom of the dam, it loses potential energy and gains kinetic energy. Impinging on the blades of a paddle wheel, the water loses kinetic energy, which is transferred to rotational kinetic energy of the wheel. The principle provides an accounting procedure for energy.

THE FIRST LAW OF THERMODYNAMICS

The first law of thermodynamics is a statement of the principle of conservation of energy involving work, thermal energy, and heat. Engines converting heat to useful work are widely employed in our society, and the first law is vital for understanding their operation. The first law of thermodynamics accounts for joules of energy in somewhat the way a person accounts for money. A person uses a bank to receive and disperse money. If money comes into a bank and nothing is dispersed, the bank account increases. If money is dispersed and nothing comes in, the bank account decreases. If we call U the money in the bank, Q the money coming in or out, and W the money dispersed then the change in the account (ΔU) can be summarized as Algebraically, Q is positive for money coming in, negative for money going out. W is positive for money going out. If a person put $100 into the bank and took $200 out, the change in the account would be The account has decreased $100.

The energy content of a gas is called internal energy, symbol U. The gas can be put in contact with something at a higher temperature, and heat will flow in. If that something is at a lower temperature, heat will flow out. If the gas is contained in a cylinder containing a movable piston, then the gas can expand and push against the piston doing work. In principle, the work is equivalent to a person pushing and moving a box on a floor. Some agent can push on the piston doing work on the gas. If heat enters the gas and no work is done, the internal energy increases. If no heat enters or leaves the gas and if the gas expands doing work, the internal energy decreases. All of this is summarized in the equation if 100 J of energy (heat) entered the gas and the gas expanded and did 200 J of work, then The internal energy decreased 100 J.

The internal energy of all gases depends on the temperature of the gas. For an ideal gas, the internal energy depends only on the temperature. The temperature is most appropriately measured on the Kelvin scale. The contribution to the internal energy from the random kinetic energy of the molecules in the gas is called thermal energy.

In a monetary bank, money is added and withdrawn, but if in the end the bank is in exactly the same state, the net amount of money in the bank has not changed. Similarly, a gas may expand or be compressed and its temperature may undergo changes, but if at the end it is in exactly the same thermodynamic condition, the internal energy has not changed.

CONSERVATION OF ENERGY AND HEAT ENGINES

Heat engines work in cycles. During each cycle, heat is absorbed, work is done, and heat is rejected. At the beginning of the next cycle the gas is in exactly the same state as at the beginning of the previous cycle; the change in internal energy is zero. To illustrate, consider a conventional automobile engine. The cycle (Figure 2) starts with the piston moving down and pulling a mixture of gasoline vapor and air through the open intake valve and into the cylinder. The intake valve closes at the bottom of the downward motion, the piston moves up, and the gaseous mixture is compressed. At the top of the stroke the mixture is ignited; the gas expands doing work. At the end of the stroke the exhaust valve opens, the cooler gas is forced out the automobile's exhaust, and a new cycle begins.

Because the change in internal energy in a cycle is zero, the first law of thermodynamics requires Q=W. In words the net heat exchanged equals the work done in the cycle. The difference between the heat absorbed and the heat rejected has gone into useful work. The energy absorbed by a gas always takes place at a temperature higher than the temperature at which energy is exhausted. An automobile engine exhausts energy into the environment at a temperature of about 300 K, which is much lower than the temperature of the gasoline vapor-air mixture at the moment of ignition, about 1,000 K.

Engines are used to do work, and a quantitative measure of the performance of an engine is efficiency: For a given amount of energy absorbed at a high temperature, the more work obtained in a cycle, the more efficient the engine. In symbols, The first law of thermodynamics requires W = Q_high–Q_low so the efficiency may be written If all the heat absorbed were converted into work, the efficiency would be 1, or 100 percent. If none of the heat absorbed was converted into work, the efficiency would be 0. The first law of thermodynamics limits the efficiency of any heat engine to 1 but does not prevent an efficiency of 1. The efficiency of practical heat engines is always less than 1. For example, the efficiency of a large steam turbine in an electric power plant is about 0.5, which is considerably more efficient than the typical 0.35 efficiency of an auto engine.

When two objects at different temperatures are in contact, heat always flows from the hotter one to the cooler one. There is nothing in the first law of thermodynamics that prevents the opposite. The first law only requires that energy be conserved. A heat engine relies on heat flowing from some reservoir to a reservoir at a lower temperature. It is somewhat like a hydroelectric system relying on water flowing from a higher level to a lower level. The efficiency of a hydroelectric system increases as the difference in heights of the water levels increases. The efficiency of a heat engine increases as the difference in temperatures of the two reservoirs increases. In 1824, at age twenty-eight, the French engineer Sadi Carnot reasoned that the maximum efficiency of a heat engine depends only on the Kelvin temperatures of the two reservoirs. Formally, For a heat engine like a steam turbine in an electric power plant the low temperature is determined by the outdoor environment. This temperature is about 300 K. Engineering considerations limit the high temperature to about 800 K. The maximum efficiency according to Carnot is 0.63 or 63 percent. No matter how skilled the builders of a steam turbine, if the temperatures are 300 K and 800 K, the efficiency will never exceed 63 percent. When you realize that the efficiency can never be larger than about 63 percent, a realizable efficiency of 50 percent looks quite good.

An important message in this discussion is that all the thermal energy extracted from a reservoir is not available to do work. Some will always be lost, never to be recoverable. The principle of conservation of energy guarantees that thermal energy exhausted to the environment is not lost. But the principle does not say that the energy can be recovered. To recover the energy there must be a reservoir at a lower temperature for the heat to flow into. When everything in the environment comes to the same temperature, there is no reservoir at a lower temperature for heat to flow into.

CONSERVATION OF ENERGY AND REFRIGERATORS

Water never flows spontaneously from the bottom of a dam to the top. Water can be forced to flow to a higher level, but it requires a pump doing work. Similarly, heat never flows spontaneously from a lower temperature to a higher temperature. Heat can be forced to flow from a lower temperature to a higher temperature, but it requires work. A household refrigerator is a good example. The noise emanating from a refrigerator is due to an electric motor doing work, resulting in heat flowing from the cool interior to the warmer surroundings. In principle, a refrigerator is a heat engine running backward. The refrigerator operates in cycles and subscribes to the first law of thermodynamics. Work must be done on the working substance in order for heat (Q_low) to flow from the lower temperature. At the end of each cycle this heat as well as the work done (W) is rejected at the higher temperature. Conserving energy the first law requires

Work is invested to force heat to flow from the interior of the refrigerator. A measure of the performance, called coefficient of performance (COP), is If for every joule of work done, 2 joules of heat flowed out of the refrigerator then the performance would be 2. Using Q_high = Q_low + W, the performance equation can be written It is quite appropriate to think of a refrigerator as a heat pump. It pumps energy from one region and dumps it into another region at a higher temperature. A commercial heat pump does just this to warm a building during the heating season. There is a lot of energy outside of a building, even though the temperature may be 30°F (16.7°C) lower. This is because there are a lot of molecules in the outdoors, and each molecule contributes to the total energy. The performance of a heat pump decreases as the outside temperature decreases, but if the temperature remains above the freezing point of water (0°C or 32°F) commercial heat pumps can achieve performances between 2 and 4. If it were 2, this means that 2 joules of energy are deposited in the building for every joule of work. This is a significant gain. The temperature below the surface of the ground is several degrees higher than at the surface. Heat pumps drawing energy from the subsurface tend to be more efficient than an above-ground heat pump. This is because it is easier to extract heat from the solid sub-surface than it is from air, and the subsurface temperature is higher.

CONSERVATION OF ENERGY IN CHEMICAL REACTIONS

Conservation of energy is mandatory in chemical and nuclear reactions. To illustrate, consider the combustion of gasoline. One type of gasoline, isooctane, has the molecular form C₈H₁₈. Thermal energy is released when isooctane combines with oxygen according to From an energy standpoint the arrow is an equals sign. The total energy on the left side of the equation must equal the total energy on the right side. The sixteen carbon dioxide (CO₂) molecules and eighteen water (H₂O) molecules have lower total energy than the two isooctane (C₈H₁₈) molecules and twenty-five oxygen (O₂) molecules from which they were formed. The difference in energy is liberated as heat. It is possible to have chemical reactions in which the total energy of the molecules on the left side of the equation is less than the total energy of the molecules on the right. But for the reaction to proceed, energy must be added on the left side of the equation.

CONSERVATION OF ENERGY IN NUCLEAR REACTIONS

No name is more universally known in science than that of Albert Einstein, and no equation of physics is more recognizable than E = mc². In this equation E, m, and c stand for energy, mass, and the speed of light (300,000,000 m/s). Taken literally, the equation suggests that anything having mass has energy. It does not mean the energy of a mass moving with the speed of light. The energy, mc², is intrinsic to any mass whether or not it is moving. Because the speed of light is such a large number, the energy of anything is huge. For example, the mass-energy of a penny having a mass of 0.003 kg (3 grams) is 270 trillion joules. This is roughly 10,000 times the energy liberated from burning a ton of coal. As strange as it may seem, mass and energy, are equivalent and it is demonstrated routinely in nuclear reactions that liberate energy in a nuclear power plant.

The nucleus of an atom consists of protons and neutrons that are bound together by a nuclear force. Neutrons and protons are rearranged in a nuclear reaction in a manner somewhat akin to rearranging atoms in a chemical reaction. The nuclear reaction liberating energy in a nuclear power plant is called nuclear fission. The word "fission" is derived from "fissure," which means a crack or a separation. A nucleus is separated (fissioned) into two major parts by bombardment with a neutron.

Uranium in the fuel of a nuclear power plant is designated²³⁵U. The 92 protons and 143 neutrons in a 235U nucleus sum to 235, the number in the 235U notation. Through interaction with a neutron the 92 protons and 144 neutrons involved are rearranged into other nuclei. Typically, this rearrangement is depicted as A barium (Ba) nucleus has 56 protons and 87 neutrons; a krypton (Kr) nucleus has 36 protons and 54 neutrons. The 92 protons and 144 neutrons being rearranged are accounted for after the rearrangement. But the mass of a 235U nucleus, for example, differs from the sum of the masses of 92 free protons and 143 neutrons. When you account for the actual masses involved in the reaction, the total mass on the left side of the arrow is less than the total mass on the right. The energy released on the right side of the equation is about 32 trillionths of a joule. The energy equivalent of the mass difference using E = mc² accounts precisely for the energy released. In the reaction described by the equation above, one-thousandth of the mass of the 235U has been converted into energy. It is energy from reactions like this that ultimately is converted into electric energy in a nuclear power plant. The illustration of a nuclear fission reaction using an arrow becomes an equality for energy when the equivalence of mass and energy is taken into account.

The energy liberated in nuclear reactions is of such magnitude that mass differences are relatively easy to detect. The same mass-energy considerations pertain to chemical reactions. However, the energies involved are millions of times smaller, and mass differences are virtually impossible to detect. Discussions of energies involved in chemical reactions do not include mass energy. Nevertheless, there is every reason to believe that mass energy is involved.

A MODEL FOR CONSERVATION OF ENERGY

Big or small, simple or complex, energy converters must all subscribe to the principle of conservation of energy. Each one converts energy into some form regarded as useful, and each one diverts energy that is not immediately useful and may never be useful. Because energy is diverted, the efficiency defined as can never be 100 percent. Generally, several energy conversions are involved in producing the desired form. The food we convert to energy in our bodies involves several energy conversions prior to the one a person performs. Energy conversions are involved in our sun to produce light. Photosynthesis producing carbohydrates for the food entails energy conversions. Even the carbon dioxide and water require energy conversions for their formation. Tracking energy conversions is facilitated with the descriptive model shown in Figure 3.

An energy analysis of the production of electric energy in a coal-burning power plant provides an opportunity to illustrate the model. Generating electricity requires burning coal for heat to vaporize water to steam, a turbine driven by steam to drive an electric generator, and an electric generator to produce electric energy.

The diagram in Figure 4, with typical efficiencies for the three converters, describes the fate of 1 J of energy extracted from burning coal. It is important to note that only 0.39 J of electric energy was derived from the 1 J. More energy in the form of heat (0.61 J) was rejected to the environment than was delivered as electric energy (0.39 J). Thinking of the three converters as one unit converting the 1 J into 0.39 J of electric energy the efficiency of the converter is 0.39. This is just the product of the efficiencies of the three converters: from burning coal into electric energy.

The overall efficiency is smaller than the lowest efficiency in the chain. No matter how efficient all converters in a chain are, the efficiency will always be smaller than the lowest efficiency. As long as the steam turbine is used in the commercial production of electricity, the overall efficiency will be relatively low. Electric power plants using nuclear reactors also use steam turbines, and their efficiency is essentially the same as for a coal-burning plant. Energy in the form of heat is lost in the transmission of electricity to consumers, which reduces the overall efficiency to about 0.33. This means that every joule of electric energy paid for by a consumer requires 3 joules of energy at the input of the chain of energy converters. It also means that every joule of energy saved by turning off lights when not needed saves 3 joules at the input.

The scientific principle of conservation of energy is imbedded in the model for evaluating the performance of a chain of energy converters. Applying the model, we find that the overall efficiency of a chain of energy converters is always less than the smallest efficiency in the chain. Realizing this, strong arguments can be made for conserving energy in the sense of saving. If the overall efficiency of conversion is 10 percent, as it is for an incandescent lightbulb, then saving one unit of energy at the end of the chain saves ten units of energy at the beginning of the chain.

Joseph Priest

See also: Carnot, Nicolas Leonard Sadi; Climatic Effects; Engines; Matter and Energy; Nuclear Energy; Nuclear Fission; Refrigerators and Freezers; Thermal Energy.

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