Laws of Conservation

Contents

Laws of Conservation
The Law of Conservation of Matter
Law of Conservation of Energy
Law of Conservation of Mass-Energy – Mass-Energy Equivalence
Mass defect directly from E=mc2

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. These properties are sometimes called “constants of the motion“. These quantities are said to be “conserved” and the conservation laws which result can be considered to be the most fundamental principles of mechanics. In mechanics, examples of conserved quantities are energy, momentum, and angular momentum. The laws of conservation are exact for an isolated system.

Laws of Conservation in Nuclear Reactions

A nuclear reaction is considered to be the process in which two nuclear particles (two nuclei or a nucleus and a nucleon) interact to produce two or more nuclear particles or ˠ-rays (gamma rays). Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. Sometimes if a nucleus interacts with another nucleus or particle without changing the nature of any nuclide, the process is referred to a nuclear scattering, rather than a nuclear reaction.

In analyzing nuclear reactions, we apply the many conservation laws. Nuclear reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy (including rest energies). Additional conservation laws, not anticipated by classical physics, are are electric charge, lepton number and baryon number. Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.

Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.

For purposes of this article it is sufficient to note four of the fundamental laws governing these reactions.

Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.

The Law of Conservation of Matter

The law of conservation of matter or principle of matter conservation states that the mass of an object or collection of objects never changes over time, no matter how the constituent parts rearrange themselves.

The mass can neither be created nor destroyed.

The law requires that during any nuclear reaction, radioactive decay or chemical reaction in an isolated system, the total mass of the reactants or starting materials must be equal to the mass of the products.

The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. In chemistry the law of conservation of matter may be explained in the following way (see the picture of combustion of methane). The masses of a methane and oxygen together must be equal to the masses of carbon dioxide and water. In other words, during a chemical reaction, everything you start with, you must end up with, but it might look different.

Historically, already the ancient Greeks proposed the idea that the total amount of matter in the universe is constant. The principle of conservation of mass was first outlined by Mikhail Lomonosov in 1748. However, the law of conservation of matter (or the principle of mass/matter conservation) as a fundamental principle of physics was discovered in by Antoine Lavoisier in the late 18th century. It was of great importance in progressing from alchemy to modern chemistry. Before this discovery, there were questions like:

Why a piece of wood weighs less after burning?
Can a matter or some of its part disappear?

In the case of burned wood the problem was the measurement of the weight of released gases. Measurements of the weight of released gases was complicated, because of the buoyancy effect of the Earth’s atmosphere on the weight of gases. Once understood, the conservation of matter was of crucial importance in the progress from alchemy to the modern natural science of chemistry.

Law of Conservation of Energy

The law of conservation of energy is one of the basic laws of physics along with the conservation of mass and the conservation of momentum. The law of conservation of energy states that energy can change from one form into another, but it cannot be created or destroyed. Or the general definition is:

The total energy of an isolated system remains constant over time.

law of conservation of energy - pendulum — Newton’s cradle. A device that demonstrates the Law of Conservation of Mechanical Energy and Momentum.

Energy can be defined as the capacity for doing work. It may exist in a variety of forms and may be transformed from one type of energy to another in hundreds of ways.

For example, burning gasoline to power cars is an energy conversion process we rely on. The chemical energy in gasoline is converted to thermal energy, which is then converted to mechanical energy that makes the car move. The mechanical energy has been converted to kinetic energy. When we use the brakes to stop a car, that kinetic energy is converted by friction back to heat, or thermal energy.

A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind, which produces work without the input of energy, cannot exist.

The concept of energy conservation is widely used in many fields. In this article the following fields are discussed:

Law of Conservation of Mass-Energy – Mass-Energy Equivalence

At the beginning of the 20th century, the notion of mass underwent a radical revision. Mass lost its absoluteness. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc². In words, energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen to be equivalent to energy, to be interconvertible with energy, and to increase significantly at exceedingly high speeds near that of light. The total energy of an object was understood to comprise its rest mass as well as its increase of mass caused by increase in kinetic energy.

In special theory of relativity certain types of matter may be created or destroyed, but in all of these processes, the mass and energy associated with such matter remains unchanged in quantity. It was found the rest mass an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E = mc²) this binding energy is proportional to this mass difference and it is known as the mass defect.

Example: Mass defect of a 63Cu

Calculate the mass defect of a ⁶³Cu nucleus if the actual mass of ⁶³Cu in its nuclear ground state is 62.91367 u.

⁶³Cu nucleus has 29 protons and also has (63 – 29) 34 neutrons.

The mass of a proton is 1.00728 u and a neutron is 1.00867 u.

The combined mass is: 29 protons x (1.00728 u/proton) + 34 neutrons x (1.00867 u/neutron) = 63.50590 u

The mass defect is Δm = 63.50590 u – 62.91367 u = 0.59223 u

Convert the mass defect into energy (nuclear binding energy).

(0.59223 u/nucleus) x (1.6606 x 10^-27 kg/u) = 9.8346 x 10^-28 kg/nucleus

ΔE = Δmc²

ΔE = (9.8346 x 10^-28 kg/nucleus) x (2.9979 x 10⁸ m/s)² = 8.8387 x 10^-11 J/nucleus

The energy calculated in the previous example is the nuclear binding energy. However, the nuclear binding energy may be expressed as kJ/mol (for better understanding).

Calculate the nuclear binding energy of 1 mole of ⁶³Cu:

(8.8387 x 10^-11 J/nucleus) x (1 kJ/1000 J) x (6.022 x 10²³ nuclei/mol) = 5.3227 x 10¹⁰ kJ/mol of nuclei.

One mole of ⁶³Cu (~63 grams) is bound by the nuclear binding energy (5.3227 x 10¹⁰ kJ/mol) which is equivalent to:

14.8 million kilowatt-hours (≈ 15 GW·h)
336,100 US gallons of automotive gasoline

Example: Mass defect of the reactor core

Calculate the mass defect of the 3000MW_th reactor core after one year of operation.

It is known the average recoverable energy per fission is about 200 MeV, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The reaction rate per entire 3000MW_th reactor core is about 9.33×10¹⁹ fissions / second.

The overall energy release in the units of joules is:

200×10⁶ (eV) x 1.602×10^-19 (J/eV) x 9.33×10¹⁹ (s^-1) x 31.5×10⁶ (seconds in year) = 9.4×10¹⁶ J/year

The mass defect is calculated as:

Δm = ΔE/c²

Δm = 9.4×10¹⁶ / (2.9979 x 10⁸)² = 1.046 kg

That means in a typical 3000MWth reactor core about 1 kilogram of matter is converted into pure energy.

Note that, a typical annual uranium load for a 3000MWth reactor core is about 20 tonnes of enriched uranium (i.e. about 22.7 tonnes of UO₂). Entire reactor core may contain about 80 tonnes of enriched uranium.

Mass defect directly from E=mc²

The mass defect can be calculated directly from the Einstein relationship (E = mc²) as:

Δm = ΔE/c²

Δm = 3000×10⁶ (W = J/s) x 31.5×10⁶ (seconds in year) / (2.9979 x 10⁸)²= 1.051 kg

Nuclear binding energy curve.
Source: hyperphysics.phy-astr.gsu.edu

During the nuclear splitting or nuclear fusion, some of the mass of the nucleus gets converted into huge amounts of energy and thus this mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. The nuclear binding energies are enormous, they are of the order of a million times greater than the electron binding energies of atoms.

Generally, in both chemical and nuclear reactions, some conversion between rest mass and energy occurs, so that the products generally have smaller or greater mass than the reactants. Therefore the new conservation principle is the conservation of mass-energy.