Sodium chloride

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Sodium_chloride

Sodium chloride

Crystal structure with sodium in purple and chloride in green
Names
IUPAC name Sodium chloride
Other names common salt, regular salt halite, rock salt table salt, sea salt saline

Properties
Chemical formula	NaCl
Molar mass	58.443 g/mol
Appearance	Colorless cubic crystals
Odor	Odorless
Density	2.17 g/cm3
Melting point	800.7 °C (1,473.3 °F; 1,073.8 K)
Boiling point	1,465 °C (2,669 °F; 1,738 K)
Solubility in water	360 g/L (25°C)
Solubility in ammonia	21.5 g/L
Solubility in methanol	14.9 g/L
Magnetic susceptibility (χ)	−30.2·10−6 cm3/mol
Refractive index (nD)	1.5441 (at 589 nm)
Structure
Crystal structure	Face-centered cubic (see text), cF8
Space group	Fm3m (No. 225)
Lattice constant	a = 564.02 pm
Formula units (Z)	4
Coordination geometry	octahedral at Na+ octahedral at Cl−
Thermochemistry
Heat capacity (C)	50.5 J/(K·mol)
Std molar entropy (S⦵298)	72.10 J/(K·mol)
Std enthalpy of formation (ΔfH⦵298)	−411.120 kJ/mol
Hazards
Lethal dose or concentration (LD, LC):
LD50 (median dose)	3 g/kg (oral, rats)

Sodium chloride /ˌsoʊdiəm ˈklɔːraɪd/, commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. Sodium chloride is the salt most responsible for the salinity of seawater and of the extracellular fluid of many multicellular organisms. In its edible form, salt (also known as table salt) is commonly used as a condiment and food preservative. Large quantities of sodium chloride are used in many industrial processes, and it is a major source of sodium and chlorine compounds used as feedstocks for further chemical syntheses. Another major application of sodium chloride is deicing of roadways in sub-freezing weather.

Uses

In addition to the familiar domestic uses of salt, more dominant applications of the approximately 250 million tonnes per year production (2008 data) include chemicals and de-icing.

Chemical functions

Salt is used, directly or indirectly, in the production of many chemicals, which consume most of the world's production.

Chlor-alkali industry

See also: Chloralkali process

It is the starting point for the chloralkali process, the industrial process to produce chlorine and sodium hydroxide, according to the chemical equation

${\displaystyle 2\text{NaCl}+2{\text{H}}_2^{}\text{O}\stackrel{electrolysis}{\to }{\text{Cl}}_2^{}+{\text{H}}_2^{}+2\text{NaOH}}$

This electrolysis is conducted in either a mercury cell, a diaphragm cell, or a membrane cell. Each of those uses a different method to separate the chlorine from the sodium hydroxide. Other technologies are under development due to the high energy consumption of the electrolysis, whereby small improvements in the efficiency can have large economic paybacks. Some applications of chlorine include PVC thermoplastics production, disinfectants, and solvents.

Sodium hydroxide is extensively used in many different industries enabling production of paper, soap, and aluminium etc.

Soda-ash industry

Sodium chloride is used in the Solvay process to produce sodium carbonate and calcium chloride. Sodium carbonate, in turn, is used to produce glass, sodium bicarbonate, and dyes, as well as a myriad of other chemicals. In the Mannheim process, sodium chloride is used for the production of sodium sulfate and hydrochloric acid.

Standard

Sodium chloride has an international standard that is created by ASTM International. The standard is named ASTM E534-13 and is the standard test methods for chemical analysis of sodium chloride. These methods listed provide procedures for analyzing sodium chloride to determine whether it is suitable for its intended use and application.

Miscellaneous industrial uses

Sodium chloride is heavily used, so even relatively minor applications can consume massive quantities. In oil and gas exploration, salt is an important component of drilling fluids in well drilling. It is used to flocculate and increase the density of the drilling fluid to overcome high downwell gas pressures. Whenever a drill hits a salt formation, salt is added to the drilling fluid to saturate the solution in order to minimize the dissolution within the salt stratum. Salt is also used to increase the curing of concrete in cemented casings.

In textiles and dyeing, salt is used as a brine rinse to separate organic contaminants, to promote "salting out" of dyestuff precipitates, and to blend with concentrated dyes to standardize them. One of its main roles is to provide the positive ion charge to promote the absorption of negatively charged ions of dyes.

It is also used in processing aluminium, beryllium, copper, steel, and vanadium. In the pulp and paper industry, salt is used to bleach wood pulp. It also is used to make sodium chlorate, which is added along with sulfuric acid and water to manufacture chlorine dioxide, an excellent oxygen-based bleaching chemical. The chlorine dioxide process, which originated in Germany after World War I, is becoming more popular because of environmental pressures to reduce or eliminate chlorinated bleaching compounds. In tanning and leather treatment, salt is added to animal hides to inhibit microbial activity on the underside of the hides and to attract moisture back into the hides.

In rubber manufacture, salt is used to make buna, neoprene, and white rubber types. Salt brine and sulfuric acid are used to coagulate an emulsified latex made from chlorinated butadiene.

Salt also is added to secure the soil and to provide firmness to the foundation on which highways are built. The salt acts to minimize the effects of shifting caused in the subsurface by changes in humidity and traffic load.

Sodium chloride is sometimes used as a cheap and safe desiccant because of its hygroscopic properties, making salting an effective method of food preservation historically; the salt draws water out of bacteria through osmotic pressure, keeping it from reproducing, a major source of food spoilage. Even though more effective desiccants are available, few are safe for humans to ingest.

Water softening

Hard water contains calcium and magnesium ions that interfere with action of soap and contribute to the buildup of a scale or film of alkaline mineral deposits in household and industrial equipment and pipes. Commercial and residential water-softening units use ion-exchange resins to remove ions that cause the hardness. These resins are generated and regenerated using sodium chloride.

Road salt

The second major application of salt is for deicing and anti-icing of roads, both in grit bins and spread by winter service vehicles. In anticipation of snowfall, roads are optimally "anti-iced" with brine (concentrated solution of salt in water), which prevents bonding between the snow-ice and the road surface. This procedure obviates the heavy use of salt after the snowfall. For de-icing, mixtures of brine and salt are used, sometimes with additional agents such as calcium chloride and/or magnesium chloride. The use of salt or brine becomes ineffective below −10 °C (14 °F).

Mounds of road salt for use in winter

Salt for de-icing in the United Kingdom predominantly comes from a single mine in Winsford in Cheshire. Prior to distribution it is mixed with <100 ppm of sodium ferrocyanide as an anticaking agent, which enables rock salt to flow freely out of the gritting vehicles despite being stockpiled prior to use. In recent years this additive has also been used in table salt. Other additives had been used in road salt to reduce the total costs. For example, in the US, a byproduct carbohydrate solution from sugar-beet processing was mixed with rock salt and adhered to road surfaces about 40% better than loose rock salt alone. Because it stayed on the road longer, the treatment did not have to be repeated several times, saving time and money.

In the technical terms of physical chemistry, the minimum freezing point of a water-salt mixture is −21.12 °C (−6.02 °F) for 23.31 wt% of salt. Freezing near this concentration is however so slow that the eutectic point of −22.4 °C (−8.3 °F) can be reached with about 25 wt% of salt.

Environmental effects

Road salt ends up in fresh-water bodies and could harm aquatic plants and animals by disrupting their osmoregulation ability. The omnipresence of salt in coastal areas poses a problem in any coating application, because trapped salts cause great problems in adhesion. Naval authorities and ship builders monitor the salt concentrations on surfaces during construction. Maximal salt concentrations on surfaces are dependent on the authority and application. The IMO regulation is mostly used and sets salt levels to a maximum of 50 mg/m² soluble salts measured as sodium chloride. These measurements are done by means of a Bresle test. Salinization (increasing salinity, aka freshwater salinization syndrome) and subsequent increased metal leaching is an ongoing problem throughout North America and European fresh waterways.

In highway de-icing, salt has been associated with corrosion of bridge decks, motor vehicles, reinforcement bar and wire, and unprotected steel structures used in road construction. Surface runoff, vehicle spraying, and windblown salt also affect soil, roadside vegetation, and local surface water and groundwater supplies. Although evidence of environmental loading of salt has been found during peak usage, the spring rains and thaws usually dilute the concentrations of sodium in the area where salt was applied. A 2009 study found that approximately 70% of the road salt being applied in the Minneapolis-St Paul metro area is retained in the local watershed.

Substitution

Some agencies are substituting beer, molasses, and beet juice instead of road salt. Airlines utilize more glycol and sugar rather than salt-based solutions for deicing.

Food industry and agriculture

Main article: Salt

Many microorganisms cannot live in a salty environment: water is drawn out of their cells by osmosis. For this reason salt is used to preserve some foods, such as bacon, fish, or cabbage.

Salt is added to food, either by the food producer or by the consumer, as a flavor enhancer, preservative, binder, fermentation-control additive, texture-control agent, and color developer. The salt consumption in the food industry is subdivided, in descending order of consumption, into other food processing, meat packers, canning, baking, dairy, and grain mill products. Salt is added to promote color development in bacon, ham and other processed meat products. As a preservative, salt inhibits the growth of bacteria. Salt acts as a binder in sausages to form a binding gel made up of meat, fat, and moisture. Salt also acts as a flavor enhancer and as a tenderizer.

In many dairy industries, salt is added to cheese as a color-, fermentation-, and texture-control agent. The dairy subsector includes companies that manufacture creamery butter, condensed and evaporated milk, frozen desserts, ice cream, natural and processed cheese, and specialty dairy products. In canning, salt is primarily added as a flavor enhancer and preservative. It also is used as a carrier for other ingredients, dehydrating agent, enzyme inhibitor and tenderizer. In baking, salt is added to control the rate of fermentation in bread dough. It also is used to strengthen the gluten (the elastic protein-water complex in certain doughs) and as a flavor enhancer, such as a topping on baked goods. The food-processing category also contains grain mill products. These products consist of milling flour and rice and manufacturing cereal breakfast food and blended or prepared flour. Salt is also used a seasoning agent, e.g. in potato chips, pretzels, and cat and dog food.

Sodium chloride is used in veterinary medicine as emesis-causing agent. It is given as warm saturated solution. Emesis can also be caused by pharyngeal placement of small amount of plain salt or salt crystals.

Medicine

Main article: Saline (medicine)

Sodium chloride is used together with water as one of the primary solutions for intravenous therapy. Nasal spray often contains a saline solution.

Firefighting

A class-D fire extinguisher for various metals

Sodium chloride is the principal extinguishing agent in fire extinguishers (Met-L-X, Super D) used on combustible metal fires such as magnesium, potassium, sodium, and NaK alloys (Class D). Thermoplastic powder is added to the mixture, along with waterproofing (metal stearates) and anticaking agents (tricalcium phosphate) to form the extinguishing agent. When it is applied to the fire, the salt acts like a heat sink, dissipating heat from the fire, and also forms an oxygen-excluding crust to smother the fire. The plastic additive melts and helps the crust maintain its integrity until the burning metal cools below its ignition temperature. This type of extinguisher was invented in the late 1940s as a cartridge-operated unit, although stored pressure versions are now popular. Common sizes are 30 pounds (14 kg) portable and 350 pounds (160 kg) wheeled.

Cleanser

Since at least medieval times, people have used salt as a cleansing agent rubbed on household surfaces. It is also used in many brands of shampoo, toothpaste, and popularly to de-ice driveways and patches of ice.

Optical usage

Defect-free NaCl crystals have an optical transmittance of about 90% for infrared light, specifically between 200 nm and 20 µm. They were therefore used in optical components (windows and prisms) operating in that spectral range, where few non-absorbing alternatives exist and where requirements for absence of microscopic inhomogeneities are less strict than in the visible range. While inexpensive, NaCl crystals are soft and hygroscopic – when exposed to the ambient air, they gradually cover with "frost". This limits application of NaCl to dry environments, vacuum-sealed assembly areas or for short-term uses such as prototyping. Nowadays materials like zinc selenide (ZnSe), which are stronger mechanically and are less sensitive to moisture, are used instead of NaCl for the infrared spectral range.

Chemistry

Sodium chloride crystal under microscope.

 

NaCl octahedra. The yellow stipples represent the electrostatic force between the ions of opposite charge

Solid sodium chloride

See also: Cubic crystal system

In solid sodium chloride, each ion is surrounded by six ions of the opposite charge as expected on electrostatic grounds. The surrounding ions are located at the vertices of a regular octahedron. In the language of close-packing, the larger chloride ions (167 pm in size) are arranged in a cubic array whereas the smaller sodium ions (116 pm) fill all the cubic gaps (octahedral voids) between them. This same basic structure is found in many other compounds and is commonly known as the NaCl structure or rock salt crystal structure. It can be represented as a face-centered cubic (fcc) lattice with a two-atom basis or as two interpenetrating face centered cubic lattices. The first atom is located at each lattice point, and the second atom is located halfway between lattice points along the fcc unit cell edge.

Solid sodium chloride has a melting point of 801 °C and liquid sodium chloride boils at 1465 °C. Atomic-resolution real-time video imaging allows visualization of the initial stage of crystal nucleation of sodium chloride.

The Thermal conductivity of sodium chloride as a function of temperature has a maximum of 2.03 W/(cm K) at 8 K (−265.15 °C; −445.27 °F) and decreases to 0.069 at 314 K (41 °C; 106 °F). It also decreases with doping.

View of one slab of hydrohalite, NaCl·2H₂O. (red = O, white = H, green = Cl, purple = Na).

From cold (sub-freezing) solutions, salt crystallises with water of hydration as hydrohalite (the dihydrate NaCl·2H₂O NaCl·2H₂O).

In 2023, it was discovered that under pressure, sodium chloride can form the hydrates NaCl·8.5H₂O and NaCl·13H₂O.

Aqueous solutions

Phase diagram of water–NaCl mixture

The attraction between the Na⁺ and Cl⁻ ions in the solid is so strong that only highly polar solvents like water dissolve NaCl well.

When dissolved in water, the sodium chloride framework disintegrates as the Na⁺ and Cl⁻ ions become surrounded by polar water molecules. These solutions consist of metal aquo complex with the formula [Na(H₂O)₈]⁺, with the Na–O distance of 250 pm. The chloride ions are also strongly solvated, each being surrounded by an average of six molecules of water. Solutions of sodium chloride have very different properties from pure water. The eutectic point is −21.12 °C (−6.02 °F) for 23.31% mass fraction of salt, and the boiling point of saturated salt solution is near 108.7 °C (227.7 °F).^[11]

pH of sodium chloride solutions

The pH of a sodium chloride solution remains ≈7 due to the extremely weak basicity of the Cl⁻ ion, which is the conjugate base of the strong acid HCl. In other words, NaCl has no effect on system pH in diluted solutions where the effects of ionic strength and activity coefficients are negligible.

Solubility of NaCl (g NaCl / 1 kg of solvent at 25 °C (77 °F))
Water	360
Formamide	94
Glycerin	83
Propylene glycol	71
Formic acid	52
Liquid ammonia	30.2
Methanol	14
Ethanol	0.65
Dimethylformamide	0.4
Propan-1-ol	0.124
Sulfolane	0.05
Butan-1-ol	0.05
Propan-2-ol	0.03
Pentan-1-ol	0.018
Acetonitrile	0.003
Acetone	0.00042

Stoichiometric and structure variants

Common salt has a 1:1 molar ratio of sodium and chlorine. In 2013, compounds of sodium and chloride of different stoichiometries have been discovered; five new compounds were predicted (e.g., Na₃Cl, Na₂Cl, Na₃Cl₂, NaCl₃, and NaCl₇). The existence of some of them has been experimentally confirmed at high pressures and other conditions: cubic and orthorhombic NaCl₃, two-dimensional metallic tetragonal Na₃Cl and exotic hexagonal NaCl. This indicates that compounds violating chemical intuition are possible, in simple systems under non-ambient conditions.

Occurrence

Most of the world's salt is dissolved in the ocean. A lesser amount is found in the Earth's crust as the water-soluble mineral halite (rock salt), and a tiny amount exists as suspended sea salt particles in the atmosphere. These particles are the dominant cloud condensation nuclei far out at sea, which allow the formation of clouds in otherwise non-polluted air.

Production

Salt is currently mass-produced by evaporation of seawater or brine from brine wells and salt lakes. Mining of rock salt is also a major source. China is the world's main supplier of salt. In 2017, world production was estimated at 280 million tonnes, the top five producers (in million tonnes) being China (68.0), United States (43.0), India (26.0), Germany (13.0), and Canada (13.0). Salt is also a byproduct of potassium mining.

• 

  Modern rock salt mine near Mount Morris, New York, United States

• 

  Jordanian and Israeli salt evaporation ponds at the south end of the Dead Sea.

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  Mounds of salt, Salar de Uyuni, Bolivia.

Two-body problem

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Two-body_problem

 

Top: Two bodies with similar mass orbiting a common barycenter external to both bodies, with elliptic orbits—typical of binary stars . Bottom: Two bodies with a "slight" difference in mass orbiting a common barycenter. The sizes, and this type of orbit are similar to the Pluto–Charon system (in which the barycenter is external to both bodies), and to the Earth–Moon system—where the barycenter is internal to the larger body.

In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.

The most prominent case of the classical two-body problem is the gravitational case (see also Kepler problem), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as satellites, planets, and stars. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.

A simpler "one body" model, the "central-force problem", treats one object as the immobile source of a force acting on the other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary).

However, the one-body approximation is usually unnecessary except as a stepping stone. For many forces, including gravitational ones, the general version of the two-body problem can be reduced to a pair of one-body problems, allowing it to be solved completely, and giving a solution simple enough to be used effectively.

By contrast, the three-body problem (and, more generally, the n-body problem for n ≥ 3) cannot be solved in terms of first integrals, except in special cases.

Results for prominent cases

Gravitation and other inverse-square examples

The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside influences will be small enough to be ignored safely).

Under the force of gravity, each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar conic sections. If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass. The mutual center of mass may even be inside the larger object.

For the derivation of the solutions to the problem, see Classical central-force problem or Kepler problem.

In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive scalar force field obeying an inverse-square law, with electrostatic attraction being the obvious physical example. In practice, such problems rarely arise. Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which are isolated enough from their surroundings.

The dynamical system of a two-body system under the influence of torque turns out to be a Sturm-Liouville equation.

Inapplicability to atoms and subatomic particles

Although the two-body model treats the objects as point particles, classical mechanics only apply to systems of macroscopic scale. Most behavior of subatomic particles cannot be predicted under the classical assumptions underlying this article or using the mathematics here.

Electrons in an atom are sometimes described as "orbiting" its nucleus, following an early conjecture of Niels Bohr (this is the source of the term "orbital"). However, electrons don't actually orbit nuclei in any meaningful sense, and quantum mechanics are necessary for any useful understanding of the electron's real behavior. Solving the classical two-body problem for an electron orbiting an atomic nucleus is misleading and does not produce many useful insights.

Reduction to two independent, one-body problems

The complete two-body problem can be solved by re-formulating it as two one-body problems: a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved.

Jacobi coordinates for two-body problem; Jacobi coordinates are ${\displaystyle \mathit{R}=\frac{m_1}{M}{\mathit{x}}_1+\frac{m_2}{M}{\mathit{x}}_2}$ and ${\displaystyle \mathit{r}={\mathit{x}}_1-{\mathit{x}}_2}$ with ${\displaystyle M=m_1+m_2}$.

Let x₁ and x₂ be the vector positions of the two bodies, and m₁ and m₂ be their masses. The goal is to determine the trajectories x₁(t) and x₂(t) for all times t, given the initial positions x₁(t = 0) and x₂(t = 0) and the initial velocities v₁(t = 0) and v₂(t = 0).

When applied to the two masses, Newton's second law states that

$${\displaystyle {\mathbf{F}}_{12}({\mathbf{x}}_1,{\mathbf{x}}_2)=m_1{\ddot{\mathbf{x}}}_1}$$

(Equation 1)

$${\displaystyle {\mathbf{F}}_{21}({\mathbf{x}}_1,{\mathbf{x}}_2)=m_2{\ddot{\mathbf{x}}}_2}$$

(Equation 2)

where F₁₂ is the force on mass 1 due to its interactions with mass 2, and F₂₁ is the force on mass 2 due to its interactions with mass 1. The two dots on top of the x position vectors denote their second derivative with respect to time, or their acceleration vectors.

Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x₁ − x₂ between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x₁(t) and x₂(t).

Center of mass motion (1st one-body problem)

Let ${\displaystyle \mathbf{R}}$ be the position of the center of mass (barycenter) of the system. Addition of the force equations (1) and (2) yields

$${\displaystyle m_1{\ddot{\mathbf{x}}}_1+m_2{\ddot{\mathbf{x}}}_2=(m_1+m_2)\ddot{\mathbf{R}}={\mathbf{F}}_{12}+{\mathbf{F}}_{21}=0}$$

where we have used Newton's third law F₁₂ = −F₂₁ and where

$${\displaystyle \ddot{\mathbf{R}}\equiv \frac{m_1{\ddot{\mathbf{x}}}_1+m_2{\ddot{\mathbf{x}}}_2}{m_1+m_2}.}$$

The resulting equation:

$${\displaystyle \ddot{\mathbf{R}}=0}$$

shows that the velocity ${\displaystyle \mathbf{v}=\frac{dR}{dt}}$ of the center of mass is constant, from which follows that the total momentum m₁ v₁ + m₂ v₂ is also constant (conservation of momentum). Hence, the position R(t) of the center of mass can be determined at all times from the initial positions and velocities.

Displacement vector motion (2nd one-body problem)

Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation

$${\displaystyle \ddot{\mathbf{r}}={\ddot{\mathbf{x}}}_1-{\ddot{\mathbf{x}}}_2=\left(\frac{{\mathbf{F}}_{12}}{m_1}-\frac{{\mathbf{F}}_{21}}{m_2}\right)=\left(\frac{1}{m_1}+\frac{1}{m_2}\right){\mathbf{F}}_{12}}$$

where we have again used Newton's third law F₁₂ = −F₂₁ and where r is the displacement vector from mass 2 to mass 1, as defined above.

The force between the two objects, which originates in the two objects, should only be a function of their separation r and not of their absolute positions x₁ and x₂; otherwise, there would not be translational symmetry, and the laws of physics would have to change from place to place. The subtracted equation can therefore be written:

$${\displaystyle \mu \ddot{\mathbf{r}}={\mathbf{F}}_{12}({\mathbf{x}}_1,{\mathbf{x}}_2)=\mathbf{F}(\mathbf{r})}$$

where ${\displaystyle \mu }$ is the reduced mass

$${\displaystyle \mu =\frac{1}{\frac{1}{m_1}+\frac{1}{m_2}}=\frac{m_1{m}_2}{m_1+m_2}.}$$

Solving the equation for r(t) is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by ${\displaystyle \mathbf{F}(\mathbf{r})}$. For the case where ${\displaystyle \mathbf{F}(\mathbf{r})}$ follows an inverse-square law, see the Kepler problem.

Once R(t) and r(t) have been determined, the original trajectories may be obtained

$${\displaystyle {\mathbf{x}}_1(t)=\mathbf{R}(t)+\frac{m_2}{m_1+m_2}\mathbf{r}(t)}$$

$${\displaystyle {\mathbf{x}}_2(t)=\mathbf{R}(t)-\frac{m_1}{m_1+m_2}\mathbf{r}(t)}$$

as may be verified by substituting the definitions of R and r into the right-hand sides of these two equations.

Two-body motion is planar

The motion of two bodies with respect to each other always lies in a plane (in the center of mass frame).

Proof: Defining the linear momentum p and the angular momentum L of the system, with respect to the center of mass, by the equations

$${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}=\mathbf{r}\times \mu \frac{d\mathbf{r}}{dt},}$$

where μ is the reduced mass and r is the relative position r₂ − r₁ (with these written taking the center of mass as the origin, and thus both parallel to r) the rate of change of the angular momentum L equals the net torque N

$${\displaystyle \mathbf{N}=\frac{d\mathbf{L}}{dt}=\dot{\mathbf{r}}\times \mu \dot{\mathbf{r}}+\mathbf{r}\times \mu \ddot{\mathbf{r}},}$$

and using the property of the vector cross product that v × w = 0 for any vectors v and w pointing in the same direction,

$${\displaystyle \mathbf{N}=\frac{d\mathbf{L}}{dt}=\mathbf{r}\times \mathbf{F},}$$

with F = μ d²r/dt².

Introducing the assumption (true of most physical forces, as they obey Newton's strong third law of motion) that the force between two particles acts along the line between their positions, it follows that r × F = 0 and the angular momentum vector L is constant (conserved). Therefore, the displacement vector r and its velocity v are always in the plane perpendicular to the constant vector L.

Energy of the two-body system

If the force F(r) is conservative then the system has a potential energy U(r), so the total energy can be written as

$${\displaystyle E_{\text{tot}}=\frac{1}{2}{m}_1{\dot{\mathbf{x}}}_1^2+\frac{1}{2}{m}_2{\dot{\mathbf{x}}}_2^2+U(\mathbf{r})=\frac{1}{2}(m_1+m_2){\dot{\mathbf{R}}}^2+\frac{1}{2}\mu {\dot{\mathbf{r}}}^2+U(\mathbf{r})}$$

In the center of mass frame the kinetic energy is the lowest and the total energy becomes

$${\displaystyle E=\frac{1}{2}\mu {\dot{\mathbf{r}}}^2+U(\mathbf{r})}$$

The coordinates x₁ and x₂ can be expressed as

$${\displaystyle {\mathbf{x}}_1=\frac{\mu }{m_1}\mathbf{r}}$$

$${\displaystyle {\mathbf{x}}_2=-\frac{\mu }{m_2}\mathbf{r}}$$

and in a similar way the energy E is related to the energies E₁ and E₂ that separately contain the kinetic energy of each body:

$${\displaystyle \begin{array}{rl}{E}_1& =\frac{\mu }{m_1}E=\frac{1}{2}{m}_1{\dot{\mathbf{x}}}_1^2+\frac{\mu }{m_1}U(\mathbf{r})\\ E_2& =\frac{\mu }{m_2}E=\frac{1}{2}{m}_2{\dot{\mathbf{x}}}_2^2+\frac{\mu }{m_2}U(\mathbf{r})\\ E_{\text{tot}}& =E_1+E_2\end{array}}$$

Central forces

Main article: Classical central-force problem

For many physical problems, the force F(r) is a central force, i.e., it is of the form

$${\displaystyle \mathbf{F}(\mathbf{r})=F(r)\hat{\mathbf{r}}}$$

where r = |r| and r̂ = r/r is the corresponding unit vector. We now have:

$${\displaystyle \mu \ddot{\mathbf{r}}=F(r)\hat{\mathbf{r}},}$$

where F(r) is negative in the case of an attractive force.