coordination number

Examples of coordination number in the following topics:

• Coordination Number, Ligands, and Geometries

  • The coordination number determines the number of ligands attached to the central ion and the overall shape of the complex.
  • In coordination chemistry, the coordination number is the number of ligands attached to the central ion (more specifically, the number of donor atoms).
  • Coordination numbers are normally between two and nine.
  • Different ligand structural arrangements result from the coordination number.
  • Calculate the coordination number of the metal in a coordination complex.

• Complex Numbers in Polar Coordinates

  • Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$.
  • The previous geometric idea where the number $z=a+bi$ is associated with the point $(a,b)$ on the usual $xy$-coordinate system is called rectangular coordinates.
  • The alternative way to picture things is called polar coordinates.
  • In polar coordinates, the parameters are $r$ and $\phi$.
  • Explain how to represent complex numbers in polar coordinates and why it is useful to do so

• Metal Cations that Act as Lewis Acids

  • Ligands create a complex when forming coordinate bonds with transition metals ions; the transition metal ion acts as a Lewis acid, and the ligand acts as a Lewis base.
  • The number of coordinate bonds is known as the complex's coordination number.
  • For instance, Mg2+ can coordinate with ammonia in solutions, as shown below:
  • The product is known as a complex ion, and the study of these ions is known as coordination chemistry.
  • Examples of several metals (V, Mn, Re, Fe, Ir) in coordination complexes with various ligands.

• Three-Dimensional Coordinate Systems

  • The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).
  • Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters $x$, $y$, and $z$.
  • This is a three dimensional space represented by a Cartesian coordinate system.
  • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.
  • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system

• Cylindrical and Spherical Coordinates

  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • The latter distance is given as a positive or negative number, depending on which side of the reference plane faces the point.
  • A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
  • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Introduction to the Polar Coordinate System

  • The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.
  • The radial coordinate is often denoted by $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.
  • Adding any number of full turns ($360^{\circ}$) to the angular coordinate does not change the corresponding direction.
  • Therefore, the same point can be expressed with an infinite number of different polar coordinates($r, ϕ ± n×360°$) or ($−r, ϕ ± (2n + 1)180°$), where $n$ is any integer.
  • In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$.

• Increasing Coordination

  • Increasing coordination helps organizations to maintain efficient operations through communication and control.
  • Coordination is simply the managerial ability to maintain operations and ensure they are properly integrated with one another; therefore, increasing coordination is closely related to improving managerial skills.
  • The management team must pay special attention to issues related to coordination and governance and be able to improve upon coordination through effective management.
  • There are a number of ways to improve upon the coordination of different departments, work groups, teams, or functional specialists.
  • In practice, coordination involves a delicate balance between centralization and decentralization.

• The Cartesian System

  • On the x-axis, numbers increase toward the right and decrease toward the left.
  • On the       y-axis, numbers increase going upward and decrease going downward.
  • For example there is a relationship between the number of cars washed and the revenue obtained.  
  • The revenue, or output, depends upon the number of cars, or input, that have their cars washed.  
  • The revenue is plotted on the y-axis and the number of cars washed is plotted on the x-axis.  

• Biomolecules

  • Coordination complexes (also called coordination compounds) and transition metals are widespread in nature.
  • In addition to donor groups that are provided by amino acid residues, a large number of organic cofactors function as ligands.
  • Metalloenzymes contain a metal ion bound to the protein with one labile coordination site.
  • The structure of the active site in carbonic anhydrases is well known from a number of crystal structures.
  • The fourth coordination site is occupied by a water molecule.

• Conics in Polar Coordinates

  • Polar coordinates allow conic sections to be expressed in an elegant way.
  • In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation: $r=\frac{ep}{1\: \pm\: e\: \cos\theta}$
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation: $r=\frac{ep}{1\: \pm\: e\: \sin\theta}$