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Complexation Equilibria: Overview

Complexation Equilibria: Overview

Complexation reactions take place when dative or coordinate covalent bonds form between metal ions and ligands. The compounds formed in these reactions are called coordination compounds. The number of bonds formed between the metal ion and the ligands is called its coordination number. Generally, most metal ions in an aqueous solution are solvated by water molecules and thus exist as aqua complexes.
The equilibrium constant of the complexation reaction is represented as the formation constant...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

Network Covalent Solids

Network Covalent Solids

Network covalent solids contain a three-dimensional network of covalently bonded atoms as found in the crystal structures of nonmetals like diamond, graphite, silicon, and some covalent compounds, such as silicon dioxide (sand) and silicon carbide (carborundum, the abrasive on sandpaper). Many minerals have networks of covalent bonds.
To break or to melt a covalent network solid, covalent bonds must be broken. Because covalent bonds are relatively strong, covalent network solids are typically...

Ladder Diagrams: Complexation Equilibria

Ladder Diagrams: Complexation Equilibria

Ladder diagrams are useful for evaluating equilibria involving metal-ligand complexes. The vertical scale of the ladder diagram represents the concentration of unreacted or free ligand, pL. The horizontal lines on the scale depict the log of stepwise formation constants for metal-ligand complexes and indicate the dominant species in all the regions.
The formation constant, K1, for the formation of Cd(NH3)2+ complex from cadmium and ammonia is 3.55 × 102. Log K1 (i.e. pNH3) is 2.55, and...

The Entropy as a State Function

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

¹H NMR: Long-Range Coupling

¹H NMR: Long-Range Coupling

The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene π orbitals.

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Related Experiment Video

Updated: Jun 14, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Entanglement percolation in quantum complex networks.

Martí Cuquet¹, John Calsamiglia

¹Grup de Física Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.

Physical Review Letters

|April 7, 2010

Summary

This summary is machine-generated.

We explored entanglement percolation to create long-distance quantum entanglement in complex networks. Our quantum strategies significantly improve the entanglement threshold in random and real-world networks.

Related Experiment Videos

Last Updated: Jun 14, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

Quantum Information Science
Network Science
Complex Systems

Background:

Quantum networks are crucial for distributed quantum applications.
Effective communication and entanglement distribution are key challenges.
Complex network properties remain largely unexplored in quantum contexts.

Purpose of the Study:

To investigate entanglement percolation for establishing long-distance entanglement in quantum complex networks.
To analyze the impact of network structure on quantum communication.
To enhance entanglement distribution between arbitrary network nodes.

Main Methods:

Development of a theoretical framework for analytical study of random graphs with arbitrary degree distributions.
Application of entanglement percolation principles to quantum network models.
Numerical simulations to validate theoretical findings and assess performance on various network types.

Main Results:

Analytical results derived for specific random graph models.
Demonstrated substantial enhancement of the percolation threshold using proposed quantum strategies.
Numerical simulations confirmed significant improvements in small-world and real-world networks.

Conclusions:

Entanglement percolation is a viable strategy for robust long-distance entanglement in quantum networks.
The proposed quantum strategies offer a significant advantage in overcoming distance limitations for entanglement.
Findings pave the way for more efficient and scalable quantum network architectures.