Deadly Venom From Spiders and Snakes May Cure What Ails You

Efforts to tease apart the vast swarm of proteins in venom — a field called venomics — have burgeoned in recent years, leading to important drug discoveries.

By Jim Robbins

TUCSON, Ariz. — In a small room in a building at the Arizona-Sonora Desert Museum, the invertebrate keeper, Emma Califf, lifts up a rock in a plastic box. “This is one of our desert hairies,” she said, exposing a three-inch-long scorpion, its tail arced over its back. “The largest scorpion in North America.”

This captive hairy, along with a swarm of inch-long bark scorpions in another box, and two dozen rattlesnakes of varying species and sub- species across the hall, are kept here for the coin of the realm: their venom.

Efforts to tease apart the vast swarm of proteins in venom — a field called venomics — have burgeoned in recent years, and the growing catalog of compounds has led to a number of drug discoveries. As the components of these natural toxins continue to be assayed by evolving technologies, the number of promising molecules is also growing.

“A century ago we thought venom had three or four components, and now we know just one type of venom can have thousands,” said Leslie V. Boyer, a professor emeritus of pathology at the University of Arizona. “Things are accelerating because a small number of very good laboratories have been pumping out information that everyone else can now use to make discoveries.”

She added, “There’s a pharmacopoeia out there waiting to be explored.”

It is a striking case of modern-day scientific alchemy: The most highly evolved of natural poisons on the planet are creating a number of effective medicines with the potential for many more.

Leslie V. Boyer, founder of the Venom Immunochemistry, Pharmacology, and Emergency Response Institute, calls Arizona “venom central.”

A giant hairy scorpion at the Arizona-Sonora Desert Museum.

One of the most promising venom-derived drugs to date comes from the deadly Fraser Island funnel web spider of Australia, which halts cell death after a heart attack.

Blood flow to the heart is reduced after a heart attack, which makes the cell environment more acidic and leads to cell death. The drug, a protein called Hi1A, is scheduled for clinical trials next year. In the lab, it was tested on the cells of beating human hearts. It was found to block their ability to sense acid, “so the death message is blocked, cell death is reduced, and we see improved heart cell survival,” said Nathan Palpant, a researcher at the University of Queensland in Australia who helped make the discovery.

If proven in trials, it could be administered by emergency medical workers, and might prevent the damage that occurs after heart attacks and possibly improve outcomes in heart transplants by keeping the donor heart healthier longer.

“It looks like it’s going to be a heart attack wonder drug,” said Bryan Fry, an associate professor of toxicology at the University of Queensland, who is familiar with the research but was not involved in it. “And it’s from one of the most vilified creatures” in Australia.

The techniques used to process venom compounds have become so powerful that they are creating new opportunities. “We can do assays nowadays using only a couple of micrograms of venom that 10 or 15 years ago would have required hundreds of micrograms,” or more, Dr. Fry said. “What this has done is open up all the other venomous lineages out there that produce tiny amounts of material.”

There is an enormous natural library to sort through. Hundreds of thousands of species of reptile, insect, spider, snail and jellyfish, among other creatures, have mastered the art of chemical warfare with venom. Moreover, the makeup of venom varies from animal to animal. There is a kind of toxic terroir: Venom differs in quantity, potency and proportion and types of toxin, according to habitat and diet, and even by changing temperatures due to climate change.

Venom is made of a complex mix of toxins, which are composed of proteins with unique characteristics. They are so deadly because evolution has honed their effectiveness for so long — some 54 million years for snakes and 600 million for jellyfish.

Howard Byrne, a curator at the Arizona desert museum, handling a Gila monster, from which the drug exenatide, for Type 2 diabetes, is derived.

A tiger rattlesnake at the Arizona Sonoran Desert Museum.

Venom is the product of a biological arms race over that time; as venom becomes more deadly, victims evolve more resistance, which in turn makes venom even deadlier. Humans are included in that dynamic. “We are made of protein and our protein has little complex configurations on it that make us human,” said Dr. Boyer, who founded the Venom Immunochemistry, Pharmacology, and Emergency Response Institute, or VIPER. “And those little configurations are targets of the venom.”

The specific cellular proteins that the venom molecules have evolved to target with pinpoint accuracy are what make the drugs derived from them — which use the same pathways — so effective. Some proteins, however, have inherent problems that can make new drugs from them unworkable.

There is usually no need to gather venom to make these drugs. Once they are identified, they can be synthesized.

There are three main effects from venom. Neurotoxins attack the nervous system, paralyzing the victim. Hemotoxins target the blood and local tissue toxins attack the area around the site of poison exposure.

Numerous venom-derived drugs are on the market. Captopril, the first, was created in the 1970s from the venom of a Brazilian jararaca pit viper to treat high blood pressure. It has been successful commercially. Another drug, exenatide, is derived from Gila monster venom and is prescribed for Type 2 diabetes. Draculin is an anticoagulant from vampire bat venom and is used to treat stroke and heart attack.

The venom of the Israeli deathstalker scorpion is the source of a compound in clinical trials that finds and illuminates breast and colon tumors.

“Things are accelerating because a small number of very good laboratories have been pumping out information that everyone else can now use,” Dr. Boyer said.

Some proteins have been flagged as potential candidates for new drugs, but they have to journey through the long process of manufacture and clinical trials, which can take many years and cost millions of dollars. In March, researchers at the University of Utah announced that they had discovered a fast-acting molecule in cone snails. Cone snails fire their venom into fish, which causes the victims’ insulin levels to drop so rapidly it kills them. It holds promise as a drug for diabetes. Bee venom appears to work with a wide range of pathologies and has recently been found to kill aggressive breast cancer cells.

In Brazil researchers have been looking at the venom of the Brazilian wandering spider as a possible source of a new drug for erectile dysfunction — because of what happens to human victims when they are bit. “A characteristic of their envenomation is that males get extraordinary painful, incredibly long-lasting erections,” Dr. Fry said. “They have to separate it from its lethal factor, of course, and find a way to dial it back.”

Some scientists have long suspected that important secrets are locked up in venom. Scientific interest first surfaced in the 17th century. In the mid-18th century the Italian physician and polymath Felice Fontana added to the body of knowledge with his treatise, and in 1860 the first research to look at venom components was conducted by S. Weir Mitchell in Philadelphia.

The medicinal use of venom has a long history, often without scientific support. Venom-dipped needles are a traditional form of acupuncture. Bee sting therapy, in which a swarm of bees is placed on the skin, is used by some natural healers. The rock musician Steve Ludwin claims to have routinely injected himself with diluted venom, believing it to be a tonic that builds his immune system and boosts his energy.

The demand for venom is increasing. Ms. Califf of the Arizona-Sonora Desert Museum said she had to travel to the desert to find more bark scorpions, which she hunts at night with a black light because they glow in the dark. Arizona, Dr. Boyer said, is “venom central,” with more venomous creatures than in any other U.S. state, making it well suited for this kind of production.

Venom, made of a complex mix of toxins, are so deadly because evolution has honed their effectiveness for so long — some 54 million years for snakes.

The Arizona bark scorpion, which glows in the dark is hunted at night with a black light.

Scorpion venom is harvested by applying a tiny electrical current to the arachnid, which causes it to excrete a small drop of the amber liquid at the tip of its tail. With snakes, venom glands are gently massaged as they bare their fangs over a martini glass. After they surrender their venom, the substance is sent to researchers around the globe.

Pit vipers, including rattlesnakes, have other unusual adaptations. The “pit” is the site of the biological equipment that allows snakes to sense the heat of their prey. “You can blindfold a snake and it will still strike the target,” Dr. Boyer said.

But it’s not just venom that’s far better understood these days. In the last few years, there has been a well-heeled and concerted search for antivenom.

In 2019 the Wellcome Trust created a $100 million fund toward the pursuit. Since then there have been numerous research efforts around the world looking for a single universal treatment — one that can be carried into remote areas to immediately help someone bitten by any type of venomous snake. Currently, different types of snakebites have different antivenom.

It has been difficult. The wide array of ingredients in venom that benefit new drug research has also made it difficult to find a drug that can neutralize them. One promising universal antivenom, varespladib, is in clinical trials.

Experts hope the role of venom will lead to more respect for the fear-inducing creatures who create them. Dr. Fry, for his work on anticoagulants, is studying the venom of Komodo dragons, which, at 10 feet long and more than 300 pounds, is the largest lizard in the world. It is also highly endangered.

Work on the Komodo, “allows us to talk about the broader conservation message,” he said.

“You want nature around because it’s a biobank,” he added. “We can only find these interesting compounds from these magnificent creatures if they are not extinct.”

Chemical potential

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Chemical_potential 

In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the partial molar Gibbs free energy. At chemical equilibrium or in phase equilibrium, the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum. In a system in diffusion equilibrium, the chemical potential of any chemical species is uniformly the same everywhere throughout the system.

In semiconductor physics, the chemical potential of a system of electrons at zero absolute temperature is known as the Fermi energy.

Overview

Particles tend to move from higher chemical potential to lower chemical potential because this reduces the free energy. In this way, chemical potential is a generalization of "potentials" in physics such as gravitational potential. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential for work) to a lower gravitational potential (lower internal energy). In the same way, as molecules move, react, dissolve, etc., they will always tend naturally to go from a higher chemical potential to a lower one, changing the particle number, which is conjugate variable to chemical potential.

A simple example is a system of dilute molecules diffusing in a homogeneous environment. In this system, the molecules tend to move from areas with high concentration to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based on kinetic theory and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area and a lower chemical potential in a low concentration area. Movement of molecules from higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a spontaneous process.

Another example, not based on concentration but on phase, is an ice cube on a plate above 0 °C. An H₂O molecule that is in the solid phase (ice) has a higher chemical potential than a water molecule that is in the liquid phase (water) above 0 °C. When some of the ice melts, H₂O molecules convert from solid to the warmer liquid where their chemical potential is lower, so the ice cube shrinks. At the temperature of the melting point, 0 °C, the chemical potentials in water and ice are the same; the ice cube neither grows nor shrinks, and the system is in equilibrium.

A third example is illustrated by the chemical reaction of dissociation of a weak acid HA (such as acetic acid, A = CH₃COO⁻):

HA ⇌ H⁺ + A⁻

Vinegar contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules (HA) decreases and the concentrations of the product ions (H⁺ and A⁻) increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H⁺ and A⁻ increases. When the sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction. This explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing hydrogen ions into the solution.

Chemical potentials are important in many aspects of multi-phase equilibrium chemistry, including melting, boiling, evaporation, solubility, osmosis, partition coefficient, liquid-liquid extraction and chromatography. In each case the chemical potential of a given species at equilibrium is the same in all phases of the system.

In electrochemistry, ions do not always tend to go from higher to lower chemical potential, but they do always go from higher to lower electrochemical potential. The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything except the electric force. (See below for more on this terminology.)

Thermodynamic definition

The chemical potential μ_i of species i (atomic, molecular or nuclear) is defined, as all intensive quantities are, by the phenomenological fundamental equation of thermodynamics expressed in the form, which holds for both reversible and irreversible infinitesimal processes:

${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i},}$

where dU is the infinitesimal change of internal energy U, dS the infinitesimal change of entropy S, and dV is the infinitesimal change of volume V for a thermodynamic system in thermal equilibrium, and dN_i is the infinitesimal change of particle number N_i of species i as particles are added or subtracted. T is absolute temperature, S is entropy, P is pressure, and V is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added.

From the above equation, the chemical potential is given by

${\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}$

This is an inconvenient expression for condensed-matter systems, such as chemical solutions, as it is hard to control the volume and entropy to be constant while particles are added. A more convenient expression may be obtained by making a Legendre transformation to another thermodynamic potential: the Gibbs free energy ${\displaystyle G=U+PV-TS}$. From the differential ${\displaystyle \mathrm {d} G=\mathrm {d} U+P\,\mathrm {d} V+V\,\mathrm {d} P-T\,\mathrm {d} S-S\,\mathrm {d} T}$ and using the above expression for ${\displaystyle \mathrm {d} U}$, a differential relation for ${\displaystyle \mathrm {d} G}$ is obtained:

${\displaystyle \mathrm {d} G=-S\,\mathrm {d} T+V\,\mathrm {d} P+\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}$

As a consequence, another expression for ${\displaystyle \mu _{i}}$ results:

${\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P,N_{j\neq i}},}$

and the change in Gibbs free energy of a system that is held at constant temperature and pressure is simply

${\displaystyle \mathrm {d} G=\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}$

In thermodynamic equilibrium, when the system concerned is at constant temperature and pressure but can exchange particles with its external environment, the Gibbs free energy is at its minimum for the system, that is ${\displaystyle \mathrm {d} G=0}$. It follows that

${\displaystyle \mu _{1}\,\mathrm {d} N_{1}+\mu _{2}\,\mathrm {d} N_{2}+\dots =0.}$

Use of this equality provides the means to establish the equilibrium constant for a chemical reaction.

By making further Legendre transformations from U to other thermodynamic potentials like the enthalpy ${\displaystyle H=U+PV}$ and Helmholtz free energy ${\displaystyle F=U-TS}$, expressions for the chemical potential may be obtained in terms of these:

${\displaystyle \mu _{i}=\left({\frac {\partial H}{\partial N_{i}}}\right)_{S,P,N_{j\neq i}},}$

${\displaystyle \mu _{i}=\left({\frac {\partial F}{\partial N_{i}}}\right)_{T,V,N_{j\neq i}}.}$

These different forms for the chemical potential are all equivalent, meaning that they have the same physical content and may be useful in different physical situations.

Applications

The Gibbs–Duhem equation is useful because it relates individual chemical potentials. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by

${\displaystyle d\mu _{\text{B}}=-{\frac {n_{\text{A}}}{n_{\text{B}}}}\,d\mu _{\text{A}}}$

where ${\displaystyle n_{\text{A}}}$ is the number of moles of A and ${\displaystyle n_{\text{B}}}$ is the number of moles of B. Every instance of phase or chemical equilibrium is characterized by a constant. For instance, the melting of ice is characterized by a temperature, known as the melting point at which solid and liquid phases are in equilibrium with each other. Chemical potentials can be used to explain the slopes of lines on a phase diagram by using the Clapeyron equation, which in turn can be derived from the Gibbs–Duhem equation. They are used to explain colligative properties such as melting-point depression by the application of pressure. Henry's law for the solute can be derived from Raoult's law for the solvent using chemical potentials.

History

Chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. He defined it as follows:

If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.

Gibbs later noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body. This freedom to choose the boundary of the system allows the chemical potential to be applied to a huge range of systems. The term can be used in thermodynamics and physics for any system undergoing change. Chemical potential is also referred to as partial molar Gibbs energy (see also partial molar property). Chemical potential is measured in units of energy/particle or, equivalently, energy/mole.

In his 1873 paper A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional volume–entropy–internal energy graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other. In his own words from the aforementioned paper, Gibbs states:

If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure P and temperature T, this equation may be written:

${\displaystyle \displaystyle \delta (\epsilon -T\eta +P\nu )=0}$

Where δ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.

In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body.

Electrochemical, internal, external, and total chemical potential

The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called total chemical potential. If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (electric potential energy, gravitational potential energy, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) Therefore, the total chemical potential can be split into internal chemical potential and external chemical potential:

${\displaystyle \mu _{\text{tot}}=\mu _{\text{int}}+\mu _{\text{ext}},}$

where

${\displaystyle \mu _{\text{ext}}=qV_{\text{ele}}+mgh+\cdots ,}$

i.e., the external potential is the sum of electric potential, gravitational potential, etc. (where q and m are the charge and mass of the species, V_ele and h are the electric potential and height of the container, respectively, and g is the acceleration due to gravity). The internal chemical potential includes everything else besides the external potentials, such as density, temperature, and enthalpy. This formalism can be understood by assuming that the total energy of a system, ${\displaystyle U}$, is the sum of two parts: an internal energy, ${\displaystyle U_{\text{int}}}$, and an external energy due to the interaction of each particle with an external field, ${\displaystyle U_{\text{ext}}=N(qV_{\text{ele}}+mgh+\cdots )}$. The definition of chemical potential applied to ${\displaystyle U_{\text{int}}+U_{\text{ext}}}$ yields the above expression for ${\displaystyle \mu _{\text{tot}}}$.

The phrase "chemical potential" sometimes means "total chemical potential", but that is not universal. In some fields, in particular electrochemistry, semiconductor physics, and solid-state physics, the term "chemical potential" means internal chemical potential, while the term electrochemical potential is used to mean total chemical potential.

Systems of particles

Electrons in solids

Main article: Fermi level

Electrons in solids have a chemical potential, defined the same way as the chemical potential of a chemical species: The change in free energy when electrons are added or removed from the system. In the case of electrons, the chemical potential is usually expressed in energy per particle rather than energy per mole, and the energy per particle is conventionally given in units of electronvolt (eV).

Chemical potential plays an especially important role in solid-state physics and is closely related to the concepts of work function, Fermi energy, and Fermi level. For example, n-type silicon has a higher internal chemical potential of electrons than p-type silicon. In a p–n junction diode at equilibrium the chemical potential (internal chemical potential) varies from the p-type to the n-type side, while the total chemical potential (electrochemical potential, or, Fermi level) is constant throughout the diode.

As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction band. It may also be specified "relative to vacuum", to yield a quantity known as work function, however, work function varies from surface to surface even on a completely homogeneous material. Total chemical potential, on the other hand, is usually specified relative to electrical ground.

In atomic physics, the chemical potential of the electrons in an atom is sometimes said to be the negative of the atom's electronegativity. Likewise, the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken electronegativity scale. By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is seen that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i.e.,

${\displaystyle \mu _{\text{Mulliken}}=-\chi _{\text{Mulliken}}=-{\frac {IP+EA}{2}}=\left[{\frac {\delta E[N]}{\delta N}}\right]_{N=N_{0}}.}$

Sub-nuclear particles

In recent years, thermal physics has applied the definition of chemical potential to systems in particle physics and its associated processes. For example, in a quark–gluon plasma or other QCD matter, at every point in space there is a chemical potential for photons, a chemical potential for electrons, a chemical potential for baryon number, electric charge, and so forth.

In the case of photons, photons are bosons and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly (at least, it occurs rapidly in the presence of dense charged matter), it is safe to assume that the photon chemical potential is never different from zero.

Electric charge is different because it is conserved, i.e. it can be neither created nor destroyed. It can, however, diffuse. The "chemical potential of electric charge" controls this diffusion: Electric charge, like anything else, will tend to diffuse from areas of higher chemical potential to areas of lower chemical potential. Other conserved quantities like baryon number are the same. In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out.

In the case of electrons, the behaviour depends on temperature and context. At low temperatures, with no positrons present, electrons cannot be created or destroyed. Therefore, there is an electron chemical potential that might vary in space, causing diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear out of the vacuum (pair production), so the chemical potential of electrons by themselves becomes a less useful quantity than the chemical potential of the conserved quantities like (electrons minus positrons).

The chemical potentials of bosons and fermions is related to the number of particles and the temperature by Bose–Einstein statistics and Fermi–Dirac statistics respectively.

Ideal vs. non-ideal solutions

The chemical potential of component i in solution for (left) ideal [incorrectly linearized] and (right) real solutions

Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution:

${\displaystyle \mu _{i}=\mu _{i}^{\text{ideal}}+\mu _{i}^{\text{excess}},}$

In an ideal solution, the chemical potential of species i (μ_i) is dependent on temperature and pressure. μ_i0(T, P) is defined as the chemical potential of pure species i. Given this definition, the chemical potential of species i in an ideal solution is

${\displaystyle \mu _{i}^{\text{ideal}}\approx \mu _{i0}+RT\ln(x_{i}),}$

where R is the gas constant, and ${\displaystyle x_{i}}$ is the mole fraction of species i contained in the solution. The chemical potential becomes negative infinity when ${\displaystyle x_{i}=0}$, but this does not lead to unphysical results because ${\displaystyle x_{i}=0}$ means that species i is not present in the system.

This equation assumes that ${\displaystyle \mu _{i}}$ only depends on the mole fraction (${\displaystyle x_{i}}$) contained in the solution. This neglects intermolecular interaction between species i with itself and other species [i–(j≠i)]. This can be corrected for by factoring in the coefficient of activity of species i, defined as γ_i. This correction yields

${\displaystyle \mu _{i}=\mu _{i0}(T,P)+RT\ln(x_{i})+RT\ln(\gamma _{i})=\mu _{i0}(T,P)+RT\ln(x_{i}\gamma _{i}).}$

The plots above give a very rough picture of the ideal and non-ideal situation.