Fundamental thermodynamic relation

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

In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G (Gibbs free energy) or H (enthalpy). The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.

${\displaystyle \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V}$

Here, U is internal energy, T is absolute temperature, S is entropy, P is pressure, and V is volume.

This is only one expression of the fundamental thermodynamic relation. It may be expressed in other ways, using different variables (e.g. using thermodynamic potentials). For example, the fundamental relation may be expressed in terms of the enthalpy H as

${\displaystyle \mathrm{d}H=T\mathrm{d}S+V\mathrm{d}P}$

in terms of the Helmholtz free energy F as

${\displaystyle \mathrm{d}F=-S\mathrm{d}T-P\mathrm{d}V}$

and in terms of the Gibbs free energy G as

${\displaystyle \mathrm{d}G=-S\mathrm{d}T+V\mathrm{d}P}$.

The first and second laws of thermodynamics

The first law of thermodynamics states that:

${\displaystyle \mathrm{d}U=\delta Q-\delta W}$

where ${\displaystyle \delta Q}$ and ${\displaystyle \delta W}$ are infinitesimal amounts of heat supplied to the system by its surroundings and work done by the system on its surroundings, respectively.

According to the second law of thermodynamics we have for a reversible process:

${\displaystyle \mathrm{d}S=\frac{\delta Q}{T}}$

Hence:

${\displaystyle \delta Q=T\mathrm{d}S}$

By substituting this into the first law, we have:

${\displaystyle \mathrm{d}U=T\mathrm{d}S-\delta W}$

Letting ${\displaystyle \delta W}$ be reversible pressure-volume work done by the system on its surroundings,

${\displaystyle \delta W=P\mathrm{d}V}$

we have:

${\displaystyle \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V}$

This equation has been derived in the case of reversible changes. However, since U, S, and V are thermodynamic state functions that depends on only the initial and final states of a thermodynamic process, the above relation holds also for non-reversible changes. If the composition, i.e. the amounts ${\displaystyle n_i}$ of the chemical components, in a system of uniform temperature and pressure can also change, e.g. due to a chemical reaction, the fundamental thermodynamic relation generalizes to:

${\displaystyle \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V+\sum _i{\mu }_i\mathrm{d}{n}_i}$

The ${\displaystyle {\mu }_i}$ are the chemical potentials corresponding to particles of type ${\displaystyle i}$.

If the system has more external parameters than just the volume that can change, the fundamental thermodynamic relation generalizes to

${\displaystyle \mathrm{d}U=T\mathrm{d}S+\sum _j{X}_j\mathrm{d}{x}_j+\sum _i{\mu }_i\mathrm{d}{n}_i}$

Here the ${\displaystyle X_j}$ are the generalized forces corresponding to the external parameters ${\displaystyle x_j}$. (The negative sign used with pressure is unusual and arises because pressure represents a compressive stress that tends to decrease volume. Other generalized forces tend to increase their conjugate displacements.)

Relationship to statistical mechanics

The fundamental thermodynamic relation and statistical mechanical principles can be derived from one another.

Derivation from statistical mechanical principles

The above derivation uses the first and second laws of thermodynamics. The first law of thermodynamics is essentially a definition of heat, i.e. heat is the change in the internal energy of a system that is not caused by a change of the external parameters of the system.

However, the second law of thermodynamics is not a defining relation for the entropy. The fundamental definition of entropy of an isolated system containing an amount of energy ${\displaystyle E}$ is:

${\displaystyle S=k\log \left[\mathrm{\Omega }\left(E\right)\right]}$

where ${\displaystyle \mathrm{\Omega }\left(E\right)}$ is the number of quantum states in a small interval between ${\displaystyle E}$ and ${\displaystyle E+\delta E}$. Here ${\displaystyle \delta E}$ is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of ${\displaystyle \delta E}$. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on ${\displaystyle \delta E}$. The entropy is thus a measure of the uncertainty about exactly which quantum state the system is in, given that we know its energy to be in some interval of size ${\displaystyle \delta E}$.

Deriving the fundamental thermodynamic relation from first principles thus amounts to proving that the above definition of entropy implies that for reversible processes we have:

${\displaystyle dS=\frac{\delta Q}{T}}$

The fundamental assumption of statistical mechanics is that all the ${\displaystyle \mathrm{\Omega }\left(E\right)}$ states at a particular energy are equally likely. This allows us to extract all the thermodynamical quantities of interest. The temperature is defined as:

${\displaystyle \frac{1}{kT}\equiv \beta \equiv \frac{d\log \left[\mathrm{\Omega }\left(E\right)\right]}{dE}}$

This definition can be derived from the microcanonical ensemble, which is a system of a constant number of particles, a constant volume and that does not exchange energy with its environment. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

The generalized force, X, corresponding to the external parameter x is defined such that ${\displaystyle Xdx}$ is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate ${\displaystyle E_r}$ is given by:

${\displaystyle X=-\frac{d{E}_r}{dx}}$

Since the system can be in any energy eigenstate within an interval of ${\displaystyle \delta E}$, we define the generalized force for the system as the expectation value of the above expression:

${\displaystyle X=-⟨\frac{d{E}_r}{dx}⟩}$

To evaluate the average, we partition the ${\displaystyle \mathrm{\Omega }(E)}$ energy eigenstates by counting how many of them have a value for ${\displaystyle \frac{d{E}_r}{dx}}$ within a range between ${\displaystyle Y}$ and ${\displaystyle Y+\delta Y}$. Calling this number ${\displaystyle {\mathrm{\Omega }}_Y\left(E\right)}$, we have:

${\displaystyle \mathrm{\Omega }(E)=\sum _Y{\mathrm{\Omega }}_Y(E)}$

The average defining the generalized force can now be written:

${\displaystyle X=-\frac{1}{\mathrm{\Omega }(E)}\sum _{Y}Y{\mathrm{\Omega }}_Y(E)}$

We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then ${\displaystyle \mathrm{\Omega }\left(E\right)}$ will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between ${\displaystyle E}$ and ${\displaystyle E+\delta E}$. Let's focus again on the energy eigenstates for which ${\displaystyle \frac{d{E}_r}{dx}}$ lies within the range between ${\displaystyle Y}$ and ${\displaystyle Y+\delta Y}$. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E − Y dx to E move from below E to above E. There are

${\displaystyle N_Y(E)=\frac{{\mathrm{\Omega }}_Y(E)}{\delta E}Ydx}$

such energy eigenstates. If ${\displaystyle Ydx\le \delta E}$, all these energy eigenstates will move into the range between ${\displaystyle E}$ and ${\displaystyle E+\delta E}$ and contribute to an increase in ${\displaystyle \mathrm{\Omega }}$. The number of energy eigenstates that move from below ${\displaystyle E+\delta E}$ to above ${\displaystyle E+\delta E}$ is, of course, given by ${\displaystyle N_Y\left(E+\delta E\right)}$. The difference

${\displaystyle N_Y(E)-N_Y(E+\delta E)}$

is thus the net contribution to the increase in ${\displaystyle \mathrm{\Omega }}$. Note that if Y dx is larger than ${\displaystyle \delta E}$ there will be energy eigenstates that move from below ${\displaystyle E}$ to above ${\displaystyle E+\delta E}$. They are counted in both ${\displaystyle N_Y(E)}$ and ${\displaystyle N_Y(E+\delta E)}$, therefore the above expression is also valid in that case.

Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:

${\displaystyle {\left(\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }x}\right)}_E=-\sum _{Y}Y{\left(\frac{\mathrm{\partial }{\mathrm{\Omega }}_Y}{\mathrm{\partial }E}\right)}_x={\left(\frac{\mathrm{\partial }(\mathrm{\Omega }X)}{\mathrm{\partial }E}\right)}_x}$

The logarithmic derivative of ${\displaystyle \mathrm{\Omega }}$ with respect to x is thus given by:

${\displaystyle {\left(\frac{\mathrm{\partial }\log \left(\mathrm{\Omega }\right)}{\mathrm{\partial }x}\right)}_E=\beta X+{\left(\frac{\mathrm{\partial }X}{\mathrm{\partial }E}\right)}_x}$

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and thus vanishes in the thermodynamic limit. We have thus found that:

${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }x}\right)}_E=\frac{X}{T}}$

Combining this with

${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }E}\right)}_x=\frac{1}{T}}$

Gives:

${\displaystyle dS={\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }E}\right)}_{x}dE+{\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }x}\right)}_{E}dx=\frac{dE}{T}+\frac{X}{T}dx}$

which we can write as:

${\displaystyle dE=TdS-Xdx}$

Derivation of statistical mechanical principles from the fundamental thermodynamic relation

It has been shown that the fundamental thermodynamic relation together with the following three postulates^[2]

1. The probability density function is proportional to some function of the ensemble parameters and random variables.
2. Thermodynamic state functions are described by ensemble averages of random variables.
3. The entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.

is sufficient to build the theory of statistical mechanics without the equal a priori probability postulate.

For example, in order to derive the Boltzmann distribution, we assume the probability density of microstate i satisfies $Pr(i)\propto f(E_i,T)$. The normalization factor (partition function) is therefore

${\displaystyle Z=\sum _{i}f(E_i,T).}$

The entropy is therefore given by

${\displaystyle S=k_B\sum _i\frac{f(E_i,T)}{Z}\log \left(\frac{f(E_i,T)}{Z}\right).}$

If we change the temperature T by dT while keeping the volume of the system constant, the change of entropy satisfies

${\displaystyle dS={\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_{V}dT}$

where

${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_V=-k_B\sum _i\frac{Z\cdot \frac{\mathrm{\partial }f(E_i,T)}{\mathrm{\partial }T}\cdot \log f(E_i,T)-\frac{\mathrm{\partial }Z}{\mathrm{\partial }T}\cdot f(E_i,T)\cdot \log f(E_i,T)}{Z^2}=-k_B\sum _i\frac{\mathrm{\partial }}{\mathrm{\partial }T}\left(\frac{f(E_i,T)}{Z}\right)\cdot \log f(E_i,T)}$

Considering that

${\displaystyle ⟨E⟩=\sum _i\frac{f(E_i,T)}{Z}\cdot E_i}$

we have

${\displaystyle d⟨E⟩=\sum _i\frac{\mathrm{\partial }}{\mathrm{\partial }T}\left(\frac{f(E_i,T)}{Z}\right)\cdot E_i\cdot dT}$

From the fundamental thermodynamic relation, we have

${\displaystyle -\frac{dS}{k_B}+\frac{d⟨E⟩}{k_{B}T}+\frac{P}{k_{B}T}dV=0}$

Since we kept V constant when perturbing T, we have $dV=0$. Combining the equations above, we have

${\displaystyle \sum _i\frac{\mathrm{\partial }}{\mathrm{\partial }T}\left(\frac{f(E_i,T)}{Z}\right)\cdot \left[\log f(E_i,T)+\frac{E_i}{k_{B}T}\right]\cdot dT=0}$

Physics laws should be universal, i.e., the above equation must hold for arbitrary systems, and the only way for this to happen is

${\displaystyle \log f(E_i,T)+\frac{E_i}{k_{B}T}=0}$

That is

${\displaystyle f(E_i,T)=\exp \left(-\frac{E_i}{k_{B}T}\right).}$

It has been shown that the third postulate in the above formalism can be replaced by the following:

3. At infinite temperature, all the microstates have the same probability.

However, the mathematical derivation will be much more complicated.

Nucleophile

From Wikipedia, the free encyclopedia

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

 

A hydroxide ion acting as a nucleophile in an S_N2 reaction, converting a haloalkane into an alcohol

In chemistry, a nucleophile is a chemical species that forms bonds by donating an electron pair. All molecules and ions with a free pair of electrons or at least one pi bond can act as nucleophiles. Because nucleophiles donate electrons, they are Lewis bases.

Nucleophilic describes the affinity of a nucleophile to bond with positively charged atomic nuclei. Nucleophilicity, sometimes referred to as nucleophile strength, refers to a substance's nucleophilic character and is often used to compare the affinity of atoms. Neutral nucleophilic reactions with solvents such as alcohols and water are named solvolysis. Nucleophiles may take part in nucleophilic substitution, whereby a nucleophile becomes attracted to a full or partial positive charge, and nucleophilic addition. Nucleophilicity is closely related to basicity.

History

The terms nucleophile and electrophile were introduced by Christopher Kelk Ingold in 1933, replacing the terms anionoid and cationoid proposed earlier by A. J. Lapworth in 1925. The word nucleophile is derived from nucleus and the Greek word φιλος, philos, meaning friend.

Properties

In general, in a group across the periodic table, the more basic the ion (the higher the pK_a of the conjugate acid) the more reactive it is as a nucleophile. Within a series of nucleophiles with the same attacking element (e.g. oxygen), the order of nucleophilicity will follow basicity. Sulfur is in general a better nucleophile than oxygen.

Nucleophilicity

Many schemes attempting to quantify relative nucleophilic strength have been devised. The following empirical data have been obtained by measuring reaction rates for many reactions involving many nucleophiles and electrophiles. Nucleophiles displaying the so-called alpha effect are usually omitted in this type of treatment.

Swain–Scott equation

The first such attempt is found in the Swain–Scott equation derived in 1953:

${\displaystyle {\log }_{10}\left(\frac{k}{k_0}\right)=sn}$

This free-energy relationship relates the pseudo first order reaction rate constant (in water at 25 °C), k, of a reaction, normalized to the reaction rate, k₀, of a standard reaction with water as the nucleophile, to a nucleophilic constant n for a given nucleophile and a substrate constant s that depends on the sensitivity of a substrate to nucleophilic attack (defined as 1 for methyl bromide).

This treatment results in the following values for typical nucleophilic anions: acetate 2.7, chloride 3.0, azide 4.0, hydroxide 4.2, aniline 4.5, iodide 5.0, and thiosulfate 6.4. Typical substrate constants are 0.66 for ethyl tosylate, 0.77 for β-propiolactone, 1.00 for 2,3-epoxypropanol, 0.87 for benzyl chloride, and 1.43 for benzoyl chloride.

The equation predicts that, in a nucleophilic displacement on benzyl chloride, the azide anion reacts 3000 times faster than water.

Ritchie equation

The Ritchie equation, derived in 1972, is another free-energy relationship:

${\displaystyle {\log }_{10}\left(\frac{k}{k_0}\right)=N^{+}}$

where N⁺ is the nucleophile dependent parameter and k₀ the reaction rate constant for water. In this equation, a substrate-dependent parameter like s in the Swain–Scott equation is absent. The equation states that two nucleophiles react with the same relative reactivity regardless of the nature of the electrophile, which is in violation of the reactivity–selectivity principle. For this reason, this equation is also called the constant selectivity relationship.

In the original publication the data were obtained by reactions of selected nucleophiles with selected electrophilic carbocations such as tropylium or diazonium cations:

or (not displayed) ions based on malachite green. Many other reaction types have since been described.

Typical Ritchie N⁺ values (in methanol) are: 0.5 for methanol, 5.9 for the cyanide anion, 7.5 for the methoxide anion, 8.5 for the azide anion, and 10.7 for the thiophenol anion. The values for the relative cation reactivities are −0.4 for the malachite green cation, +2.6 for the benzenediazonium cation, and +4.5 for the tropylium cation.

Mayr–Patz equation

In the Mayr–Patz equation (1994):

${\displaystyle \log (k)=s(N+E)}$

The second order reaction rate constant k at 20 °C for a reaction is related to a nucleophilicity parameter N, an electrophilicity parameter E, and a nucleophile-dependent slope parameter s. The constant s is defined as 1 with 2-methyl-1-pentene as the nucleophile.

Many of the constants have been derived from reaction of so-called benzhydrylium ions as the electrophiles:

and a diverse collection of π-nucleophiles:

.

Typical E values are +6.2 for R = chlorine, +5.90 for R = hydrogen, 0 for R = methoxy and −7.02 for R = dimethylamine.

Typical N values with s in parenthesis are −4.47 (1.32) for electrophilic aromatic substitution to toluene (1), −0.41 (1.12) for electrophilic addition to 1-phenyl-2-propene (2), and 0.96 (1) for addition to 2-methyl-1-pentene (3), −0.13 (1.21) for reaction with triphenylallylsilane (4), 3.61 (1.11) for reaction with 2-methylfuran (5), +7.48 (0.89) for reaction with isobutenyltributylstannane (6) and +13.36 (0.81) for reaction with the enamine 7.

The range of organic reactions also include SN2 reactions:

With E = −9.15 for the S-methyldibenzothiophenium ion, typical nucleophile values N (s) are 15.63 (0.64) for piperidine, 10.49 (0.68) for methoxide, and 5.20 (0.89) for water. In short, nucleophilicities towards sp₂ or sp₃ centers follow the same pattern.

Unified equation

In an effort to unify the above described equations the Mayr equation is rewritten as:

${\displaystyle \log (k)=s_E{s}_N(N+E)}$

with s_E the electrophile-dependent slope parameter and s_N the nucleophile-dependent slope parameter. This equation can be rewritten in several ways:

• with s_E = 1 for carbocations this equation is equal to the original Mayr–Patz equation of 1994,
• with s_N = 0.6 for most n nucleophiles the equation becomes

${\displaystyle \log (k)=0.6s_{E}N+0.6s_{E}E}$

or the original Scott–Swain equation written as:

${\displaystyle \log (k)=\log (k_0)+s_{E}N}$

• with s_E = 1 for carbocations and s_N = 0.6 the equation becomes:

${\displaystyle \log (k)=0.6N+0.6E}$

or the original Ritchie equation written as:

${\displaystyle \log (k)-\log (k_0)=N^{+}}$

Types

Examples of nucleophiles are anions such as Cl⁻, or a compound with a lone pair of electrons such as NH₃ (ammonia) and PR₃.

In the example below, the oxygen of the hydroxide ion donates an electron pair to form a new chemical bond with the carbon at the end of the bromopropane molecule. The bond between the carbon and the bromine then undergoes heterolytic fission, with the bromine atom taking the donated electron and becoming the bromide ion (Br⁻), because a S_N2 reaction occurs by backside attack. This means that the hydroxide ion attacks the carbon atom from the other side, exactly opposite the bromine ion. Because of this backside attack, S_N2 reactions result in a inversion of the configuration of the electrophile. If the electrophile is chiral, it typically maintains its chirality, though the S_N2 product's absolute configuration is flipped as compared to that of the original electrophile.

An ambident nucleophile is one that can attack from two or more places, resulting in two or more products. For example, the thiocyanate ion (SCN⁻) may attack from either the sulfur or the nitrogen. For this reason, the S_N2 reaction of an alkyl halide with SCN⁻ often leads to a mixture of an alkyl thiocyanate (R-SCN) and an alkyl isothiocyanate (R-NCS). Similar considerations apply in the Kolbe nitrile synthesis.

Halogens

While the halogens are not nucleophilic in their diatomic form (e.g. I₂ is not a nucleophile), their anions are good nucleophiles. In polar, protic solvents, F⁻ is the weakest nucleophile, and I⁻ the strongest; this order is reversed in polar, aprotic solvents.

Carbon

Carbon nucleophiles are often organometallic reagents such as those found in the Grignard reaction, Blaise reaction, Reformatsky reaction, and Barbier reaction or reactions involving organolithium reagents and acetylides. These reagents are often used to perform nucleophilic additions.

Enols are also carbon nucleophiles. The formation of an enol is catalyzed by acid or base. Enols are ambident nucleophiles, but, in general, nucleophilic at the alpha carbon atom. Enols are commonly used in condensation reactions, including the Claisen condensation and the aldol condensation reactions.

Oxygen

Examples of oxygen nucleophiles are water (H₂O), hydroxide anion, alcohols, alkoxide anions, hydrogen peroxide, and carboxylate anions. Nucleophilic attack does not take place during intermolecular hydrogen bonding.

Sulfur

Of sulfur nucleophiles, hydrogen sulfide and its salts, thiols (RSH), thiolate anions (RS⁻), anions of thiolcarboxylic acids (RC(O)-S⁻), and anions of dithiocarbonates (RO-C(S)-S⁻) and dithiocarbamates (R₂N-C(S)-S⁻) are used most often.

In general, sulfur is very nucleophilic because of its large size, which makes it readily polarizable, and its lone pairs of electrons are readily accessible.

Nitrogen

Nitrogen nucleophiles include ammonia, azide, amines, nitrites, hydroxylamine, hydrazine, carbazide, phenylhydrazine, semicarbazide, and amide.

Metal centers

Although metal centers (e.g., Li⁺, Zn²⁺, Sc³⁺, etc.) are most commonly cationic and electrophilic (Lewis acidic) in nature, certain metal centers (particularly ones in a low oxidation state and/or carrying a negative charge) are among the strongest recorded nucleophiles and are sometimes referred to as "supernucleophiles." For instance, using methyl iodide as the reference electrophile, Ph₃Sn^– is about 10000 times more nucleophilic than I^–, while the Co(I) form of vitamin B₁₂ (vitamin B_12s) is about 10⁷ times more nucleophilic. Other supernucleophilic metal centers include low oxidation state carbonyl metalate anions (e.g., CpFe(CO)₂^–).