Racemic mixture

In chemistry, a racemic mixture or racemate (/reɪˈsiːmeɪt, rə-, ˈræsɪmeɪt/) is a mixture that has equal amounts (50:50) of left- and right-handed enantiomers of a chiral molecule or salt. Racemic mixtures are rare in nature, but many compounds are produced industrially as racemates.

History

The first known racemic mixture was racemic acid, which Louis Pasteur found to be a mixture of the two enantiomeric isomers of tartaric acid. He manually separated the crystals of a mixture, starting from an aqueous solution of the sodium ammonium salt of racemate tartaric acid. Pasteur benefited from the fact that ammonium tartrate salt gives enantiomeric crystals with distinct crystal forms (at 77 °F). Reasoning from the macroscopic scale down to the molecular, he reckoned that the molecules had to have non-superimposable mirror images. A sample with only a single enantiomer is an enantiomerically pure or enantiopure compound.

Etymology

The word racemic derives from Latin racemus, meaning pertaining to a bunch of grapes. Racemic acid, when naturally produced in grapes, is only the right-handed version of the molecule, better known as tartaric acid. In many Germanic languages racemic acid is called "grape acid", e.g. German Traubensäure and Swedish druvsyra. Carl von Linné gave red elderberry the scientific name Sambucus racemosa as the Swedish name, druvfläder, means 'grape elder', so called because its berries grow in a grape-like cluster.

Nomenclature

A racemic mixture is denoted by the prefix (±)- or dl- (for sugars the prefix dl- may be used), indicating an equal (1:1) mixture of dextro and levo isomers. Also the prefix rac- (or racem-) or the symbols RS and SR (all in italic letters) are used.

If the ratio is not 1:1 (or is not known), the prefix (+)/(−), d/l- or d/l- (with a slash) is used instead.

The usage of d and l is discouraged by IUPAC.

Properties

A racemate is optically inactive (achiral), meaning that such materials do not rotate the polarization of plane-polarized light. Although the two enantiomers rotate plane-polarized light in opposite directions, the rotations cancel each other out because they are present in equal amounts of negative (-) counterclockwise (levorotatory) and positive (+) clockwise (dextrorotatory) enantiomers.

In contrast to the two pure enantiomers, which have identical physical properties except for the direction of rotation of plane-polarized light, a racemate sometimes has different properties from either of the pure enantiomers. Different melting points are most common, but different solubilities and boiling points are also possible.

Pharmaceuticals may be available as a racemate or as the pure enantiomer, which might have different potencies. Because biological systems have many chiral asymmetries, pure enantiomers frequently have very different biological effects; examples include glucose and methamphetamine.

Crystallization

There are four ways to crystallize a racemate; three of which H. W. B. Roozeboom had distinguished by 1899:

Conglomerate (sometimes racemic conglomerate)

If the molecules of the substance have a much greater affinity for the same enantiomer than for the opposite one, a mechanical mixture of enantiomerically pure crystals will result. The mixture of enantiomerically pure R and S crystals forms a eutectic mixture. Consequently, the melting point of the conglomerate is always lower than that of the pure enantiomer. Addition of a small amount of one enantiomer to the conglomerate increases the melting point. Roughly 10% of racemic chiral compounds crystallize as conglomerates.

Racemic compound (sometimes true racemate)

If molecules have a greater affinity for the opposite enantiomer than for the same enantiomer, the substance forms a single crystalline phase in which the two enantiomers are present in an ordered 1:1 ratio in the elementary cell. Adding a small amount of one enantiomer to the racemic compound decreases the melting point. But the pure enantiomer can have a higher or lower melting point than the compound. A special case of racemic compounds are kryptoracemic compounds (or kryptoracemates), in which the crystal itself has handedness (is enantiomorphic), despite containing both enantiomorphs in a 1:1 ratio.

Pseudoracemate (sometimes racemic solid solution)

When there is no big difference in affinity between the same and opposite enantiomers, then in contrast to the racemic compound and the conglomerate, the two enantiomers will coexist in an unordered manner in the crystal lattice. Addition of a small amount of one enantiomer changes the melting point slightly or not at all.

Quasiracemate

A quasiracemate is a co-crystal of two similar but distinct compounds, one of which is left-handed and the other right-handed. Although chemically different, they are sterically similar (isosteric) and are still able to form a racemic crystalline phase. One of the first such racemates studied, by Pasteur in 1853, forms from a 1:2 mixture of the bis ammonium salt of (+)-tartaric acid and the bis ammonium salt of (−)-malic acid in water. Re-investigated in 2008, the crystals formed are dumbbell-shape with the central part consisting of ammonium (+)-bitartrate, whereas the outer parts are a quasiracemic mixture of ammonium (+)-bitartrate and ammonium (−)-bimalate.

Resolution

The separation of a racemate into its components, the individual enantiomers, is called a chiral resolution. Various methods exist for this separation, including crystallization, chromatography, and the use of various reagents.

Synthesis

Without a chiral influence (for example a chiral catalyst, solvent or starting material), a chemical reaction that makes a chiral product will always yield a racemate. That can make the synthesis of a racemate cheaper and easier than making the pure enantiomer, because it does not require special conditions. This fact also leads to the question of how biological homochirality evolved on what is presumed to be a racemic primordial earth.

Symmetry considerations

The reagents of, and the reactions that produce, racemic mixtures are said to be "not stereospecific" or "not stereoselective", for their indecision in a particular stereoisomerism. A frequent scenario is that of a planar species (such as an sp² carbon atom or a carbocation intermediate) acting as an electrophile. The nucleophile will have a 50% probability of 'hitting' either of the two sides of the planar grouping, thus producing a racemic mixture:

Achiral precursors lack any inherent asymmetry to favor one enantiomer over the other (no energetic bias), meaning the potential energy surfaces for forming either enantiomer are mirror images. When a reaction pathway proceeds, the laws of statistical thermodynamics ensure that the system explores all microstates equally (no entropic bias). Each enantiomer represents an energetically equivalent microstate, meaning they both have the same likelihood of formation under symmetric conditions. This symmetry ensures that the activation energies and Gibbs free energies for both enantiomers are identical. The partition function, which accounts for all possible states, treats enantiomers equally because their energy levels are degenerate (identical). Consequently, their probabilities of formation are equal.

Kinetic considerations

Considering the reaction kinetically, when an achiral precursor ($A$) transforms into either enantiomer ($L$ or $D$), the process involves passing through transition states. Let’s denote the transition state for forming $L$ as $\ddagger _{L}$ and for $D$ as $\ddagger _{D}$. Because these transition states are enantiomeric (mirror images) and the environment is achiral, their activation energies are identical. Invoking Eyring equation from transition state theory, the rate constant $k_{i}$ for the two competing pathways is equal to:

$k_{L}={\frac {\kappa k_{B}T}{h}}e^{-\Delta G_{L}^{\ddagger }/(RT)}\quad ,\quad k_{D}={\frac {\kappa k_{B}T}{h}}e^{-\Delta G_{D}^{\ddagger }/(RT)}$

where $\Delta G_{i}^{\ddagger }$ is the Gibbs free energy of activation, $R$ is the gas constant, $T$ is the temperature, $k_{B}$ is the Boltzmann constant, $\kappa$ is the transmission coefficient and $h$ is the Planck constant.

Since the activation barriers are equal, the rate constants for forming the two enantiomers are identical:

$\Delta G_{L}^{\ddagger }=\Delta G_{D}^{\ddagger }\Longrightarrow k_{L}=k_{D}$

For this reaction $A\longrightarrow L$, $A\longrightarrow D$, the rates of formation are:

${\frac {d[{\ce {L}}]}{dt}}=k_{L}[{\ce {A}}]\quad ,\quad {\frac {d[{\ce {D}}]}{dt}}=k_{D}[{\ce {A}}]$

where $[A]$ is the concentration of the achiral precursor, while $[L]$ and $[D]$ are the concentrations of the enantiomers.

The integrated rate equations are then:

$[{\ce {L}}]={\frac {k_{L}}{k_{L}+k_{D}}}{\ce {[A]0}}\left(1-e^{-(k_{L}+k_{D})t}\right)\quad ,\quad [{\ce {D}}]={\frac {k_{D}}{k_{L}+k_{D}}}{\ce {[A]0}}\left(1-e^{-(k_{L}+k_{D})t}\right)$

where $[A]_{0}$ is the concentration of the achiral precursor at the beginning of the reaction ($t=0$).

As the two pathways proceed with the same rate constant, both enantiomers accumulate at the same rate leading to equal amounts over time:

${\frac {{\ce {[L]}}}{{\ce {[D]}}}}={\frac {k_{L}}{k_{D}}}=1$

so,

$[{\ce {L}}]=[{\ce {D}}]={\frac {\ce {[A]_{0}}}{2}}\left(1-e^{-2kt}\right)$

The system reaches chemical equilibrium when $t\rightarrow \infty$, there the term $e^{-2kt}$ vanishes:

$[{\ce {L}}]=[{\ce {D}}]={\frac {\ce {[A]_{0}}}{2}}\left(1-e^{-\infty }\right)={\frac {\ce {[A]_{0}}}{2}}\left(1-{\frac {1}{e^{\infty }}}\right)={\frac {\ce {[A]_{0}}}{2}}\left(1-{\frac {1}{\infty }}\right)={\frac {\ce {[A]_{0}}}{2}}\left(1-0\right)={\frac {\ce {[A]_{0}}}{2}}$

As $[A]_{0}$ is 100%, it is thus proven kinetically that at equilibrium the product distribution is 50% / 50%. It can be seen that the rate-determining steps in the formation of stereocenters from an achiral precursor do not favor a particular orientation. The reaction kinetics is said to be mirror-symmetric.

Thermodynamic considerations

Statistical thermodynamics approach: canonical treatment

In statistical thermodynamics, each possible configuration of a system (called a microstate) is weighted by its Boltzmann factor, $\exp(-E_{i}/k_{B}T)$. In the canonical ensemble and according to the Boltzmann distribution, the probability $P_{i}$ of finding the system in a particular microstate $i$ of energy $E_{i}$ is:

$P_{i}={\frac {e^{-E_{i}/(k_{B}T)}}{Z}},\quad Z=\sum _{i}e^{-E_{i}/(k_{B}T)}$

where $E_{i}$ is the energy of the microstate, $k_{B}$ is the Boltzmann constant, $T$ is the temperature and $Z$ is the partition function. In statistical thermodynamics, the partition function encodes the statistical properties of a system in thermodynamic equilibrium. It's a sum over all possible states of the system, weighted by their Boltzmann factors.

Since every microstate of the $L$-enantiomer has a one-to-one mirror counterpart in the $D$-enantiomer of identical energy ($E_{L}=E_{D}$), their partition functions $Z_{i}$ are equal: $Z_{L}=Z_{D}$

Hence, the total partition function of the system is simply: $Z=Z_{L}+Z_{D}$

The probability of forming either enantiomer is thus given by:

$P_{i}={\frac {Z_{i}}{Z}}={\frac {Z_{i}}{2Z_{i}}}={\frac {1}{2}}$

This equality reflects that, in the absence of any external biases (chiral catalysts or chiral fields such as circularly polarized light), the statistical weight (and so the likelihood) of each enantiomer being formed is exactly equal. The system will evolve to maximize entropy, which corresponds to the most probable distribution of microstates. For two equally probable microstates, the most probable distribution is effectively (50:50).

Statistical thermodynamics approach: grand-canonical treatment

The canonical treatment assumes that the system exchanges its average finite energy in contact with a heat bath at constant temperature, while keeping the total number of particles fixed in a given volume. The same results from above can be obtained through the grand-canonical approach, now allowing the system to exchange the number of particles with an external reservoir. The grand-canonical partition function $\Xi$ is given by:

$\Xi =\sum _{N=0}^{\infty }\sum _{i}e^{-\beta (E_{i}-\mu _{i}N_{i})}$

where $N_{i}$ is the number of molecules $i$, $\beta =1/k_{B}T$ and $\mu _{i}$ is the chemical potential of the molecules $i$.

For a single enantiomer $i$ in the grand-canonical ensemble, the partition function can be written considering all possible states with varying numbers of $i$ molecules:

$\Xi _{i}=\sum _{N_{i}=0}^{\infty }e^{\beta \mu _{i}N_{i}}Z_{i}(N_{i})$

where $Z_{i}(N_{i})$ is the canonical partition function for $N_{i}$ molecules of $i$. Considering an ideal solution, the partition function can be factored into molecular partition functions:

$Z_{i}(N_{i})={\frac {(z_{i})^{N_{i}}}{N_{i}!}}$

where $z_{i}$ is the molecular partition function of a single $i$ molecule, accounting for the internal degrees of freedom (rotations, vibrations, etc.).

As before $\Xi _{L}=\Xi _{D}$, therefore:

$\Xi _{i}=\sum _{N_{i}=0}^{\infty }{\frac {(e^{\beta \mu _{i}}z_{i})^{N_{i}}}{N_{i}!}}=\exp(e^{\beta \mu _{i}}z_{i})$

This sum is just the expansion in Taylor series of the exponential function $\exp(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

Since the precursor is achiral and the environment is symmetric, the molecular partition functions for both enantiomers are identical:

$z=z_{L}=z_{D}$

By using the grand-canonical partition function, the average number of molecules of each species can be computed as:

$\langle N_{i}\rangle ={\frac {1}{\beta }}{\frac {\partial \ln \Xi _{i}}{\partial \mu }}={\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\left(e^{\beta \mu }z\right)=e^{\beta \mu }z$

This implies that: $\langle N_{L}\rangle =\langle N_{D}\rangle$

The total partition function for the system is the product of the individual partition functions (assuming they are non-interacting):

$\Xi =\prod _{i}{}\Xi _{i}=\Xi _{L}\Xi _{D}=\exp(2e^{\beta \mu }z)$

The total average number of product molecules and the individual averages for each enantiomer are then:

${\frac {\langle N_{i}\rangle }{\sum _{i}^{}\langle N_{i}\rangle }}={\frac {e^{\beta \mu }z}{2e^{\beta \mu }z}}={\frac {1}{2}}$

This equality arises because, in an achiral environment, there's no thermodynamic reason for one enantiomer to be more stable than the other. In essence, the lack of chirality in the starting materials and the reaction environment means there's no energetic bias to favor one stereoisomeric pathway over the other. The system explores the potential energy surface, and the pathways leading to the two enantiomers are energetically equivalent at the transition state. Statistical probability then dictates that equal numbers of molecules will traverse these equivalent energy barriers, leading to equal amounts (50:50) of both enantiomers.

Classical chemical thermodynamics approach

Macroscopically speaking, the extents of reaction can be defined for $A\longrightarrow L$ as $\xi _{L}$ and for $A\longrightarrow D$ as $\xi _{D}$. The extent of reaction $\xi _{i}$ is a measure of how far a reaction has proceeded towards the products.

Initially, at the beginning of the reaction,

$n_{A}^{0}=n_{0}$

$n_{L}^{0}=n_{D}^{0}=0$

At equilibrium,

$n_{A}=n_{0}-\xi _{L}-\xi _{D}$

$n_{L}=\xi _{L}$

$n_{D}=\xi _{D}$

The molar fractions are defined by using the total number of moles $n_{0}$, as:

$x_{A}={\frac {n_{0}-\xi _{L}-\xi _{D}}{n_{0}}}$

$x_{L}={\frac {\xi _{L}}{n_{0}}}$

$x_{D}={\frac {\xi _{D}}{n_{0}}}$

The equilibrium condition can be derived by using the chemical potential $\mu _{i}$ of species $i$

$\mu _{i}=\mu _{i}^{0}+RT\ln x_{i}$

For both pathways,

$\mu _{L}-\mu _{A}=(\mu _{L}^{0}-\mu _{A}^{0})+RT\ln {\frac {x_{L}}{x_{A}}}=0\quad ,\quad \mu _{D}-\mu _{A}=(\mu _{D}^{0}-\mu _{A}^{0})+RT\ln {\frac {x_{D}}{x_{A}}}=0$

${\frac {x_{L}}{x_{A}}}=\exp \left(-{\frac {\Delta G_{L}^{0}}{RT}}\right)=K_{L}\quad ,\quad {\frac {x_{D}}{x_{A}}}=\exp \left(-{\frac {\Delta G_{D}^{0}}{RT}}\right)=K_{D}$

Since $L$ and $D$ are enantiomers, $\mu _{L}^{0}=\mu _{D}^{0}$ , so ${\Delta G_{L-A}^{0}}={\Delta G_{D-A}^{0}}$ , and thus:

${\frac {x_{L}}{x_{A}}}=K_{L}={\frac {x_{D}}{x_{A}}}=K_{D}=K$

This implies that $x_{L}=x_{D}$

Applying the equilibrium condition and solving for the extent of reaction,

Thus, the enantiomers constitute 50% each of the product mixture, regardless of the value of K. The equilibrium constant K determines how much A is converted to products, but the symmetry ensures that L and D are always produce in equal amounts.

The key equality (chemical potentials estandars iguales) ensures K1=K2. This symmetry means the system has no prference for forming L over D or vicevers, leading to extensiones iguales. The (50:50) proportion among the enantiomers holds because the equilibrium constants are identical and the stoihciometric coefficientd balance the production evenly.

This theoretical framework holds under conditions where the reaction system is free from external chiral influences. Once a chiral perturbation is introduced, the inherent symmetry is broken. The activation energies are altered, favoring one reaction pathway over the other and thus leading to enantiomeric excess. This is the cornerstone behind asymmetric synthesis and enantioselective catalysis.

Racemic pharmaceuticals

See also: Enantiopure drug

Some drug molecules are chiral, and the enantiomers have different effects on biological entities. They can be sold as one enantiomer or as a racemic mixture. Examples include thalidomide, ibuprofen, cetirizine and salbutamol. A well known drug that has different effects depending on its ratio of enantiomers is amphetamine. Adderall is an unequal mixture of both amphetamine enantiomers. A single Adderall dose combines the neutral sulfate salts of dextroamphetamine and amphetamine, with the dextro isomer of amphetamine saccharate and D/L-amphetamine aspartate monohydrate. The original Benzedrine was a racemic mixture, and isolated dextroamphetamine was later introduced to the market as Dexedrine. The prescription analgesic tramadol is also a racemate.

In some cases (e.g., ibuprofen and thalidomide), the enantiomers interconvert or racemize in vivo. This means that preparing a pure enantiomer for medication is largely pointless. However, sometimes samples containing pure enantiomers may be made and sold at a higher cost in cases where the use requires specifically one isomer (e.g., for a stereospecific reagent); compare omeprazole and esomeprazole. Moving from a racemic drug to a chiral specific drug may be done for a better safety profile or an improved therapeutic index. This process is called chiral switching and the resulting enantiopure drug is called a chiral switch. As examples, esomeprazole is a chiral switch of (±)-omeprazole and levocetirizine is a chiral switch of (±)-cetirizine.

While often only one enantiomer of the drug may be active, there are cases in which the other enantiomer is harmful, like salbutamol and thalidomide. The (R) enantiomer of thalidomide is effective against morning sickness, while the (S) enantiomer is teratogenic, causing birth defects. Since the drug racemizes, the drug cannot be considered safe for use by women of child-bearing age, and its use is tightly controlled when used for treating other illness.

Methamphetamine is available by prescription under the brand name Desoxyn. The active component of Desoxyn is dextromethamphetamine hydrochloride. This is the right-handed isomer of methamphetamine. The left-handed isomer of methamphetamine, levomethamphetamine, is an OTC drug that is less centrally-acting and more peripherally-acting. Methedrine during the 20th century was a 50:50 racemic mixture of both methamphetamine isomers (levo and dextro).

Wallach's rule

Wallach's rule (first proposed by Otto Wallach) states that racemic crystals tend to be denser than their chiral counterparts. This rule has been substantiated by crystallographic database analysis.

See also

• Chiral switch
• Chirality (same as optical isomerism)
• Descriptor (chemistry)
• Racemic (protein) crystallography
• Racemization
• Skeletal formula § Stereochemistry which describes how stereochemistry is denoted in skeletal formulae