Chemical thermodynamics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Chemical_thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measurements of various thermodynamic properties, but also the application of mathematical methods to the study of chemical questions and the spontaneity of processes.

The structure of chemical thermodynamics is based on the first two laws of thermodynamics. Starting from the first and second laws of thermodynamics, four equations called the "fundamental equations of Gibbs" can be derived. From these four, a multitude of equations, relating the thermodynamic properties of the thermodynamic system can be derived using relatively simple mathematics. This outlines the mathematical framework of chemical thermodynamics.

History

J. Willard Gibbs - founder of chemical thermodynamics

In 1865, the German physicist Rudolf Clausius, in his Mechanical Theory of Heat, suggested that the principles of thermochemistry, e.g. the heat evolved in combustion reactions, could be applied to the principles of thermodynamics. Building on the work of Clausius, between the years 1873-76 the American mathematical physicist Willard Gibbs published a series of three papers, the most famous one being the paper On the Equilibrium of Heterogeneous Substances. In these papers, Gibbs showed how the first two laws of thermodynamics could be measured graphically and mathematically to determine both the thermodynamic equilibrium of chemical reactions as well as their tendencies to occur or proceed. Gibbs’ collection of papers provided the first unified body of thermodynamic theorems from the principles developed by others, such as Clausius and Sadi Carnot.

During the early 20th century, two major publications successfully applied the principles developed by Gibbs to chemical processes and thus established the foundation of the science of chemical thermodynamics. The first was the 1923 textbook Thermodynamics and the Free Energy of Chemical Substances by Gilbert N. Lewis and Merle Randall. This book was responsible for supplanting the chemical affinity with the term free energy in the English-speaking world. The second was the 1933 book Modern Thermodynamics by the methods of Willard Gibbs written by E. A. Guggenheim. In this manner, Lewis, Randall, and Guggenheim are considered as the founders of modern chemical thermodynamics because of the major contribution of these two books in unifying the application of thermodynamics to chemistry.

Overview

The primary objective of chemical thermodynamics is the establishment of a criterion for determination of the feasibility or spontaneity of a given transformation. In this manner, chemical thermodynamics is typically used to predict the energy exchanges that occur in the following processes:

1. Chemical reactions
2. Phase changes
3. The formation of solutions

The following state functions are of primary concern in chemical thermodynamics:

• Internal energy (U)
• Enthalpy (H)
• Entropy (S)
• Gibbs free energy (G)

Most identities in chemical thermodynamics arise from application of the first and second laws of thermodynamics, particularly the law of conservation of energy, to these state functions.

The three laws of thermodynamics (global, unspecific forms):

1. The energy of the universe is constant.

2. In any spontaneous process, there is always an increase in entropy of the universe.

3. The entropy of a perfect crystal (well ordered) at 0 Kelvin is zero.

Chemical energy

Main article: Chemical energy

Chemical energy is the energy that can be released when chemical substances undergo a transformation through a chemical reaction. Breaking and making chemical bonds involves energy release or uptake, often as heat that may be either absorbed by or evolved from the chemical system.

Energy released (or absorbed) because of a reaction between chemical substances ("reactants") is equal to the difference between the energy content of the products and the reactants. This change in energy is called the change in internal energy of a chemical system. It can be calculated from ${\displaystyle {\mathrm{\Delta }}_{\mathrm{f}}{U}_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{s}}^{\mathrm{o}}}$, the internal energy of formation of the reactant molecules related to the bond energies of the molecules under consideration, and ${\displaystyle {\mathrm{\Delta }}_{\mathrm{f}}{U}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{s}}^{\mathrm{o}}}$, the internal energy of formation of the product molecules. The change in internal energy is equal to the heat change if it is measured under conditions of constant volume (at STP condition), as in a closed rigid container such as a bomb calorimeter. However, at constant pressure, as in reactions in vessels open to the atmosphere, the measured heat is usually not equal to the internal energy change, because pressure-volume work also releases or absorbs energy. (The heat change at constant pressure is called the enthalpy change; in this case the widely tabulated enthalpies of formation are used.)

A related term is the heat of combustion, which is the chemical energy released due to a combustion reaction and of interest in the study of fuels. Food is similar to hydrocarbon and carbohydrate fuels, and when it is oxidized, its energy release is similar (though assessed differently than for a hydrocarbon fuel — see food energy).

In chemical thermodynamics, the term used for the chemical potential energy is chemical potential, and sometimes the Gibbs-Duhem equation is used.

Chemical reactions

Main article: Chemical reaction

In most cases of interest in chemical thermodynamics there are internal degrees of freedom and processes, such as chemical reactions and phase transitions, which create entropy in the universe unless they are at equilibrium or are maintained at a "running equilibrium" through "quasi-static" changes by being coupled to constraining devices, such as pistons or electrodes, to deliver and receive external work. Even for homogeneous "bulk" systems, the free-energy functions depend on the composition, as do all the extensive thermodynamic potentials, including the internal energy. If the quantities { N_i }, the number of chemical species, are omitted from the formulae, it is impossible to describe compositional changes.

Gibbs function or Gibbs Energy

For an unstructured, homogeneous "bulk" system, there are still various extensive compositional variables { N_i } that G depends on, which specify the composition (the amounts of each chemical substance, expressed as the numbers of molecules present or the numbers of moles). Explicitly,

${\displaystyle G=G(T,P,\{N_i\}).}$

For the case where only PV work is possible,

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

a restatement of the fundamental thermodynamic relation, in which μ_i is the chemical potential for the i-th component in the system

${\displaystyle {\mu }_i={\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }{N}_i}\right)}_{T,P,N_{j\ne i},etc.}.}$

The expression for dG is especially useful at constant T and P, conditions, which are easy to achieve experimentally and which approximate the conditions in living creatures

${\displaystyle (\mathrm{d}G{)}_{T,P}=\sum _i{\mu }_i\mathrm{d}{N}_i.}$

Chemical affinity

Main article: Chemical affinity

While this formulation is mathematically defensible, it is not particularly transparent since one does not simply add or remove molecules from a system. There is always a process involved in changing the composition; e.g., a chemical reaction (or many), or movement of molecules from one phase (liquid) to another (gas or solid). We should find a notation which does not seem to imply that the amounts of the components ( N_i ) can be changed independently. All real processes obey conservation of mass, and in addition, conservation of the numbers of atoms of each kind.

Consequently, we introduce an explicit variable to represent the degree of advancement of a process, a progress variable ξ for the extent of reaction (Prigogine & Defay, p. 18; Prigogine, pp. 4–7; Guggenheim, p. 37.62), and to the use of the partial derivative ∂G/∂ξ (in place of the widely used "ΔG", since the quantity at issue is not a finite change). The result is an understandable expression for the dependence of dG on chemical reactions (or other processes). If there is just one reaction

${\displaystyle (\mathrm{d}G{)}_{T,P}={\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }\xi }\right)}_{T,P}\mathrm{d}\xi .}$

If we introduce the stoichiometric coefficient for the i-th component in the reaction

${\displaystyle {\nu }_i=\mathrm{\partial }{N}_i/\mathrm{\partial }\xi }$

(negative for reactants), which tells how many molecules of i are produced or consumed, we obtain an algebraic expression for the partial derivative

${\displaystyle {\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }\xi }\right)}_{T,P}=\sum _i{\mu }_i{\nu }_i=-\mathbb{A}}$

where we introduce a concise and historical name for this quantity, the "affinity", symbolized by A, as introduced by Théophile de Donder in 1923.(De Donder; Progogine & Defay, p. 69; Guggenheim, pp. 37, 240) The minus sign ensures that in a spontaneous change, when the change in the Gibbs free energy of the process is negative, the chemical species have a positive affinity for each other. The differential of G takes on a simple form that displays its dependence on composition change

${\displaystyle (\mathrm{d}G{)}_{T,P}=-\mathbb{A}d\xi .}$

If there are a number of chemical reactions going on simultaneously, as is usually the case,

${\displaystyle (\mathrm{d}G{)}_{T,P}=-\sum _k{\mathbb{A}}_{k}d{\xi }_k.}$

with a set of reaction coordinates { ξ_j }, avoiding the notion that the amounts of the components ( N_i ) can be changed independently. The expressions above are equal to zero at thermodynamic equilibrium, while they are negative when chemical reactions proceed at a finite rate, producing entropy. This can be made even more explicit by introducing the reaction rates dξ_j/dt. For every physically independent process (Prigogine & Defay, p. 38; Prigogine, p. 24)

${\displaystyle \mathbb{A}\dot{\xi }\le 0.}$

This is a remarkable result since the chemical potentials are intensive system variables, depending only on the local molecular milieu. They cannot "know" whether temperature and pressure (or any other system variables) are going to be held constant over time. It is a purely local criterion and must hold regardless of any such constraints. Of course, it could have been obtained by taking partial derivatives of any of the other fundamental state functions, but nonetheless is a general criterion for (−T times) the entropy production from that spontaneous process; or at least any part of it that is not captured as external work. (See Constraints below.)

We now relax the requirement of a homogeneous "bulk" system by letting the chemical potentials and the affinity apply to any locality in which a chemical reaction (or any other process) is occurring. By accounting for the entropy production due to irreversible processes, the equality for dG is now replaced by

${\displaystyle \mathrm{d}G=-S\mathrm{d}T+V\mathrm{d}P-\sum _k{\mathbb{A}}_k\mathrm{d}{\xi }_k+\delta W^{\prime }}$

or

${\displaystyle \mathrm{d}{G}_{T,P}=-\sum _k{\mathbb{A}}_k\mathrm{d}{\xi }_k+\delta W^{\prime }.}$

Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as T times a corresponding increase in the entropy of the system and its surrounding. Or it may go partly toward doing external work and partly toward creating entropy. The important point is that the extent of reaction for a chemical reaction may be coupled to the displacement of some external mechanical or electrical quantity in such a way that one can advance only if the other also does. The coupling may occasionally be rigid, but it is often flexible and variable.

Solutions

In solution chemistry and biochemistry, the Gibbs free energy decrease (∂G/∂ξ, in molar units, denoted cryptically by ΔG) is commonly used as a surrogate for (−T times) the global entropy produced by spontaneous chemical reactions in situations where no work is being done; or at least no "useful" work; i.e., other than perhaps ± P dV. The assertion that all spontaneous reactions have a negative ΔG is merely a restatement of the second law of thermodynamics, giving it the physical dimensions of energy and somewhat obscuring its significance in terms of entropy. When no useful work is being done, it would be less misleading to use the Legendre transforms of the entropy appropriate for constant T, or for constant T and P, the Massieu functions −F/T and −G/T, respectively.

Non-equilibrium

Main article: non-equilibrium thermodynamics

Generally the systems treated with the conventional chemical thermodynamics are either at equilibrium or near equilibrium. Ilya Prigogine developed the thermodynamic treatment of open systems that are far from equilibrium. In doing so he has discovered phenomena and structures of completely new and completely unexpected types. His generalized, nonlinear and irreversible thermodynamics has found surprising applications in a wide variety of fields.

The non-equilibrium thermodynamics has been applied for explaining how ordered structures e.g. the biological systems, can develop from disorder. Even if Onsager's relations are utilized, the classical principles of equilibrium in thermodynamics still show that linear systems close to equilibrium always develop into states of disorder which are stable to perturbations and cannot explain the occurrence of ordered structures.

Prigogine called these systems dissipative systems, because they are formed and maintained by the dissipative processes which take place because of the exchange of energy between the system and its environment and because they disappear if that exchange ceases. They may be said to live in symbiosis with their environment.

The method which Prigogine used to study the stability of the dissipative structures to perturbations is of very great general interest. It makes it possible to study the most varied problems, such as city traffic problems, the stability of insect communities, the development of ordered biological structures and the growth of cancer cells to mention but a few examples.

System constraints

In this regard, it is crucial to understand the role of walls and other constraints, and the distinction between independent processes and coupling. Contrary to the clear implications of many reference sources, the previous analysis is not restricted to homogeneous, isotropic bulk systems which can deliver only PdV work to the outside world, but applies even to the most structured systems. There are complex systems with many chemical "reactions" going on at the same time, some of which are really only parts of the same, overall process. An independent process is one that could proceed even if all others were unaccountably stopped in their tracks. Understanding this is perhaps a "thought experiment" in chemical kinetics, but actual examples exist.

A gas-phase reaction at constant temperature and pressure which results in an increase in the number of molecules will lead to an increase in volume. Inside a cylinder closed with a piston, it can proceed only by doing work on the piston. The extent variable for the reaction can increase only if the piston moves out, and conversely if the piston is pushed inward, the reaction is driven backwards.

Similarly, a redox reaction might occur in an electrochemical cell with the passage of current through a wire connecting the electrodes. The half-cell reactions at the electrodes are constrained if no current is allowed to flow. The current might be dissipated as Joule heating, or it might in turn run an electrical device like a motor doing mechanical work. An automobile lead-acid battery can be recharged, driving the chemical reaction backwards. In this case as well, the reaction is not an independent process. Some, perhaps most, of the Gibbs free energy of reaction may be delivered as external work.

The hydrolysis of ATP to ADP and phosphate can drive the force-times-distance work delivered by living muscles, and synthesis of ATP is in turn driven by a redox chain in mitochondria and chloroplasts, which involves the transport of ions across the membranes of these cellular organelles. The coupling of processes here, and in the previous examples, is often not complete. Gas can leak slowly past a piston, just as it can slowly leak out of a rubber balloon. Some reaction may occur in a battery even if no external current is flowing. There is usually a coupling coefficient, which may depend on relative rates, which determines what percentage of the driving free energy is turned into external work, or captured as "chemical work", a misnomer for the free energy of another chemical process.

Transcendental number

From Wikipedia, the free encyclopedia

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

In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. The quality of a number being transcendental is called transcendence.

Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set.

All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic non-rational, and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x² − 2 = 0. The golden ratio (denoted ${\displaystyle \phi }$ or ${\displaystyle \varphi }$) is another irrational number that is not transcendental, as it is a root of the polynomial equation x² − x − 1 = 0.

History

The name "transcendental" comes from Latin trānscendere 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x. Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the modern sense.

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch proof that π is transcendental.

Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant

$${\displaystyle \begin{array}{rl}{L}_b& =\sum _{n=1}^{\mathrm{\infty }}{10}^{-n!}\\ & ={10}^{-1}+{10}^{-2}+{10}^{-6}+{10}^{-24}+{10}^{-120}+{10}^{-720}+{10}^{-5040}+{10}^{-40320}+\dots \\ & =0.\mathbf{\text{1}}\mathbf{\text{1}}000\mathbf{\text{1}}00000000000000000\mathbf{\text{1}}00000000000000000000000000000000000000000000000000000\dots \end{array}}$$

in which the nth digit after the decimal point is 1 if n is equal to k! (k factorial) for some k and 0 otherwise. In other words, the nth digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the Liouville numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental.

The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873.

In 1874 Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers. Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers.

In 1882 Ferdinand von Lindemann published the first complete proof that π is transcendental. He first proved that e^a is transcendental if a is a non-zero algebraic number. Then, since e^iπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, π must therefore be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of π implies that geometric constructions involving compass and straightedge only cannot produce certain results, for example squaring the circle.

In 1900 David Hilbert posed a question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is a^b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).

Properties

A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number is the root of an integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable.

No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals.

Applying any non-constant single-variable algebraic function to a transcendental argument yields a transcendental value. For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as ${\displaystyle 5\pi }$, ${\displaystyle \frac{\pi -3}{\sqrt{2}}}$, ${\displaystyle (\sqrt{\pi }-\sqrt{3}{)}^8}$, and ${\displaystyle \sqrt[4]{{\pi }^5+7}}$ are transcendental as well.

However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent. For example, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether e + π, for example, is transcendental, though at least one of e + π and eπ must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x² − (a + b) x + a b . If (a + b) and a b were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.

The non-computable numbers are a strict subset of the transcendental numbers.

All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion. Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.

Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that π is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms, that have a "simple" structure, and that are not eventually periodic are transcendental (in other words, algebraic irrational roots of at least third degree polynomials do not have apparent pattern in their continued fraction expansions, since eventually periodic continued fractions correspond to quadratic irrationals, see Hermite's problem).

Numbers proven to be transcendental

Numbers proven to be transcendental:

• π (by the Lindemann–Weierstrass theorem).
• ${\displaystyle e^a}$ if ${\displaystyle a}$ is algebraic and nonzero (by the Lindemann–Weierstrass theorem), in particular Euler's number e.
• ${\displaystyle e^{\pi \sqrt{n}}}$ where ${\displaystyle n}$ is a positive integer; in particular Gelfond's constant ${\displaystyle e^{\pi }}$ (by the Gelfond–Schneider theorem).
• Algebraic combinations of ${\displaystyle \pi }$ and ${\displaystyle e^{\pi \sqrt{n}},n\in {\mathbb{Z}}^{+}}$ such as ${\displaystyle \pi +e^{\pi }}$ and ${\displaystyle \pi e^{\pi }}$ (following from their algebraic independence).
• ${\displaystyle a^b}$ where ${\displaystyle a}$ is algebraic but not 0 or 1, and ${\displaystyle b}$ is irrational algebraic, in particular the Gelfond–Schneider constant ${\displaystyle 2^{\sqrt{2}}}$ (by the Gelfond–Schneider theorem).
• The natural logarithm ${\displaystyle \ln (a)}$ if ${\displaystyle a}$ is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
• ${\displaystyle {\log }_b(a)}$ if ${\displaystyle a}$ and ${\displaystyle b}$ are positive integers not both powers of the same integer, and ${\displaystyle a}$ is not equal to 1 (by the Gelfond–Schneider theorem).
• All numbers of the form ${\displaystyle \pi +{\beta }_1\ln (a_1)+\cdots +{\beta }_n\ln (a_n)}$ are transcendental, where ${\displaystyle {\beta }_j}$ are algebraic for all ${\displaystyle 1\le j\le n}$ and ${\displaystyle a_j}$ are non-zero algebraic for all ${\displaystyle 1\le j\le n}$ (by Baker's theorem).

• The trigonometric functions ${\displaystyle \sin (x),\cos (x),...}$ and their hyperbolic counterparts, for any nonzero algebraic number ${\displaystyle x}$, expressed in radians (by the Lindemann–Weierstrass theorem).
• Non-zero results of the inverse trigonometric functions ${\displaystyle \arcsin (x),\arccos (x),...}$ and their hyperbolic counterparts, for any algebraic number ${\displaystyle x}$ (by the Lindemann–Weierstrass theorem).
• ${\displaystyle {\pi }^{-1}\arctan (x)}$, for rational ${\displaystyle x}$ such that ${\displaystyle x\notin \{0,\pm 1\}}$.
• The fixed point of the cosine function (also referred to as the Dottie number ${\displaystyle d}$) – the unique real solution to the equation ${\displaystyle \cos (x)=x}$, where ${\displaystyle x}$ is in radians (by the Lindemann–Weierstrass theorem).
• ${\displaystyle W(a)}$ if ${\displaystyle a}$ is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular the omega constant Ω.
• ${\displaystyle W(r,a)}$ if both ${\displaystyle a}$ and the order ${\displaystyle r}$ are algebraic such that ${\displaystyle a\ne 0}$, for any branch of the generalized Lambert W function.
• ${\displaystyle {\sqrt{x}}_s}$, the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem).
• Values of the gamma function of rational numbers that are of the form ${\displaystyle \mathrm{\Gamma }(n/2),\mathrm{\Gamma }(n/3),\mathrm{\Gamma }(n/4)}$ or ${\displaystyle \mathrm{\Gamma }(n/6)}$.
• Algebraic combinations of ${\displaystyle \pi }$ and ${\displaystyle \mathrm{\Gamma }(1/3)}$ or of ${\displaystyle \pi }$ and ${\displaystyle \mathrm{\Gamma }(1/4)}$ such as the lemniscate constant ${\displaystyle \varpi }$ (following from their respective algebraic independences).
• The values of Beta function ${\displaystyle \mathrm{B}(a,b)}$ if ${\displaystyle a,b}$ and ${\displaystyle a+b}$ are non-integer rational numbers.
• The Bessel function of the first kind ${\displaystyle J_{\nu }(x)}$, its first derivative, and the quotient ${\displaystyle \frac{J_{\nu }^{\prime }(x)}{J_{\nu }(x)}}$ are transcendental when ${\displaystyle \nu }$ is rational and ${\displaystyle x}$ is algebraic and nonzero, and all nonzero roots of ${\displaystyle J_{\nu }(x)}$ and ${\displaystyle J_{\nu }^{\prime }(x)}$ are transcendental when ${\displaystyle \nu }$ is rational.
• The number ${\displaystyle \frac{\pi }{2}\frac{Y_0(2)}{J_0(2)}-\gamma }$, where ${\displaystyle Y_{\alpha }(x)}$ and ${\displaystyle J_{\alpha }(x)}$ are Bessel functions and ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.
• Any Liouville number, in particular: Liouville's constant.
• Numbers with large irrationality measure, such as the Champernowne constant ${\displaystyle C_{10}}$ (by Roth's theorem).
• Numbers artificially constructed not to be algebraic periods.
• Any non-computable number, in particular: Chaitin's constant.
• Constructed irrational numbers which are not simply normal in any base.
• Any number for which the digits with respect to some fixed base form a Sturmian word.
• The Prouhet–Thue–Morse constant and the related rabbit constant.
• The Komornik–Loreti constant.
• The paperfolding constant (also named as "Gaussian Liouville number").
• The values of the infinite series with fast convergence rate as defined by Y. Gao and J. Gao, such as ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{3^n}{2^{3^n}}}$.
• Numbers of the form ${\displaystyle \sum _{k=0}^{\mathrm{\infty }}{10}^{-b^k}}$ and ${\displaystyle \sum _{k=0}^{\mathrm{\infty }}{10}^{-\lfloor b^k\rfloor }}$ For b > 1 where ${\displaystyle b\mapsto \lfloor b\rfloor }$ is the floor function.

• Any number of the form ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{E_n({\beta }^{r^n})}{F_n({\beta }^{r^n})}}$ (where ${\displaystyle E_n(z)}$, ${\displaystyle F_n(z)}$ are polynomials in variables ${\displaystyle n}$ and ${\displaystyle z}$, ${\displaystyle \beta }$ is algebraic and ${\displaystyle \beta \ne 0}$, ${\displaystyle r}$ is any integer greater than 1).
• The numbers ${\displaystyle \alpha =\mathrm{3.3003300000...}}$ and ${\displaystyle {\alpha }^{-1}=\mathrm{0.3030000030...}}$ with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.
• The values of the Rogers-Ramanujan continued fraction ${\displaystyle R(q)}$ where ${\displaystyle q\in \mathbb{C}}$ is algebraic and ${\displaystyle 0<|q|<1}$. The lemniscatic values of theta function ${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}}$ (under the same conditions for ${\displaystyle q}$) are also transcendental.
• j(q) where ${\displaystyle q\in \mathbb{C}}$ is algebraic but not imaginary quadratic (i.e, the exceptional set of this function is the number field whose degree of extension over ${\displaystyle \mathbb{Q}}$ is 2).
• The constants ${\displaystyle {ϵ}_k}$ and ${\displaystyle {\nu }_k}$ in the formula for first index of occurrence of Gijswijt's sequence, where k is any integer greater than 1.

Conjectured transcendental numbers

Numbers which have yet to be proven to be either transcendental or algebraic:

• Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational: eπ, e + π, π^π, e^e, π^e, π^√2, e^π2. It has been shown that both e + π and π/e do not satisfy any polynomial equation of degree ${\displaystyle \le 8}$ and integer coefficients of average size 10⁹. At least one of the numbers e^e and e^e2 is transcendental. Schanuel's conjecture would imply that all of the above numbers are transcendental and algebraically independent.
• The Euler–Mascheroni constant γ: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes γ/4) contains at most one algebraic number. In 2012 it was shown that at least one of γ and the Gompertz constant δ is transcendental.
• The values of the Riemann zeta function ζ(n) at odd positive integers ${\displaystyle n\ge 3}$; in particular Apéry's constant ζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.
• The values of the Dirichlet beta function β(n) at even positive integers ${\displaystyle n\ge 2}$; in particular Catalan's Constant β(2). (none of them are known to be irrational).
• Values of the Gamma Function Γ(1/n) for positive integers ${\displaystyle n=5}$ and ${\displaystyle n\ge 7}$ are not known to be irrational, let alone transcendental. For ${\displaystyle n\ge 2}$ at least one the numbers Γ(1/n) and Γ(2/n) is transcendental.
• Any number given by some kind of limit that is not obviously algebraic.

Proofs for specific numbers

A proof that e is transcendental

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following:

Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients c₀, c₁, ..., c_n satisfying the equation: $${\displaystyle c_0+c_{1}e+c_2{e}^2+\cdots +c_n{e}^n=0,c_0,c_n\ne 0.}$$ It is difficult to make use of the integer status of these coefficients when multiplied by a power of the irrational e, but we can absorb those powers into an integral which “mostly” will assume integer values. For a positive integer k, define the polynomial $${\displaystyle f_k(x)=x^k{\left[(x-1)\cdots (x-n)\right]}^{k+1},}$$ and multiply both sides of the above equation by $${\displaystyle {\int }_0^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x,}$$ to arrive at the equation: $${\displaystyle c_0\left({\int }_0^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)+c_{1}e\left({\int }_0^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)+\cdots +c_n{e}^n\left({\int }_0^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)=0.}$$

By splitting respective domains of integration, this equation can be written in the form $${\displaystyle P+Q=0}$$ where $${\displaystyle \begin{array}{rl}P& =c_0\left({\int }_0^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)+c_{1}e\left({\int }_1^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)+c_2{e}^2\left({\int }_2^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)+\cdots +c_n{e}^n\left({\int }_n^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x\right)\\ Q& =c_{1}e\left({\int }_0^1{f}_k(x)e^{-x}\mathrm{d}x\right)+c_2{e}^2\left({\int }_0^2{f}_k(x)e^{-x}\mathrm{d}x\right)+\cdots +c_n{e}^n\left({\int }_0^n{f}_k(x)e^{-x}\mathrm{d}x\right)\end{array}}$$ Here P will turn out to be an integer, but more importantly it grows quickly with k.

Lemma 1

There are arbitrarily large k such that ${\displaystyle \frac{P}{k!}}$ is a non-zero integer.

Proof. Recall the standard integral (case of the Gamma function) $${\displaystyle {\int }_0^{\mathrm{\infty }}{t}^j{e}^{-t}\mathrm{d}t=j!}$$ valid for any natural number ${\displaystyle j}$. More generally,

if ${\displaystyle g(t)=\sum _{j=0}^m{b}_j{t}^j}$ then ${\displaystyle {\int }_0^{\mathrm{\infty }}g(t)e^{-t}\mathrm{d}t=\sum _{j=0}^m{b}_{j}j!}$.

This would allow us to compute ${\displaystyle P}$ exactly, because any term of ${\displaystyle P}$ can be rewritten as $${\displaystyle c_a{e}^a{\int }_a^{\mathrm{\infty }}{f}_k(x)e^{-x}\mathrm{d}x=c_a{\int }_a^{\mathrm{\infty }}{f}_k(x)e^{-(x-a)}\mathrm{d}x=\left\{\begin{array}{rl}t& =x-a\\ x& =t+a\\ \mathrm{d}x& =\mathrm{d}t\end{array}\right\}=c_a{\int }_0^{\mathrm{\infty }}{f}_k(t+a)e^{-t}\mathrm{d}t}$$ through a change of variables. Hence $${\displaystyle P=\sum _{a=0}^n{c}_a{\int }_0^{\mathrm{\infty }}{f}_k(t+a)e^{-t}\mathrm{d}t={\int }_0^{\mathrm{\infty }}(\sum _{a=0}^n{c}_a{f}_k(t+a))e^{-t}\mathrm{d}t}$$ That latter sum is a polynomial in ${\displaystyle t}$ with integer coefficients, i.e., it is a linear combination of powers ${\displaystyle t^j}$ with integer coefficients. Hence the number ${\displaystyle P}$ is a linear combination (with those same integer coefficients) of factorials ${\displaystyle j!}$; in particular ${\displaystyle P}$ is an integer.

Smaller factorials divide larger factorials, so the smallest ${\displaystyle j!}$ occurring in that linear combination will also divide the whole of ${\displaystyle P}$. We get that ${\displaystyle j!}$ from the lowest power ${\displaystyle t^j}$ term appearing with a nonzero coefficient in ${\displaystyle \sum _{a=0}^n{c}_a{f}_k(t+a)}$, but this smallest exponent ${\displaystyle j}$ is also the multiplicity of ${\displaystyle t=0}$ as a root of this polynomial. ${\displaystyle f_k(x)}$ is chosen to have multiplicity ${\displaystyle k}$ of the root ${\displaystyle x=0}$ and multiplicity ${\displaystyle k+1}$ of the roots ${\displaystyle x=a}$ for ${\displaystyle a=1,\dots ,n}$, so that smallest exponent is ${\displaystyle t^k}$ for ${\displaystyle f_k(t)}$ and ${\displaystyle t^{k+1}}$ for ${\displaystyle f_k(t+a)}$ with ${\displaystyle a>0}$. Therefore ${\displaystyle k!}$ divides ${\displaystyle P}$.

To establish the last claim in the lemma, that ${\displaystyle P}$ is nonzero, it is sufficient to prove that ${\displaystyle k+1}$ does not divide ${\displaystyle P}$. To that end, let ${\displaystyle k+1}$ be any prime larger than ${\displaystyle n}$ and ${\displaystyle |c_0|}$. We know from the above that ${\displaystyle (k+1)!}$ divides each of ${\displaystyle c_a{\int }_0^{\mathrm{\infty }}{f}_k(t+a)e^{-t}\mathrm{d}t}$ for ${\displaystyle 1⩽a⩽n}$, so in particular all of those are divisible by ${\displaystyle k+1}$. It comes down to the first term ${\displaystyle c_0{\int }_0^{\mathrm{\infty }}{f}_k(t)e^{-t}\mathrm{d}t}$. We have (see falling and rising factorials) $${\displaystyle f_k(t)=t^k[(t-1)\cdots (t-n){]}^{k+1}=[(-1{)}^n(n!){]}^{k+1}{t}^k+\text{higher degree terms}}$$ and those higher degree terms all give rise to factorials ${\displaystyle (k+1)!}$ or larger. Hence $${\displaystyle P\equiv c_0{\int }_0^{\mathrm{\infty }}{f}_k(t)e^{-t}\mathrm{d}t\equiv c_0[(-1{)}^n(n!){]}^{k+1}k!(\mathrm{mod}(k+1))}$$ That right hand side is a product of nonzero integer factors less than the prime ${\displaystyle k+1}$, therefore that product is not divisible by ${\displaystyle k+1}$, and the same holds for ${\displaystyle P}$; in particular ${\displaystyle P}$ cannot be zero.

Lemma 2

For sufficiently large k, ${\displaystyle \left|\frac{Q}{k!}\right|<1}$.

Proof. Note that

$${\displaystyle \begin{array}{rl}{f}_k{e}^{-x}& =x^k{\left[(x-1)(x-2)\cdots (x-n)\right]}^{k+1}{e}^{-x}\\ & ={\left(x(x-1)\cdots (x-n)\right)}^k\cdot \left((x-1)\cdots (x-n)e^{-x}\right)\\ & =u(x{)}^k\cdot v(x)\end{array}}$$

where u(x), v(x) are continuous functions of x for all x, so are bounded on the interval [0, n]. That is, there are constants G, H > 0 such that

$${\displaystyle \left|f_k{e}^{-x}\right|\le |u(x){|}^k\cdot |v(x)|<G^{k}H\text{ for }0\le x\le n.}$$

So each of those integrals composing Q is bounded, the worst case being

$${\displaystyle \left|{\int }_0^n{f}_k{e}^{-x}\mathrm{d}x\right|\le {\int }_0^n\left|f_k{e}^{-x}\right|\mathrm{d}x\le {\int }_0^n{G}^{k}H\mathrm{d}x=n{G}^{k}H.}$$

It is now possible to bound the sum Q as well:

$${\displaystyle |Q|<G^k\cdot nH\left(|c_1|e+|c_2|e^2+\cdots +|c_n|e^n\right)=G^k\cdot M,}$$

where M is a constant not depending on k. It follows that

$${\displaystyle \left|\frac{Q}{k!}\right|<M\cdot \frac{G^k}{k!}\to 0\text{ as }k\to \mathrm{\infty },}$$

finishing the proof of this lemma.

Conclusion

Choosing a value of k that satisfies both lemmas leads to a non-zero integer ${\displaystyle \left(\frac{P}{k!}\right)}$ added to a vanishingly small quantity ${\displaystyle \left(\frac{Q}{k!}\right)}$ being equal to zero: an impossibility. It follows that the original assumption, that e can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e is transcendental.

The transcendence of π

A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof.

For detailed information concerning the proofs of the transcendence of π and e, see the references and external links.