In an adiabatic irreversible process, dQ = 0 is not equal to TdS ( TdS > 0 ) where Q is thermal energy, T is temperature, and S is entropy.

Clausius defined this transformation heat as dQ = TdS.

( dP / dQ ) =

: R ' = < R >+ ( dP / dQ ) ( Q / P )

** Reciprocity relation or exact differential, a mathematical differential of the form dQ, for some differentiable function Q

In multivariate calculus, a differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.

So, in order for a differential dQ, that is a function of four variables to be an exact differential, there are six conditions to satisfy.

* 1854 – Clausius establishes the importance of dQ / T ( Clausius's theorem ), but does not yet name the quantity.

( P + dP * Q / dQ ) can also be written as P ( 1 + dPQ / dQP ). Here, ( dP / dQ ) ( Q / P ) is 1 / E < sub > P </ sub >, where E < sub > P </ sub > is price elasticity of demand.

In thermodynamics, when dQ is exact, the function Q is a state function of the system.

So it suffices to show that ( dQ / ds ) Q < sup > T </ sup > is a skew-symmetric matrix.

Hence the entries κ and τ of ( dQ / ds ) Q < sup > T </ sup > are invariants of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has the same curvature and torsion.

In a homogeneous system only reversible processes can take place so the second law of thermodynamics gives δQ = TdS with T the absolute temperature of the system.

As such a quasistatic process in a homogeneous system is reversible, the total amount of heat added to a closed system can be expressed as δQ = TdS where T is the temperature and S the entropy of the system.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number l = 0, 1, 2 and 3 respectively.

10, 000 square metres = 0. 01 square kilometres

In the IERS numerical standards, the speed of light in a vacuum is defined as c < sub > 0 </ sub > = in a vacuum, in accordance with the SI units.

The time to traverse an AU is found to be τ < sub > A </ sub > =, resulting in the astronomical unit in metres as c < sub > 0 </ sub > τ < sub > A </ sub > =.

At temperatures near 0 K, nearly all molecular motion ceases and, when entropy = S, ΔS = 0 for any adiabatic process.

The Nernst postulate identifies the isotherm T = 0 as coincident with the adiabat S = 0, although other isotherms and adiabats are distinct.

As no two adiabats intersect, no other adiabat can intersect the T = 0 isotherm.

The question remains whether both can have zero entropy at T = 0 even though each is perfectly ordered.

( Guggenheim, p. 111 ) These quantities drop toward their T = 0 limiting values and approach with zero slopes.

Moreover, the slopes of the derivatives of ΔG and ΔH converge and are equal to zero at T = 0.

?& 2AC = 0

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

Modern physics explains this by noting that the n = 1 state holds 2 electrons, the n = 2 state holds 8 electrons, and the n = 3 state holds 8 electrons ( in argon ).

These aberrations are avoided if, according to Abbe, the sine condition, sin u ' 1 / sin u1 = sin u ' 2 / sin u2, holds for all rays reproducing the point O.

; Associativity: For all a, b and c in A, the equation ( a • b ) • c = a • ( b • c ) holds.

The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set

The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F ( w x ) = f ( x ); here w x represents the concatenation of the two strings w and x.

f ( gh ) = ( fg ) h. This holds for all functions over all domains.

: For all a, b and c in F, the following equality holds: a · ( b + c ) = ( a · b ) + ( a · c ).

#* We extend this result to more and more complex and lengthy sentences, D < sub > n </ sub > ( n = 1, 2 ...), built out from B, so that either any of them is refutable and therefore so is φ, or all of them are not refutable and therefore each holds in some model.

A server holds a subtree starting from a specific entry, e. g. " dc = example, dc = com " and its children.

Servers may also hold references to other servers, so an attempt to access " ou = department, dc = example, dc = com " could return a referral or continuation reference to a server that holds that part of the directory tree.

; Associativity: For all a, b and c in S, the equation ( a • b ) • c = a • ( b • c ) holds.

For example if a player holds 7 ♠ 8 ♠ on a board of 5 ♣ 6 ♠ < font color = red > 9 ♥</ font > he can be said to have the nuts, however if the next card comes < font color = red > 7 ♥</ font > then 8-10 becomes the nuts.

The number of neutrons, N, is known as the neutron number of the atom ; thus, A = Z + N. Since protons and neutrons have approximately the same mass ( and the mass of the electrons is negligible for many purposes ), and the mass defect is usually very small compared to the mass, the atomic mass of an atom is roughly equal to A.

This central charge would thus be approximately half the atomic weight ( though it was almost 25 % off the figure for the atomic number in gold ( Z = 79, A = 197 ), the single element from which Rutherford made his guess ).

: Precise instructions ( in language understood by " the computer ") for a fast, efficient, " good " process that specifies the " moves " of " the computer " ( machine or human, equipped with the necessary internally contained information and capabilities ) to find, decode, and then process arbitrary input integers / symbols m and n, symbols + and =

The binding energy needed for a nucleon to escape the nucleus, for various isotopes | deadurl = no

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

The advantage of spherical coordinates ( for atoms ) is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ ( r, θ, φ ) = R ( r ) Θ ( θ ) Φ ( φ ).

The best current ( 2009 ) estimate of the International Astronomical Union ( IAU ) for the value of the astronomical unit in meters is A = m, based on a comparison of JPL and IAA – RAS ephemerides.

The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

Anagrams are also often used in fiction for character aliases, such as Tom Marvolo Riddle = I am Lord Voldemort, Leonardo da vinci = O Draconian Devil, The Mona Lisa

Under Ambrose's major influence, emperors Gratian, Valentinian II and Theodosius I carried on a persecution of Paganism .< ref name = " MacMullen1984p100 "> MacMullen ( 1984 ) p. 100: ‘ The law of June 391, issued by Theodosius [...] was issued from Milan and represented the will of its bishop, Ambrose ; for Theodosius — recently excommunicated by Ambrose, penitent, and very much under his influence < sup > 43 </ sup > — was no natural zealot.

100 cc and P = 2. 50E6 pascals, so we can solve for temperature by simple algebra:

In the later play Frogs, Aristophanes softens his criticisms, but even so it may be only for the sake of punning on Agathon's name ( ἁγαθός = " good ") that he makes Dionysus call him a " good poet ".

Ayckbourn joined Wolfit on tour to the Edinburgh Festival Fringe as an acting assistant stage manager ( meaning a role that involved both acting and stage management ) for three weeks, with his first role on the professional stage being various parts in The Strong are Lonely by Fritz Hochwälder .< ref name = acting >

These points form a line, and y = x is said to be the equation for this line.

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.