UPB
Applications of Taylor series in chemistry
Student:Musat Isabelle Miruna 1112B Teacher:Grosu Corina
Keywords Taylor series
Introduction Polynomial functions are easy to understand but complicated functions, infinite polynomials, are not obvious. Infinite polynomials are made easier when represented using series: complicated functions are easily represented using Taylor’s series. This representation make some functions properties easy to study such as the asymptotic behavior. Differential equations are made easy with Taylor series. Taylor’s series is an essential theoretical tool in computational science and approximation. This paper points out and attempts to illustrate some of the many applications of Taylor’s series expansion. Concrete examples in the physical science division and various engineering fields are used to paint the applications pointed out. Taylors series is an expansion of a function into an infinite series of a variable x or into a finite series plus a remainder term[1]. The coefficients of the expansion or of the subsequent terms of the series involve the successive derivatives of the function. The function to be expanded should have a nth derivative in the interval of expansion. The series resulting from Taylors expansion is referred to as the Taylor series. The series is finite and the only concern is the magnitude of the remainder. Taylor series could also be written in the context of a complex variable .Some functions have no antiderivative which can be expressed in terms of familiar functions. This makes evaluating definite integrals of these functions difficult because the fundamental theorem of calculus cannot be used. However, a series representation of this function eases things up.
Applications In the field of physics and chemistry, there is a great need for geometric optimization of physical systems. In chemistry, as an example, the quasi-newton method make use of a two variable Taylor’s series to approximate the equilibrium geometry of a cluster of atoms [3]. Consider U,the geometry of a molecule, and assume it is a function of only two variables, x and y, let x1 and y1 be the initial coordinates, if terms higher than the quadratic terms are neglected then the Taylor series is as follows:
The U(x, y) function fits well around the equilibrium position, the quadratic approximation works well around the minimum. If U were accurately a quadratic function of the coordinates in the region near (x1, y1), then the elements of the Hessian matrix (second partial derivatives) will be constant in this region. Accurate ab initio self consistent field calculation of the second derivatives is very time consuming, thus the optimization usually starts with an approximation of the Hessian and then proceeds to improve on this approximation. If U(x,y) is written in the form below then, the first approximation to the Hessian matrix element at (or near) the equilibrium geometry can be computed.
The superscript denotes the first approximation to the Hessian matrix elements at or near the equilibrium geometry. The molecular geometry has a 3N-6 dimensional vector when internal coordinates are considered and by 3N when only cartesian coordinates are used. N is the number of atoms in the molecule. In cartesian coordinates, rotation and translation accounts for the six in 3N-6. Once the Hessian matrix elements are determined, the molecular properties can be extracted via the Taylor’s series expansion. The position of the atom in the molecule constantly shift from the equilibrium position. an overall atomic behavior , in the course of vibration, is modeled on the Lennard-Jones 6-12 potential. The dynamics of the vibrations can be study by expanding the potential in a Taylor’s series. The second derivative of the Taylor’s series expansion correspond to the gradient of the,(harmonic) potential curve of a short range vibration around the equilibrium position,re.
In many textbooks of physical chemistry the approximate formulas valid in some limiting cases are usually derived by the applications of Taylor series. It appears however, that in some cases Taylor expansions are “too much”, and a simpler solution, namely taking some limits, may be sufficient. We investigate here two examples. We start with the “usual” considerations in both cases, and then try to reformulate and solve the problems using some limits.
Osmotic pressure
In an osmotic experiment the pure solvent enters the solution accross a semipermeable membrane increasing thereby the inner pressure of the solution. This increase in the pressure makes the chemical potential of the solvent greater in the solution, and finally – when the chemical potential of the pure solvent will be equal to that of the solvent in the solution – an osmotic equilibrium will be established. The equilibrium value of the pressure excess in the solution is called osmotic pressure.
The osmotic pressure Π can be described by the function defined by: where 0 < x2 < 1 is the mole fraction of the solute, R > 0 is the universal gas constant and T > 0 is the thermodynamical temperature also held at a constant value. To obtain a simpler formula for x2 ≪ 1 we usually apply the Taylor series expansion
with -1 < x ≤ 1. When x2 ≪ 1, we can neglect the higher-order terms in the Taylor expansion and using
we obtain
and
.
We now attack the problem by calculating some limits. A rearrangement of the starting formula
yields
and we are interested in the limit of the right-hand side for x2 → 0. This appears to be simple, but, unfortunately, it turns out that we must calculate a 0/0 type limit. Applying L’ Hospital’s rule:
and so
A slightly different formulation is
The last two formulas have the same chemical meaning as Eqn (1).
Solubility of a sparingly soluble salt in an electrolyte solution Suppose that a sparingly soluble salt and a well-soluble electrolyte have a common anion. If the sparingly soluble salt dissolves ideally in the electrolyte solution then in the saturated solution
where S(c) > 0 is the solubility of the sparingly soluble salt, c > 0 is the concentration of the electrolyte solution and Ksol > 0 is the solubility constant. First of all, we must check whether S as a function of c really exists. A simple calculation shows
that from which we see that only one S(c) > 0 belongs to any c > 0. Thus we can speak of S as a function of c.
To deal with the case c ≫ S(c) we usually neglect S(c) compared to c and take S(c) + c ≈ c. Then Eqn (2) yields
A more definite mathematical procedure uses the Taylor expansion
where -1 < x ≤ 1. For x ≪ 1 we obtain
that is Using the last formula in Eqn (3) we find Eqn (4). In terms of limits we use
From this and Eqn (2)
that is The chemical meaning of this equation coincides with that of Eqn (4).
The things become even simpler if we notice that Eqn (2) yields
that is,
> S(c). Thus, S(c) has an upper bound, which is enough to see that
This leads again to Eqn (5).
Conclusion We have probe through the complexity of Taylor series and shown evidence of its extensive and very effective applications. The effectiveness in error determination, function optimization, definite integral resolution, and limit determination is evidence of the Taylor series being an enormous tool in physical sciences and in Computational science as well as an effective way of epresenting complicated functions. The two examples analyzed show that the explicite use of limits, although not typical, may lead to formulas with the desired chemical meanings. It would be interesting to see whether an extensive application of this formalism would or would not facilitate the assimilation of the material.
Bibliography http://sces.phys.utk.edu/~moreo/mm08/fosso.pdf http://puma.unideb.hu/~potagy/mathchem/serieslimits/serieslimits.html
