Point Charge Approximation

Source: Laboratory of Jonathan Flombaum—Johns Hopkins University A common carnival game is to ask people to guess the number of jellybeans packed into a jar. The chances that anyone will get the exact number right are low. But what about the chances that someone will guess 17 or 147,000? Probably even less than the chances of guessing the correct answer; 17 and 147,000 just seem irrational. Why? After all, if the beans cannot be taken out and counted one-at-a-time, how can someone tell that an...

In some cases, there are seemingly more than one valid Lewis structures for molecules and polyatomic ions. The concept of formal charges can be used to help predict the most appropriate Lewis structure when more than one reasonable structure exists. Calculating Formal Charge The formal charge of an atom in a molecule is the hypothetical charge the atom would have if the electrons in the bonds are evenly distributed between the atoms. Alternatively, formal charge results when from the number...

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

In ordinary chemical reactions, the nucleus — which contains the protons and neutrons of each atom and thus identifies the element — remains unchanged. Electrons, however, can be added to atoms by transfer from other atoms, lost by transfer to other atoms, or shared with other atoms. The transfer and sharing of electrons among atoms govern the chemistry of the elements. During the formation of some compounds, atoms gain or lose electrons to form electrically charged particles called ions. When...