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A Guidebook To Mechanism In Organic Chemistry (Descont.) Exciting News! ✨ I am thrilled to share that I am adding a new book to the DesCont series:    📘A Guidebook to Mechanism in Organic Chemistry by Peter Sykes This will also be the first impression for a theme in Chemistry in this blog. If you have followed the previous writings under the DesCont series, you now that they are meant to clarify, extend & reflect ideas from the book being explored. I hope this new addition brings you insights, clarity, and a good time reading along. 🌿 The book is now available in the DesCont page. Fig 1: A methane molecule (one of the simplest organic compounds)

Infinitely Many Primes?! Prime numbers are one of the most fascinating aspects of Number Theory. Despite being one of the simplest to define, they have amazed mathematicians for centuries. In the post, we see, how one of the earliest theorems on prime numbers was proved by Euclid about 2000 years ago! Prime Numbers:  A natural number greater than 1, that is divisible only by 1 and itself is called a prime number. In other words, if ` p \in N`, `p > 1`  has only `1` and `p` as its factor, then the number `p` is called a prime number. For example `19` has no factors except `1` and itself. Thus it's a prime number. The first `20` prime numbers are listed below:  ` 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29` `31, 37, 41, 43, 47, 53, 59, 61, 67, 71` Fig 1: The distribution of first 20 prime numbers (In x-axis, 10 units represent the gap between two primes) So Many Primes! It's then exciting to know certain properties of these special numbers, isn't it? Let's then see a few no...

(Refresh the browser once more for the equations to load properly) Transient States in RL Circuits: The Journey to Steady-State  In this article, we attempt to obtain the complete solution of an RL circuit, including the transient state. We first consider the case of a DC voltage, where the current gradually builds up and after that, we take up the special case of AC Voltage which is often skipped in elementary courses. Fig: Growth of current in cases of DC and AC voltages DC Voltage: A DC Voltage refers to a steady voltage that does not vary with time. The below diagram shows a simple DC circuit involving a resistor `R` and an inductor `L`. A cell acts as the source of a DC emf E. Fig: A simple RL circuit with DC emf From Kirchhoff's Voltage Law, we know that the magnitude of the net Voltage drop across a loop is equal to the magnitude of the net emf in that loop. The voltage drop due to the resistor is `iR` and due to the inductor is `L \frac{di}{dt}`. Thus we m...

SHM Phasors This post is only an extension to the previous article on AC Phasors. The reader is advised to read the article on AC Phasors here  first, to gain an elementary idea on phasors. In this article, we consider phasors used in describing Simple Harmonic Motion. Simple Harmonic Motion: The motion governed by a force proportional to the displacement of the particle from the mean position and directed towards the mean position is called simple harmonic motion. i.e. for an ideal particle of mass `m`, we have, ` F = -kx` i.e.   ` m \frac{\text{d}^{2}x}{\text{dt}^{2}} = -kx ` Suppose we have the mass particle `m` attached to a spring that applies force according to the above equation. We stretch it from the mean position to some distance say `a` and then release it.  The differential equation of this motion is then, ` \frac{\text{d}^{2}x}{\text{dt}^{2}} = - \frac{k}{m}x` Since both `k` and `m` are positive quantities, we can let `\frac{k}{m} = w^{2}`, for some `w...

(Refresh the browser once more for the equations to load properly) Animating AC Phasors In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation! Sinusoids: Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the ele...

Newton-Leibniz Rule Newton-Leibniz's rule helps us to find the derivative of an integral, with respect to some variable, where the limits of integration are some functions of the same variable. While there are many variations of this rule (see note 1 below), we consider here one of the simplest forms. The Theorem: Let ` f(t) ` be a continuous integrable function and let `u(x)` and `v(x)` be some functions of x. Then the derivative of the integral of `f(t)` w.r.t. x is given by,  ` \frac{\text{d}}{\text{d}x}\int_{u(x)}^{v(x)} f(t) dt = f(v(x))\cdot v'(x) - f(u(x)) \cdot u'(x) ` Proof:  The proof for this is very simple. Let g(t) be the anti-derivative of f(t). Then we can write,  ` \int_{u(x)}^{v(x)} f(t) dt = g(t) |_{u(x)}^{v(x)} = g(v(x)) - g(u(x)) ` Now taking the derivative using the chain rule, we get,  ` \frac{\text{d}}{\text{d}x} [g(v(x)) - g(u(x))] = g'(v(x)) \cdot v'(x) - g'(u(x)) \cdot u'(x)   ` However, by our assumption since` g ` is the anti-de...

An Elegant Way to Expand The Inverse Tangent In this post we shall see a method to expand the inverse tangent function ` tan^{-1}(x) ` using an infinite geometric series. The infinite series equips us with a neat and easy to remember technique to find the expansion. A Geometric Series: A series in which a term is obtained by multiplying the previous term by a constant ratio is called a geometric series. This contant ratio is also called the common ratio `r`. For example consider these series: ` 2, 4, 8, 16, 32, ... ` `1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \frac{1}{625}, ... ` In the first series, `r = 2` and in the second series `r = \frac{1}{5}`. Sum of an Infinite Geometric Series: In general, any geometric series can be written as,  `a, ar^{1}, ar^{2}, ar^{3}, ar^{4}, ... ` An infinite series however can only be summed when the series converges i.e. only when `|r| < 1 `. When this is true the sum `S` is given by ` \frac{a}{1-r} ` i.e. ` S = a + ar^{1} + ar^{2} + ar^{3}...