Activated manganese(IV) oxide can selectively oxidize allylic and benzylic alcohols via a radical intermediate mechanism. Primary allylic alcohols are oxidized to aldehydes, while secondary allylic alcohols yield ketones. The redox reaction of potassium permanganate with an Mn(II) salt such as manganese sulfate (under either alkaline or acidic conditions), followed by thorough drying, yields the oxidizing agent: activated MnO2. While MnO2 is insoluble in the solvents used for the reaction, the...

The current growth and decay in RL circuits can be understood by considering a series RL circuit consisting of a resistor, an inductor, a constant source of emf, and two switches. When the first switch is closed, the circuit is equivalent to a single-loop circuit consisting of a resistor and an inductor connected to a source of emf. In this case, the source of emf produces a current in the circuit. If there were no self-inductance in the circuit, the current would rise immediately to a steady...

Vascular plants, which account for over 90% of the Earth’s vegetation, all undergo primary growth—which lengthens roots and shoots. Many land plants, notably woody plants, also undergo secondary growth—which thickens roots and shoots.

The oxidation of an organic compound in the presence of air or oxygen is called autoxidation. For example, cumene reacts with oxygen to form hydroperoxide. Autoxidation involves initiation, propagation, and termination steps. Many organic compounds are susceptible to autoxidation—especially ethers in the presence of oxygen, which form hydroperoxides. Even though this reaction is slow, old ether bottles contain small amounts of peroxide, which leads to laboratory explosions during ether...

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...