The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is
Aldehydes, Ketones, and Carboxylic Acids are carbonyl compounds that contain a carbon-oxygen double bond. These organic compounds are very important in the field of organic chemistry and also have many industrial applications.
Aldehydes are organic compounds that have the functional group -CHO.
Preparation of Aldehydes
Acid chlorides are reduced to aldehydes with hydrogen in the presence of palladium catalyst spread on barium sulfate.
Ketones are organic compounds that have the functional group C=O and the structure R-(C=O)-R’.
Preparation of Ketones
Acid chlorides on reaction with dialkyl cadmium produce ketones. Dialkyl cadmium themselves are prepared from Grignard reagents.
Carboxylic acids are organic compounds that contain a (C=O)OH group attached to an R group (where R refers to the remaining part of the molecule).
Preparation of Carboxylic Acids
Primary alcohols are readily oxidized to carboxylic acids with common oxidizing agents such as potassium permanganate in neutral acidic or alkaline media or by potassium dichromate and chromium trioxide in acidic media.