The order of the oxidation state of the phosphorus atom in $H_{3}P{O}_2, H_{3}P{O}_4, H_{3}P{O}_3$ and $H_4{P}_2{O}_6$ is

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\(P\) and \(Q\) are isomers of dicarboxylic acid \(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{O}_{4}\). Both decolorize \(\mathrm{Br}_{2} / \mathrm{H}_{2} \mathrm{O}\). On heating, \(\boldsymbol{P}\) forms the cyclic anhydride.

Upon treatment with dilute alkaline \(\mathrm{KMnO}_{4}, \boldsymbol{P}\) as well as \(\boldsymbol{Q}\) could produce one or more than one from \(\boldsymbol{S}, \boldsymbol{T}\) and \(\boldsymbol{U}\).




Question:
Compounds formed from \(\boldsymbol{P}\) and \(\boldsymbol{Q}\) are, respectively

The treatment of galena with ${\mathrm{HNO}}_3$ produces a gas that is

The reaction of ${\mathrm{HClO}}_3$ with $\mathrm{HCl}$ gives a paramagnetic gas, which upon reaction with ${\mathrm{O}}_3$ produces

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Thermal decomposition of gaseous \(\mathrm{X}_{2}\) to gaseous \(\mathrm{X}\) at \(298 \mathrm{~K}\) takes place according to the following equation:

\(\mathrm{X}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{X}(\mathrm{g})\)

The standard reaction Gibbs energy, \(\Delta_{r} G^{\circ}\), of this reaction is positive. At the start of the reaction, there is one mole of \(\mathrm{X}_{2}\) and no \(\mathrm{X}\). As the reaction proceeds, the number of moles of \(\mathrm{X}\) formed is given by \(\beta\). Thus, \(\beta_{\text {equilibrium }}\) is the number of moles of \(\mathrm{X}\) formed at equilibrium. The reaction is carried out at a constant total pressure of \(2\) bar. Consider the gases to behave ideally. (Given : \(R=0.083 \mathrm{~L}\) bar \(\mathrm{K}^{-1} \mathrm{~mol}^{-1}\) )

Question:
The equilibrium constant \(K_{P}\) for this reaction at \(298 \mathrm{~K},\) in terms of \(\beta_{equilibrium}\) is

For the given compound X, the total number of optically active stereoisomers is ____.

The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased $4$ times and its pressure $2$ times. As a result, the diffusion coefficient of this gas increases $\mathrm{x}$ times. The value of $\mathrm{x}$ is:

Consider the lines given by
\[
L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0
\]
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II.

According to Bohr's model, the highest kinetic energy is associated with the electron in the

Let \(\left(x_0, y_0\right)\) be the solution of the following equations \((2 x)^{\log 2}=(3 y)^{\log 3}\), \(3^{\log x}=2^{\log y}\), then \(x_0\) is equal to

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Let \(f:[0,1] \rightarrow \mathbb{R}\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0)=f(1)=0\) and satisfies \(f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .\)


Question:

If the function \(\mathrm{e}^{-x} f(x)\) assumes its minimum in the interval \([0,1]\) at \(x=\frac{1}{4}\), which of the following is true?

On dissolving $0.5\mathrm{ }\mathrm{g}$ of a non-volatile non-ionic solute to $39\mathrm{ }\mathrm{g}$ of benzene, its vapor pressure decreases from $650\mathrm{ }\mathrm{m}\mathrm{m}\mathrm{ }\mathrm{H}\mathrm{g}$ to $640\mathrm{ }\mathrm{m}\mathrm{m}\mathrm{ }\mathrm{H}\mathrm{g}.$ The depression of freezing point of benzene (in $\mathrm{K}$ ) upon addition of the solute is _______
(Given data: Molar mass and the molal freezing point depression constant of benzene are $78\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ and $5.12\mathrm{ }\mathrm{K}\mathrm{ }\mathrm{k}\mathrm{g}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1},$ respectively)