Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at 25^o C. For this process, the correct statement is 

The given graph represent the variations of compressibility factor \((Z)=\frac{P V}{n R T}\) versus \(P\), for three real gases \(A, B\) and \(C\). Identify the only incorrect statement

An ideal gas is expanded from $({\mathrm{p}}_1\mathrm{ },\mathrm{ }{\mathrm{V}}_1\mathrm{ },\mathrm{ }{\mathrm{T}}_1)\mathrm{ }\mathrm{to}\mathrm{ }({\mathrm{p}}_2\mathrm{ },\mathrm{ }{\mathrm{V}}_2\mathrm{ },\mathrm{ }{\mathrm{T}}_2)$ under different conditions. The correct statement(s) among the following is(are)

Statement \(\mathbf{1}\left[\mathrm{Fe}\left(\mathrm{H}_2 \mathrm{O}\right)_5 \mathrm{NO}\right] \mathrm{SO}_4\) is paramagnetic.
Statement 2 The \(\mathrm{Fe}\) in \(\left[\mathrm{Fe}\left(\mathrm{H}_2 \mathrm{O}\right)_5 \mathrm{NO}\right] \mathrm{SO}_4\) has three unpaired electrons.

A gas has a compressibility factor of $0.5$ and a molar volume of $0.4 {\mathrm{dm}}^3 {\mathrm{mol}}^{-1}$ at a temperature of $800 \mathrm{K}$ and pressure $\mathrm{x} \mathrm{atm}$. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be  $\mathrm{y} {\mathrm{dm}}^3 {\mathrm{mol}}^{-1}$ . The value of$\mathrm{x}/\mathrm{y}$ is _________
[Use: Gas constant, $\mathrm{R}=8\times {10}^{-2} \mathrm{L} \mathrm{atm} {\mathrm{K}}^{-1} {\mathrm{mol}}^{-1}$ ]

The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element \(X\) as shown below. To which group, element \(X\) belongs in the periodic table?

\({ }_{29}^{63} \mathrm{Cu}+{ }_{1}^{1} \mathrm{H} \rightarrow 6_{0}^{1} n+{ }_{2}^{4} \alpha+2{ }_{1}^{1} \mathrm{H}+\mathrm{X}\)

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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 INCORRECT statement among the following, for this reaction, is

A thermally isolated cylindrical closed vessel of height $8\mathrm{m}$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3\mathrm{kg}$. Thus the partition is held initially at a distance of $4\mathrm{m}$ from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains $0.1$ mole of an ideal gas at temperature $300\mathrm{K}$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in $\mathrm{m}$) will be ____________ (take the acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and the universal gas constant$=8.3\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$).

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If ${\mathrm{\lambda }}_{{\mathrm{X}}^{-}}^0\approx {\mathrm{\lambda }}_{{\mathrm{Y}}^{-}}^0$ , the difference in their $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}$ values, $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{X}\right)-\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{Y}\right)$ , is (Consider degree of ionization of both acids to << 1)

Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.

The value of $R$ is ________ meter.

Half-life of a radio active substance ' \(A\) ' is 4 days. The probability that a nucleus will decay in two half-lives is

A $10\mathrm{}\mathrm{c}\mathrm{m}$ long perfectly conducting wire $\mathrm{P}\mathrm{Q}$ is moving, with a velocity $1\mathrm{}\mathrm{c}\mathrm{m}/\mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1\mathrm{}\mathrm{m}\mathrm{H}$ and a resistance $\mathrm{R}=1\mathrm{\Omega }$ as shown in figure. The horizontal rails, $\mathrm{L}$ and $\mathrm{R}$ lie in the same plane with a uniform magnetic field $\mathrm{B}=1\mathrm{}\mathrm{T}$ perpendicular to the plane. If the key $\mathrm{S}$ is closed at certain instant, the current in the circuit after $1$ milli second is $\mathrm{x}\times {10}^{-3}\mathrm{A},$ where the value of $\mathrm{x}$ is_______.
[Assume the velocity of wire $\mathrm{P}\mathrm{Q}$ remains constant $\left(1\mathrm{c}\mathrm{m}/\mathrm{s}\right)$ after key $\mathrm{S}$ is closed. Given: ${\mathrm{e}}^{-1}=0.37,$ where $\mathrm{e}$ is base of the natural logarithm]