The cyanide process of gold extraction involves leaching out gold from its ore with ${\mathrm{C}\mathrm{N}}^{-}$ in the presence of $\mathrm{Q}$ in water to form $\mathrm{R}$ . Subsequently, $\mathrm{R}$ is treated with $\mathrm{T}$ to obtain $\mathrm{A}\mathrm{u}$ and $\mathrm{Z}$ . Choose the correct option(s).

Statement 1 For every chemical reaction at equilibrium, standard Gibbs energy of reaction is zero.
Statement 2 At constant temperature and pressure, chemical reactions are spontaneous in the direction of decreasing Gibbs energy.

For 'invert sugar', the correct statement(s) is (are)
(Given: specific rotations of $\left(+\right)-$sucrose, $\left(+\right)-$maltose, $\mathrm{L}-\left(-\right)-$glucose and $\mathrm{L}-\left(+\right)-$fructose in aqueous solution are $+\mathrm{ }{66}^{°},\mathrm{ }+{140}^{°},\mathrm{ }–{52}^{°}$ and $+\mathrm{ }{92}^{°}$, respectively)

The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic \(\mathrm{KOH}\) is

$\mathrm{M}{\mathrm{X}}_2$ dissociates in ${\mathrm{M}}^{2+}$ and ${\mathrm{X}}^{-}$ ions in an aqueous solution, with a degree of dissociation $\left(\mathrm{\alpha }\right)$ of 0.5. The ratio of the observed depression of freezing point of the aqueous solution to the value of the depression of freezing point in the absence of ionic dissociation is

The sum of the spin only magnetic moment values (in B.M.) of \(\left[\mathrm{Mn}(\mathrm{Br})_6\right]^{3-}\) and \(\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-}\) is _____.

The wave function, ${\mathrm{\psi }}_{\mathrm{n},\mathrm{}\mathrm{l},\mathrm{}{\mathrm{m}}_{\mathrm{l}}}$ is a mathematical function whose value depends upon spherical polar coordinates $\left(\mathrm{r},\mathrm{}\mathrm{\theta },\mathrm{}\mathrm{\varphi }\right)$ of the electron and characterized by the quantum numbers $\mathrm{n},\mathrm{}\mathrm{l}$ and ${\mathrm{m}}_{\mathrm{l}}$ . Here $\mathrm{r}$ is the distance from the nucleus, $\mathrm{\theta }$ is colatitude and $\mathrm{\varphi }$ is azimuth. In the mathematical functions given in the Table, $\mathrm{Z}$is the atomic number and ${\mathrm{a}}_{\mathrm{o}}$ is Bohr radius

Column 1	Column 2	Column 3
(I) 1s orbital	\(\text {(i) } \psi_{ n , 1, m_1}\) \(\propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{3}{2}} e ^{-\left(\frac{ Zr }{ e _{ o }}\right)}\)	(P)
(II) 2s orbital	(ii) One radial node	(Q) Probability density at the nucleus \(\propto \frac{1}{ a _{ o }^3}\).
(III) \(2 p _{ z }\) orbital	\(\text {(iii) } \psi_{ n }, 1, m_{ l }\) \(\propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{5}{2}} re ^{-\left(\frac{ Zr }{2 a _{ o }}\right)} \cos \theta\)	(R) Probability density is maximum at the nucleus.
(IV) \(3 d_{ z }^2\) orbital	(iv) \(x y\)-plane is a nodal plane	(S) Energy needed to excite an electron from \(n =2\) state to \(n =4\) state is \(\frac{27}{32}\) times the energy needed to excite are electron from \(n =2\) state to \(n =6\) state.

For hydrogen atom, what is the only CORRECT combination?

Which is correct statement if \(\mathrm{N}_2\) is added at equilibrium condition?

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The empty space in this \(\mathrm{HCP}\) unit cell is

A real gas behaves like an ideal gas if its

Let \(a_{1}, a_{2}, a_{3}, \ldots . .\) be in harmonic progression with \(a_{1}=5\) and \(a_{20}=25\). The least positive integer \(n\) for which \(a_{n} < 0\) is

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. \(\mathrm{NaOH}\) solution, gives \(\mathbf{P}\) and \(\mathbf{Q}\). Compound \(\mathbf{P}\) does not give Tollens' test, whereas \(\mathbf{Q}\) on acidification gives positive Tollens' test. Treatment of \(\mathbf{P}\) with excess cyclohexanone in the presence of catalytic amount of \(p\)-toluenesulfonic acid (PTSA) gives product \(\mathbf{R}\).
Sum of the number of methylene groups \(\left(-\mathrm{CH}_2-\right)\) and oxygen atoms in \(\mathbf{R}\) is ________.