In thermodynamics, the $\mathrm{P}-\mathrm{V}$ work done is given by $\mathrm{w}=-\int {\mathrm{dVP}}_{\mathrm{ext}}$. For a system undergoing a particular process, the work done is,
$\mathrm{w}=-\int \mathrm{dV}\left(\frac{\mathrm{RT}}{\mathrm{V}-\mathrm{b}}-\frac{\mathrm{a}}{{\mathrm{V}}^2}\right)$. This equation is applicable to a

The compound(s) formed upon combustion of sodium metal in excess air is (are)

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 compound (s) having peroxide linkage is(are) :

Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

The standard reduction potential data at $25°C$ is given below.
\(E ^o\left( Fe ^{3+}, Fe ^{2+}\right)=+0.77 V ; \)
\( E ^o\left( Fe ^{2+}, Fe \right)=-0 . 44 V \)
\( E ^o\left( Cu ^{2+}, Cu \right)=+0.34 V ; \)
\( E ^o\left( Cu ^{+}, Cu \right)=+0.52 V \)
\( E ^o\left[ O _2(g)+4 H ^{+}+4 e ^{-} \rightarrow 2 H _2 O \right]=\) \(+1.23~V \)
\( E ^o\left[ O _2(g)+2 H _2 O +4 e^{-} \longrightarrow \text { 4OH}^{-}\right]=\) \(+0.40~V \)
\( E ^o\left( Cr ^{3+}, Cr \right)=-0.74 V ; \)
\( E ^o\left( Cr ^{2+}, Cr \right)=-0.91 V\)
Match E^o of the redox pair in List I with the values given in List II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	$E^o\left({\mathit{\text{Fe}}}^{3+},\mathit{\text{Fe}}\right)$	P.	$-0.18V$
B.	$E^o\left(4{\text{H}}_2\text{O}\rightleftharpoons 4H^{+}+4{\mathit{\text{OH}}}^{-}\right)$	Q.	$-0.4V$
C	$E^o\left({\mathit{\text{Cu}}}^{2+}+\mathit{\text{Cu}}\to 2{\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{Cu}}^{+}\right)$	R.	$-0.04V$
D.	$E^o\left({\mathit{\text{Cr}}}^{3+},{\mathit{\text{Cr}}}^{2+}\right)$	S.	$-0.83V$

Which of the given statement(s) about \(\mathbf{N}, \mathbf{O}, \mathbf{P}\) and \(\mathbf{Q}\) with respect to \(\mathbf{M}\) is (are) correct?

The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is (are)

The correct statements (s) about intermolecular forces is(are)

Let \(\mathbb{R}^2\) denote \(\mathbb{R} \times \mathbb{R}\). Let \(S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .\)
Then which of the following statements is (are) TRUE?

The number of optically active products obtained from the complete ozonolysis of the given compound is :

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

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Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\(
{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1
\)
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls to the decay which \({ }^{14} \mathrm{C}\) undergoes.
\(
{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}
\)
The half life period of \({ }^{14} \mathrm{C}\) is 5770 years. The decay constant \((\lambda)\) can be calculated by using the following formula \(\lambda=\frac{0.693}{t_{1 / 2}}\).
The comparison of the \(\beta^{-}\)activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\).
Question:
What should be the age of fossil for meaningful determination of its age?