Molar volume \(\left(V_m\right)\) of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with \(V_m\) as the variable. The ratio (in \(\mathrm{mol} \mathrm{dm}^{-3}\) ) of the coefficient of \(V_m^2\) to the coefficient of \(V_m\) for a gas having van der Waals constants \(a=6.0 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}\) and \(b=0.060 \mathrm{dm}^3 \mathrm{~mol}^{-1}\) at 300 K and 300 atm is _____ .
Use: Universal gas constant \((R)=0.082 \mathrm{dm}^3 \mathrm{~atm} \mathrm{~mol}{ }^{-1} \mathrm{~K}^{-1}\)

Under hydrolytic conditions, the compounds used for the preparation of linear polymer and for chain termination, respectively, are

Among the following, the correct statement(s) is(are)

Regarding the molecular orbital (MO) energy levels for homonuclear diatomic molecules, the INCORRECT statement(s) is(are)

${}_{92}{}^{238}\mathrm{U}$ is known to undergo radioactive decay to form ${}_{82}{}^{206}\mathrm{Pb}$ by emitting alpha and bet a particles. A rock initially contained $68\times {10}^{-6}\mathrm{g}$ of ${}_{92}{}^{238}\mathrm{U}$. If the number of alpha particles that it would emit during its radioactive decay of ${}_{92}{}^{238}\mathrm{U}$ to ${}_{82}{}^{206}\mathrm{Pb}$ in three half-lives is $\mathrm{Z}\times {10}^{18}$, then what is the value of $\mathrm{Z}$?

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at $1$ bar pressure is given. ${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{K}$, respectively. The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{K}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}=1 \mathrm{J} {\mathrm{mol}}^{-1} {\mathrm{K}}^{-1}$. Assume $\left({\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}\right)$ is independent of temperature in the range of $200$ to $700 \mathrm{K}.{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The value of entropy change, ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }$ (in ${\mathrm{Jmol}}^{-1} {\mathrm{K}}^{-1}$ ), at $300 \mathrm{K}$ is [Use: $\ln 2=0.69$0. Given: ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }=0$ at $0 \mathrm{K}$]

The number of hexagonal faces that are present in a truncated octahedron is

A tiny spherical oil drop carrying a net charge \(q\) is balanced in still air with a vertical uniform electric field of strength \(\frac{81 \pi}{7} \times 10^5 \mathrm{Vm}^{-1}\). When the field is switched off, the drop is observed to fall with terminal velocity \(2 \times 10^{-3} \mathrm{~ms}^{-1}\). Given \(g=9.8 \mathrm{~ms}^{-2}\), viscosity of the air \(=1.8 \times 10^{-5} \mathrm{Ns} \mathrm{m}^{-2}\) and the density of oil \(=900 \mathrm{~kg} \mathrm{~m}^{-3}\), the magnitude of \(q\) is

Paragraph:
The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
Question:
The structure of \(\mathrm{XeO}_3\) is

A large glass slab \(\left(\mu=\frac{5}{3}\right)\) of thickness 8 \(\mathrm{cm}\) is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius \(R \mathrm{~cm}\). What is the value of \(R\) ?

Based on VSEPR theory, the number of 90 degree \(\mathrm{F}-\mathrm{Br}-\mathrm{F}\) angles in \(\mathrm{BrF}_5\) is

The Henry's law constant for the solubility of \(\mathrm{N}_2\) gas in water at \(298 \mathrm{~K}\) is \(1.0 \times 105 \mathrm{~atm}\). The mole fraction of \(\mathrm{N}_2\) in air is \(0.8\). The number of moles of \(\mathrm{N}_2\) from air dissolved in 10 moles of water of \(298 \mathrm{~K}\) and \(5 \mathrm{~atm}\) pressure is

Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter \(D\) of a tube. The measured value of \(D\) is: