At low pressure,the van der Waals equation becomes:

The $Ca^{2+}$ and $F^{-}$ ions are located in a $CaF_2$ crystal,respectively at face-centered cubic lattice points and in:

The number of real solutions of $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$ is

Considering the gases to be ideal,the value of $\gamma = \frac{C_p}{C_v}$ for a gaseous mixture consisting of $3 \ moles$ of carbon dioxide $(CO_2)$ and $2 \ moles$ of oxygen $(O_2)$ will be $(\gamma_{O_2} = 1.4, \gamma_{CO_2} = 1.3)$.

Two soap bubbles combine to form a single bubble. In this process, the change in volume and surface area are respectively $V$ and $A$. If $P$ is the atmospheric pressure, and $T$ is the surface tension of the soap solution, the following relation is true :