Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = [X^{$$\alpha $$}]; [speed] = [X^{$$\beta $$} ]; [acceleration] = [X^p]; [linear momentum] = [X^q]; [force] = [X^r]. Then

As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate (KMnO₄) of different concentrations n₁ and n₂ (n₁ > n₂) molecules per unit volume with $$\Delta $$n = (n₁ − n₂) << n₁. When they are connected by a tube of small length l and cross-sectional area S, KMnO₄ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed v of the molecules is limited by the viscous force −$$\beta $$v on each molecule, where $$\beta $$ is a constant. Neglecting all terms of the order ($$\Delta $$n)², which of the following is/are correct? (k_B is the Boltzmann constant)

A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below. Correct shape of the emergent wavefront will be (figures are schematic and not drawn to scale)

Consider one mole of helium gas enclosed in a container at initial pressure P₁ and volume V₁. It expands isothermally to volume 4V₁. After this, the gas expands adiabatically and its volume becomes 32V₁. The work done by the gas during isothermal and adiabatic expansion processes are W_iso and W_adia, respectively. If the ratio $${{{W_{iso}}} \over {{W_{adia}}}}$$ = f ln 2, then f is ______.