According to the kinetic theory of gases,which one of the following statements is wrong?

For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the $\ln P$ versus $\ln V$ graph is plotted for an adiabatic process, as shown.

In the kinetic theory of gases,which of the following statements regarding elastic collisions of the molecules is wrong?

Statement $(I)$: Gas thermometers are less sensitive than liquid thermometers.
Statement $(II)$: The ratio of the universal gas constant and Avogadro's number is called Boltzmann's constant.
Statement $(III)$: The density of a given mass of a gas at constant pressure is inversely proportional to its absolute temperature.
The correct option among the following is:

$A$ gas has temperature $T$ and volume $V$. At constant pressure,if the temperature is increased by $\Delta T$,the volume increases by $\Delta V$. How does $\delta = \frac{\Delta V}{V\Delta T}$ vary with temperature?

As shown schematically in the figure,two vessels contain water solutions (at temperature $T$) of potassium permanganate $(KMnO_4)$ of different concentrations $n_1$ and $n_2$ $(n_1 > n_2)$ molecules per unit volume with $\Delta n = (n_1 - n_2) \ll n_1$. When they are connected by a tube of small length $\ell$ and cross-sectional area $S$,$KMnO_4$ 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)
$(A)$ the force causing the molecules to move across the tube is $\Delta n k_B T S$
$(B)$ force balance implies $n_1 \beta v \ell = \Delta n k_B T$
$(C)$ total number of molecules going across the tube per sec is $\left(\frac{\Delta n}{\ell}\right)\left(\frac{k_B T}{\beta}\right) S$
$(D)$ rate of molecules getting transferred through the tube does not change with time