Ideal gas equation

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate ($\ce{KMnO4}$) 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 $l$ and cross-sectional area $S$, $\ce{KMnO4}$ 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 thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be ______ (take the acceleration due to gravity $= 10\ \mathrm{m\,s^{-2}}$ and the universal gas constant $= 8.3\ \mathrm{J\,mol^{-1}K^{-1}}$).

The temperature of an open room of volume 30 m$^3$ increases from 17°C to 27°C due to the sunshine. The atmospheric pressure in the room remains $1\times10^5$ Pa. If $n_i$ and $n_f$ are the number of molecules in the room before and after heating, then $n_f-n_i$ will be :

For the P-V diagram given for an ideal gas,

out of the following which one correctly represents the T-P diagram ?

'n' moles of an ideal gas undergoes a process A→B as shown in the figure. The maximum temperature of the gas during the process will be :

Which of the following shows the correct relationship between the pressure 'P' and density ρ of an ideal gas at constant temperature ?

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm. What will be length of the air column above mercury in the tube now ?

(Atmospheric pressure $=76$ cm of Hg)

The equation of state for a gas is given by $PV=nRT+\alpha V$, where $n$ is the number of moles and $\alpha$ is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are $T_o$ and $P_o$ respectively. The work done by the gas when its temperature doubles isobarically will be :

Modern vacuum pumps can evacuate a vessel down to a pressure of $4.0\times10^{-15}$ atm. at room temperature (300 K). Taking $R=8.3$ JK$^{-1}$ mole$^{-1}$, 1 atm $=10^5$ Pa and $N_{\text{Avogadro}}=6\times10^{23}$ mole$^{-1}$, the mean distance between molecules of gas in an evacuated vessel will be of the order of :

An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section $8.0\times10^{-3}$ m$^2$. Initially the gas is at 300K and occupies a volume of $2.4\times10^{-3}$ m$^3$ and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by 0.1 m. The force constant of the spring is 8000 N/m and the atmospheric pressure is $1.0\times10^5$ N/m$^2$. The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be :

(Neglect the heat loss through the lead wires of the heater. The heat capacity of the heater coil is also negligible)

Two soap bubbles coalesce to form a single bubble. If V is the subsequent change in volume of contained air and S the change in total surface area, T is the surface tension and P atmospheric pressure, which of the following relation is correct ?

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is

A real gas behaves like an ideal gas if its

A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening of the orifice. [Take atmospheric pressure $=1.0\times10^5\ \mathrm{N/m^2}$, density of water $=1000\ \mathrm{kg/m^3}$ and $g=10\ \mathrm{m/s^2}$. Neglect any effect of surface tension.]

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m². The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $n_B/n_A$, where $n_A$ and $n_B$ are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity.]

An ideal gas is expanding such that $PT^2=$ constant. The coefficient of volume expansion of the gas is

The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)