If $r$ is the radius of the atom then the volume of each atom will be $(4 / 3) \pi r^{3}$ Volume of all the substance will be $=(4 / 3) \pi r^{3} \times N=M / \rho$ $M=$ atomic mass of the substance...

### Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:

### A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have a uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

where refer to number density at heights and respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

where is the density of the suspended particle, and ‘, that of surrounding medium. [ is Avogadro’s number, and the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]

Law of atmosphere states that, $\mathrm{n}_{2}=\mathrm{n}_{1} \exp \left[-\mathrm{mg}\left(\mathrm{h}_{2}-\mathrm{h}_{1}\right) / \mathrm{k}_{\mathrm{B}} T\right]$ According to Archimedes principle,...

### From a certain apparatus, the diffusion rate of hydrogen has an average value of 1. The diffusion of another gas under the same conditions is measured to have an average rate of . Identify the gas.

[Hint: Use Graham’s law of diffusion: , where are diffusion rates of gases 1 and 2 , and and their respective molecular masses. The law is a simple consequence of kinetic theory.]

Rate of diffusion of hydrogen is given as $R_{1}=28.7 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ Rate of diffusion of another gas is given as $R_{2}=7.2 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ According to...

### A metre long narrow bore held horizontally (and closed at one end) contains a long mercury thread, which traps a column of air. What happens if the tube is held vertically with the open end at the bottom?

Length of the narrow bore is given as $L=1 \mathrm{~m}=100 \mathrm{~cm}$ Length of the mercury thread is given as $\mid=76 \mathrm{~cm}$ Length of the air column between mercury and the closed end,...

### Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at atm and temperature . Take the radius of a nitrogen molecule to be roughly A. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of ).

Mean free path is given as $1.11\times10^{-7}$ Collision frequency is given as $4.58\times10^{9}s^{-1}$ Successive collision time ≅ 500 x (Collision time) Pressure inside the cylinder containing...

### At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at (atomic mass of , of ).

Temperature of the helium atom is given as $T_{\mathrm{He}}=-20^{\circ} \mathrm{C}=253 \mathrm{~K}$ Atomic mass of argon is given as $M_{\text {Ar }}=39.9 u$ Atomic mass of helium is given as $M_{H...

### Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not, in which case is the largest?

All three vessels are the same size and have the same capacity. As a result, the pressure, volume, and temperature of each gas are the same. The three vessels will each contain an equal quantity of...

### Estimate the average thermal energy of a helium atom at the temperature of 10 million kelvin (the typical core temperature in the case of a star).

At temperature, $\mathrm{T}=10^{7} \mathrm{~K}$, we have Average thermal energy will be $=(3 / 2) \mathrm{kT}$ $=(3 / 2) \times 1.38 \times 10^{-23} \times 10^{7}$ We get, $=2.07 \times 10^{-16}...

### Estimate the average thermal energy of a helium atom at

(i) room temperature ,

(ii) the temperature on the surface of the Sun (6000 K)

(i) At room temperature the temperature is $T=27^{\circ} \mathrm{C}=300 \mathrm{~K}$ Average thermal energy will be $=(3 / 2) \mathrm{kT}$ Where, $k$ is the Boltzmann constant having value $1.38...

### Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity at a temperature of and 1 atm pressure.

Volume of the room is given as $V=25.0 \mathrm{~m}^{3}$ Temperature of the room is given as $T=27^{0} \mathrm{C}=300 \mathrm{~K}$ Pressure in the room will be $P=1 \mathrm{~atm}=1 \times 1.013...

### An air bubble of volume rises from the bottom of a lake deep at a temperature of To what volume does it grow when it reaches the surface, which is at a temperature of 35 C?

Volume of the air bubble is given as $\mathrm{V}_{1}=1.0 \mathrm{~cm}^{3}$ $=1.0 \times 10^{-6} \mathrm{~m}^{3}$ Air bubble rises to height given as $d=40 \mathrm{~m}$ Temperature at a depth of 40m...

### An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to . Estimate the mass of oxygen taken out of the cylinder , molecular mass of .

Volume of gas is given as $V_{1}=30$ litres $=30 \times 10^{-3} \mathrm{~m}^{3}$ Gauge pressure is given as $\mathrm{P}_{1}=15 \mathrm{~atm}=15 \times 1.013 \times 10^{5} \mathrm{P}$ a Temperature...

### The figure shows plot of PV/T versus for of oxygen gas at two different temperatures.

(a) What is the value of PV/T where the curves meet on the $y$-axis? (b) If we obtained similar plots for $1.00 \times 10^{-3}$ kg of hydrogen, would we get the same value of PV/T at the point where...

### The figure shows plot of PV/T versus for of oxygen gas at two different temperatures.

(a) What does the dotted plot signify? (b) Which is true: $\mathbf{T}_{1}>\mathbf{T}_{2}$ or $\mathbf{T}_{1}<\mathbf{T}_{2}$ ? Solution: (a) The dotted plot is perpendicular to the x-axis,...

### Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP: 1 atmospheric pressure, Show that it is litres.

We know, The ideal gas equation is given as: $P V=n R T$ Where, $R$ is the universal gas constant having value $8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ $\mathrm{n}$ is the number of...

### Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 A.

Diameter of an oxygen molecule is given as $d=3 \AA$ Radius will be, $r=d / 2$ $r=3 / 2=1.5 \AA=1.5 \times 10^{-8} \mathrm{~cm}$ Actual volume occupied by 1 mole of oxygen gas at STP is given as...