Maxwell's Speed of Distribution Law

Q.

A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature. $T_2$ by releasing the piston suddenly. If $L_1$ and $L_2$ are the lengths of the gas column before and after expansion respectively, then $\frac{T_1}{T_2}$ is given by

1. 

2. 

3. 

4. 

Q.

Given below are two statements:

Statement 1: In a diatomic molecule, the rotational energy at a given temperature obeys Maxwells distribution.

Statement 2: In a diatomic molecule, the rotational energy at a given temperature equals to translational kinetic energy for each other.

In the light of the above statements, choose the correct answer from the options given below:

1. Both statement 1 and statement 2 are false

2. Both statement 1 and statement 2 are true

3. Statement 1 is false but statement 2 is true

4. Statement 1 is true but statement 2 is false

Q. Identify the correct order as indicated in the given diagram of different kind of speed of molecules of same gas.


$v_{mp}\to$ Most probable speed
$v_{rms}\to$ Root mean square speed
$v_{av}\to$ Average speed

1. $v_{mp}-1,v_{av}-2,v_{rms}-3$
2. $v_{mp}-2,v_{av}-1,v_{rms}-3$
3. $v_{mp}-3,v_{av}-1,v_{rms}-2$
4. $v_{mp}-3,v_{av}-2,v_{rms}-1$

Q.

The density of a mixture of ${\mathrm{O}}_2\mathrm{and}{\mathrm{N}}_2\mathrm{is}1.33\mathrm{g}/\mathrm{L}$. Calculate the partial pressure of O₂.

Q. The de Broglie wavelength of a neutron corresponding to root mean square speedat ${927}^{\circ }C\text{is}\lambda$. What will be the de Broglie wavelength of the root mean square speed at ${27}^{\circ }C$ ?

1. $\frac{\lambda }{2}$
2. $\lambda$
3. $2\lambda$
4. $4\lambda$

Q. A system consists of two types of gas molecules $A$ and $B$ having the same number density $2\times {10}^{25}{\text{m}}^{-3}$. The diameter of $A$ and $B$ are $10\dot{A}$ and $5\dot{A}$ respectively. They suffer collision at room temperature. The ratio of average distance covered by the molecule $A$ to that of $B$ between two successive collision is $\times {10}^{-2}$

Q.

An athlete covers $100m$ in $9seconds$. Determine his speed?

Q.

At what temperature the most probable speed of ${\mathrm{SO}}_2$ molecules have the same value of root mean square speed of ${\mathrm{O}}_2$ molecules?

Q. Identify the correct order of temperature as indicated in given diagram for same gas.

1. $T_1>T_2=T_3$
2. $T_1=T_2>T_3$
3. $T_1>T_2>T_3$
4. $T_3>T_2>T_1$

Q. Assuming the nitrogen molecule is moving with r.m.s. velocity at $400\text{K}$, the de-broglie wavelength of nitrogen molecule is close to :
(Given : nitrogen molecule weight : $4.64\times {10}^{-26}\text{kg},$ Boltzman constant : $1.38\times {10}^{-23}\text{J/K},$ Planck constant :$6.63\times {10}^{-34}\text{J-s}$)

1. $0.24\mathring{\text{A}}$
2. $0.20\mathring{\text{A}}$
3. $0.34\mathring{\text{A}}$
4. $0.44\mathring{\text{A}}$

Q. An ideal gas is filled in an insulated cylinder fitted with a movable piston connected with an ideal spring (of force constant $K$) as shown in the figure. Initially, the spring is relaxed and the gas in the piston has volume $V_0$ and temperature $T_0$ . Now the gas in the cylinder is heated slowly so that the piston slides towards right to a distance $x$. Find the final pressure and temperature of the gas. (Given atmospheric pressure = $P_0$ and area of cross-section of the cylinder is $A$).

Q. When we increase the temperature of the gas stored in a closed container,

1. the number of molecules of gas increase.
2. the speed and kinetic energy of the gas molecules increase.
3. the most probable speed of the gas molecules increases.
4. the number of molecules travelling at probable speed decrease.
5. the pressure of the gas increases.

Q. Identify the correct order as indicated in the given diagram of different kind of speed of molecules of same gas.


$f\left(v\right)\to$ Maxwell speed distribution function
$v_{mp}\to$ Most probable speed
$v_{rms}\to$ Root mean square speed
$v_{av}\to$ Average speed

1. $v_{mp}-3,v_{av}-2,v_{rms}-1$
2. $v_{mp}-1,v_{av}-2,v_{rms}-3$
3. $v_{mp}-2,v_{av}-1,v_{rms}-3$
4. $v_{mp}-3,v_{av}-1,v_{rms}-2$

Q. Select the incorrect statement about Maxwell’s distribution.

1. Most of the molecules move with speed $V_{mps}$
2. The area under distribution curve gives the total number of molecules of the gas
3. $V_{rms}>V_{av}>V_{mps}$
4. The distribution curve is symmetric about the most probable speed

Q. According to Maxwell's speed distribution law for a gas confined in a container, which of the following statements is/are correct:

1. All the molecules in the container will have certain non zero speed
2. Some of the molecules will have zero speed
3. There will be no molecules travelling with the same speed
4. Molecules travelling with very high speeds are lesser in number

Q. The graph shows hypothetical speed distribution for a sample of 𝑁 gas particles (for $V_0,\frac{dN}{dV}=0$)

1. The value of $a{V}_0$ is $2N$
2. The ratio $\frac{V_{avg}}{V_0}$ is equal to $\frac{2}{3}$
3. The ratio $\frac{V_{rms}}{V_0}$ is equal to $\frac{1}{\sqrt{2}}$
4. Three-fourths of the total particles have a speed between $0.5V_0$ and $V_0$

Q. According to Maxwell distribution of speeds which one is correct? a) no. of molecules having most probable speed is higher than no. of molecules moving with rms speed b) average velocity is nonzero

Q.

Mark the correct options looking to the given Maxwell’s distribution Curve.

1. $V_A=V_{rms}$
2. $V_B=V_{Avg}$
3. $V_C=V_{peak}$
4. $V_C=V_{rms}$

Q. A Carnot engine operating between temperatures $T_1$ and $T_2$ has efficiency 0.2. When $T_2$ is reduced by 60 K, its efficiency increases to 0.4. Then, $T_1$ and $T_2$ are respectively

1. 200 K, 150 K
2. 300 K, 200 K
3. 300 K, 240 K
4. 250 K, 200 K
5. 300 K, 150 K

Q.

A box of volume 22.4L filled with helium at STP, will have $N_A=6.023\times {10}^{23}$ atoms. Let us look at a much simpler, unrealistic sample of gas where we just have five atoms, having the following velocity components along a fixed direction: 1337.2 m/s, 1399.6 m/s, 1362.4m/s and1382.1m/s. Calculate the rms speed along the given direction, and the average kinetic energy of each molecule. The mass of a helium atom is $6.646\times {10}^{-27}kg$.

1. 1300 m/s, 0.5210 eV

2. 1290 m/s, 0.2412 eV

3. 1370 m/s, 0.0389 eV

4. 2239 m/s, 0.1002 eV

Q. The average speed of all the molecules in a gas at a given instant is zero, whereas the average velocity of all the molecules is zero. Explain why?

Q. Which of the Maxwell’s Distribution Curve represents its particular temperature?

1. $T_1>T_2>T_3$
2. $T_1>T_2<T_3$

3. Can’t be said

4. $T_1<T_2<T_3$

Q. Maxwell"s velocity distribution curve is given for the same quantity for two different temperatures. For the given curves then which of the folowing relation is true?

1. $T_1>T_2$
2. $T_1<T_2$
3. $T_1\le T_2$
4. $T_1=T_2$

Q. A vessel contains a mixture of one mole of $He$ & three moles of $Ar$ at 500 K. The ratio of the average translational kinetic energy per $Ar$ molecule to per $He$ molecule is

1. 5 : 3
2. 32 : 4
3. 1 : 1
4. 9: 1

Q. The molecules of a given mass of a gas have r.m.s. velocity of $200m{s}^{-1}$ at ${27}^{\circ }C$ and $1.0\times {10}^{-5}N{m}^{-2}$ pressure. When the temperature and pressure of the gas are respectively, ${127}^{\circ }C$ and $0.05\times {10}^{5}N{m}^{-2}$, the r.m.s. velocity of its molecules in $m{s}^{-1}$ is :

1. $\frac{400}{\sqrt{3}}$
2. $100\sqrt{2}$
3. $\frac{100\sqrt{2}}{3}$
4. $\frac{100}{3}$

Q. The average velocity of molecules in a gas in equilibrium is :

1. proportional to $\sqrt{T}$
2. proportional to $T$
3. proportional to $T^2$
4. equal to zero

Q. The average kinetic energy of a gas molecule at ${27}^{o}C$ is $6.21\times {10}^{-21}J$, then its average kinetic energy at ${227}^{o}C$ is:

1. $52.2\times {10}^{-21}J$
2. $10.35\times {10}^{-21}J$
3. $11.35\times {10}^{-21}J$
4. $5.22\times {10}^{-21}J$

Q. A graph of a cumulative frequency distribution is called:

1. Frequency Polygon
2. Ogive
3. None of the above
4. Histogram

Q. The given graph shows a hypothetical speed distribution for a sample of $N$ gas particles [Given that, for $V>V_0;\frac{dN}{dV}=0),\frac{dN}{dV}$ is change in number of particles with change in velocity]

Select the correct choice (s):

1. The value of $a{V}_0$ is $2N$
2. The ratio $\frac{V_{avg}}{V_0}$ is equal to $\frac{2}{3}$
3. The ratio $\frac{V_{rms}}{V_0}$ is equal to $\frac{1}{\sqrt{2}}$
4. Three fourth of the total particles has a speed between $0.5V_0$ and $V_0$

Q.

Three closed vessels A, B and C are at the same temperature T and contain gases which obey Maxwell distribution law of velocities. Vessel A contains $O_2$, B only $N_2$ and C mixture of equal quantities of $O_2$ and $N_2$. If the average speed of $O_2$ molecules in vessel A is $V_1$ and that of $N_1$ molecules in vessel B is $V_2$, then the average speed of the $O_2$ molecules in vessel C is

1. $V_1$
2. $\frac{(v_1+v_2)}{2}$
3. None of these
4. $\sqrt{v_1{v}_2}$