Kinetic Theory Test

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Kinetic Theory Test - 10

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Question 1
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Three perfect gases at absolute temperatures $$\mathrm{T}_{1},\ \mathrm{T}_{2}$$ and $$\mathrm{T}_{3}$$ are mixed. The masses of molecules are $$\mathrm{m}_{1},\ \mathrm{m}_{2}$$ and $$\mathrm{m}_{3}$$ and the number of molecules are $$\mathrm{n}_{1},\ \mathrm{n}_{2}$$ and $$\mathrm{n}_{3}$$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

A
$$\displaystyle \frac{(\mathrm{T}_{1}+\mathrm{T}_{2}+\mathrm{T}_{3})}{3}$$

B
$$\displaystyle \frac{\mathrm{n}_{1}\mathrm{T}_{\mathrm{l}}+\mathrm{n}_{2}\mathrm{T}_{2}+\mathrm{n}_{3}\mathrm{T}_{3}}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}}$$

C
$$\displaystyle \frac{\mathrm{n}_{1}\mathrm{T}_{1}^{2}+\mathrm{n}_{2}\mathrm{T}_{2}^{2}+\mathrm{n}_{3}\mathrm{T}_{3}^{3}}{\mathrm{n}_{1}\mathrm{T}_{1}+\mathrm{n}_{2}\mathrm{T}_{2}+\mathrm{n}_{3}\mathrm{T}_{3}}$$

D
$$\displaystyle \frac{\mathrm{n}_{1}^{2}\mathrm{T}_{1}^{2}+\mathrm{n}_{2}^{2}\mathrm{T}_{2}^{2}+\mathrm{n}_{3}\mathrm{T}_{3}^{3}}{\mathrm{n}_{1}\mathrm{T}_{1}+\mathrm{n}_{2}\mathrm{T}_{2}+\mathrm{n}_{3}\mathrm{T}_{3}}$$

Solution

Number of moles of first gas $$=\displaystyle \frac{\mathrm{n}_{1}}{\mathrm{N}_{\mathrm{A}}}$$
Number of moles of second gas $$=\displaystyle \frac{\mathrm{n}_{2}}{\mathrm{N}_{\mathrm{A}}}$$
Number of moles of third gas $$=\displaystyle \frac{\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}}$$
If no loss of energy then
$$\mathrm{P}_{1}\mathrm{V}_{1}+\mathrm{P}_{2}\mathrm{V}_{2}+\mathrm{P}_{3}\mathrm{V}_{3}=$$ $$PV$$

$$\displaystyle
\frac{\mathrm{n}_{1}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{1}+\frac{\mathrm{n}_{2}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{2}+\frac{\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{3}
$$ $$\displaystyle
=\frac{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{mix}
$$

$$\implies \displaystyle \mathrm{T}_{mix}=\frac{\mathrm{n}_{1}\mathrm{T}_{1}+\mathrm{n}_{2}\mathrm{T}_{2}+\mathrm{n}_{3}\mathrm{T}_{3}}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}}$$.
Question 2
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An ideal gas goes through a reversible cycle $$a\rightarrow b\rightarrow c\rightarrow d$$ has the V- T diagram shown below. Process $$d\rightarrow a $$ and $$b\rightarrow c$$ are adiabatic.
The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

A

B

C

D

Solution

In process $$a \rightarrow b$$ and $$c \rightarrow d$$ the volume is directly proportional to temperature of gas, hence the process is isobaric, however, the slope of curve is different in each case. using ideal gas equation
$$PV=nRT \Rightarrow V=\dfrac{nR}{P} T$$
slope $$m=\dfrac{nR}{P}$$ for a high pressure the slope of the V-T curve will be less. In given diagram the slope of $$c \rightarrow d$$ process is lower than $$a \rightarrow b$$ process, hence process $$a \rightarrow b$$ is at higher pressure, from which option B and D can be ruled out.
Now initial rate of change of volume w.r.t. temperature is very less from $$b \rightarrow c$$ and $$d \rightarrow a$$ as shown in V-T diagram.
hence slope of P-V curve should tend to $$\lim_{\Delta V\rightarrow 0}\dfrac{\partial P }{\partial v}=\infty$$ as initial change in volume is nearly zero, which can be verified in option A
Hence correct answer is option A.
Question 3
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In the nuclear fusion reaction, $$_{1}^{2}\mathrm{H}+_{1}^{3}\mathrm{H}\rightarrow_2^4{He}$$ $$+\ {n}$$ given that the repulsive potential energy between the two nuclei is $$7.7\times 10^{-14}\ {J}$$, the temperature at which the gases must be heated to initiate the reaction is nearly [Boltzmann's constant $$\mathrm{k}=1.38\times 10^{-23}\mathrm{J}/\mathrm{K}$$]

A
$$10^{7}\ {K}$$

B
$$10^{5}\ {K}$$

C
$$10^{3}\ {K}$$

D
$$10^{9}\ {K}$$

Solution

To initiate the reaction, the energy of the gas molecules must be equal to the repulsive potential energy.

We know that the kinetic energy of translation per molecules is given by:
$$K.E.=\dfrac32kT$$

In order to initiate the fusion reaction,

$$\dfrac{3}{2}kT=7.7\times 10^{-14}J$$

$$\implies T=\dfrac{2\times 7.7\times 10^{-14}}{3\times 1.38\times 10^{-23}}\approx3.7\times 10^{9}K$$
Question 4
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An ideal gas is enclosed in a cylinder at pressure of $$2 \,atm$$ and temperature, $$300 \,K$$. The mean time between two successive collisions is $$6 \times 10^{-8} \,s$$. If the pressure is doubled and temperature is increased to $$500 \,K$$, the mean time between two successive collisions will be close to:

A
$$4 \times 10^{-8} \,s$$

B
$$3 \times 10^{-6} \,s$$

C
$$2 \times 10^{-7} \,s$$

D
$$0.5 \times 10^{-8} \,s$$

Solution

$$t \alpha \dfrac{Volume}{Velocity}$$

Volume $$\alpha \dfrac{T}{P}$$

$$\therefore t \alpha \dfrac{\sqrt{T}}{P}$$

$$\dfrac{t_1}{6 \times 10^{-8}} = \dfrac{\sqrt{500}}{2P} \times \dfrac{P}{\sqrt{300}}$$

$$t_1 = 3.8 \times 10^{-8}$$
$$\approx 4 \times 10^{-8}$$
Question 5
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$$'N'$$ moles of a diatomic gas in a cylinder are at a temperature $$'T'$$. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas gets converted into monatomic gas. What is the change in the total kinetic energy of the gas ?

A
$$\dfrac{5}{2}nRT$$

B
$$\dfrac{1}{2}nRT$$

C
$$0$$

D
$$\dfrac{3}{2}nRT$$

Solution

$$\Rightarrow { KE }_{ initial }=\cfrac { 3 }{ 2 } nRT$$
Final moles$$({ n }_{ 2 })=2n$$
Since monoatomic gas conversion is there
$$\Rightarrow { KE }_{ final }=\cfrac { 3 }{ 2 } \times 2nRT$$
$$\Rightarrow \Delta KE=\cfrac { 3 }{ 2 } nRT$$
Question 6
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Which of the following shows the correct relationship between the pressure $$'P'$$ and density $$\rho$$ of an ideal gas at constant temperature ?

A

B

C

D

Solution

From ideal gas law:
$$PV=nRT$$
$$P=\cfrac{MnRT}{MV}, \quad M:$$ Molecular mass number
But $$\rho=Mn/V$$
$$P=\rho RT/M$$

At constant temperature, $$P \propto \rho$$
Hence, the correct graph is (d).
Question 7
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One $$kg$$ of a diatomic gas is at a pressure of $$ 8\times 10^{4}\ \mathrm{N}/\mathrm{m}^{2}$$. The density of the gas is $$4$$ $$\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$$. What is the energy of the gas due to its thermal motion ?

A
$$3\times 10^{4}\mathrm{J}$$

B
$$5\times 10^{4}\mathrm{J}$$

C
$$6\times 10^{4}\mathrm{J}$$

D
$$7\times 10^{4}\mathrm{J}$$

Solution

$$\displaystyle \mathrm{E}=\frac{5}{2}\mathrm{n}\mathrm{R}\mathrm{T}$$
$$=\displaystyle \frac{5}{2}$$ PV
$$\displaystyle =\frac{5}{2}\mathrm{P}\frac{\mathrm{m}}{\mathrm{P}}$$
$$=\displaystyle \frac{5}{2}\times\frac{(8\times 10^{4})(1)}{4}=5\times 10^{4} \mathrm{J}$$
Question 8
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The gas mixture constists of $$3$$ moles of oxygen and $$5$$ moles of argon at temperature $$T$$. Considering only translational and rotational modes, the total internal energy of the system is:

A
$$12\ RT$$

B
$$20\ RT$$

C
$$15\ RT$$

D
$$4\ RT$$

Solution

$$U=\dfrac{f_1}{2}n_1RT+\dfrac{f_2}{2}n_2RT$$
$$=\dfrac{5}{2}(3RT)+\dfrac{3}{2}\times 5RT$$
$$U=15RT$$
Question 9
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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
1 : 4

B
1 : 2

C
6 : 9

D
8 : 9

Solution

$$
Sol$$ : (D)
$$
PV
=nRT
=\dfrac{m
}{M
}RT
$$
$$
$$
$$
\Rightarrow PM
=\rho RT
$$
$$
$$
$$
\dfrac{\rho
_{1}}{\rho
_{2}}=\dfrac{P
_{1}M
_{1}}{P
_{2}M
_{2}}=(\dfrac{P
_{1}}{P
_{2}})\times(\dfrac{M
_{1}}{M
_{2}})=\dfrac{4}{3}\times\dfrac{2}{3}=\dfrac{8}{9}
$$
Here $$\rho _{1}$$ and $$\rho _{2}$$ are the densities gases in the vessel containing the mixture.
Question 10
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A gas molecule of mass $$M$$ at the surface of the Earth has kinetic energy equivalent to $$0^oC$$. If it were to go up straight without colliding with any other molecules, how high it would rise?
Assume that the height attained is much less than radius of the earth. ($$k_B$$ is Boltzmann constant)

A
$$0$$

B
$$\dfrac {273 k_B}{2 Mg}$$

C
$$\dfrac {546 k_B}{3 Mg}$$

D
$$\dfrac {819 k_B}{2 Mg}$$

Solution

Answer is D
Average translational kinetic energy of molecules=
$$\displaystyle KE= \frac{3}{2} k_B T= \frac{819}{2} k_B as T= 273K$$
Now. KE= PE
Or,
$$\displaystyle \frac{819}{2} k_B = Mgh$$
$$\displaystyle h= \frac{819k_B}{2Mg}$$