Tag: kinetic theory of gases

Questions Related to kinetic theory of gases

The diameter of oxygen molecules is $2.94 \times 10^{-10}m $. The Van der Waals gas constant in m$^3$/mol will be

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. $3.2$

  2. $32$

  3. $32\times 10^{-6}$

  4. $32 \times 10^{-3}$

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Correct Option: C
Explanation:

$\displaystyle b = 4N \times \frac{4}{3} \pi \frac{d^3}{8}$ (standard definition of boyles constant)

$\displaystyle = \frac{4 \times 6.02 \times 10^{23} \times 3.14 \times 2.94^3 \times 10^{-30}}{3 \times 8}$

$\displaystyle = 32 \times 10^{-6}$

Read the given statements and choose which is/are on the basis of kinetic theory of gases.

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. Energy of one molecule at absolute temperature is zero.

  2. $rms$ speeds of different gases are same at same temperature

  3. For one gram of all ideal gases, kinetic energy is same at same temperature.

  4. For one mole of all ideal gases, mean kinetic energy is same at same temperature.

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Correct Option: A

Work done by a system under isothermal change from a volume $V _1$ to $V _2$ for a gas, which obeys vander Waals equation $(V - \beta n) \displaystyle \left ( P + \dfrac{an^2}{V} \right ) = n RT$ is

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. $\displaystyle n RT log _e \left ( \dfrac{V _2 - n \beta}{V _1 - n \beta} \right ) + an^2 \left ( \dfrac{V _1 - V _2}{V _1 V _2} \right )$

  2. $\displaystyle n RT log _{10} \left ( \dfrac{V _2 - \alpha \beta}{V _1 - \alpha \beta} \right ) + \alpha n^2 \left ( \dfrac{V _1 - V _2}{V _1 V _2} \right )$

  3. $\displaystyle n RT log _e \left ( \dfrac{V _2 - n \alpha}{V _1 - n \alpha} \right ) + \beta n^2 \left ( \dfrac{V _1 - V _2}{V _1 V _2} \right )$

  4. $\displaystyle n RT log _e \left ( \dfrac{V _2 - n \beta}{V _1 - n \beta} \right ) + \alpha^2 \left ( \dfrac{V _1 V _2}{V _1- V _2} \right )$

Show answer
Correct Option: A
Explanation:
Given Vander Waals equation $(V - \beta n)$ ($P$ $+$ $\dfrac {a{n}^{2}} {{V}^{2}}$) $=$ $nRT$
Work done by the system($W$) = $-$ $\int _{{V} _{1}}^{{V} _{2}} {PdV}$
                                           $=$ $\int _{{V} _{2}}^{{V} _{1}} {PdV}$
From the Vander Waals equation:
$P$ $=$ $\dfrac {nRT} {(V - \beta n)}$ $-$ $\dfrac {a{n}^2} {{V}^{2}}$
Substituting $'P'$ in the Work done, we get
$W$ $=$ $\int _{{V} _{2}}^{{V} _{1}}$ ($\dfrac {nRT} {(V - \beta n)}$ $-$ $\dfrac {a{n}^2} {{V}^{2}}$) $dV$
By integrating we get,
$W$ $=$ $[$ $nRT$ $\log _{e}{(V - \beta n)}$ $-$ ($(-)\dfrac {a{n}^{2}} {V}$) ] $ _{{V} _{2} \rightarrow {V} _{1}}$
$W$ $=$ $nRT$ $\log _{e}{}$ $($$\dfrac { {V} _{2} - \beta n} {{V} _{1} - \beta n} $ $)$ $+$ $a{n}^{2}$$($ $\dfrac {{V} _{1} - {V} _{2}} { {V} _{1}{V} _{2}}$ $)$
Hence, the Correct Option is $'A'$.

An ideal gas is at a temperature  $T$  having molecules each of mass  $m .$  If  $k$  is the Boltzmann's constant and  $2 \mathrm { kT } / \mathrm { m } = 1.40 \times 10 ^ { 5 } \mathrm { m } ^ { 2 } / \mathrm { s } ^ { 2 } .$  Find the percentage of the fraction of molecules whose speed lie in the range  $324\mathrm { m } / \mathrm { s }$  to  $326\mathrm { m } / \mathrm { s } .$

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. $0.52 \%$

  2. $0.43 \%$

  3. $0.21 \%$

  4. $0.14 \%$

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Correct Option: A

In Vander Waal's equation the critical $P _{c}$ is given by

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. 3b

  2. $\displaystyle\ \frac{a}{27b^{2}}$

  3. $\displaystyle\ \frac{27a}{b^{2}}$

  4. $\displaystyle\ \frac{b^{2}}{a}$

Show answer
Correct Option: B
Explanation:

The Vander Wall's equation of state is 
$\left(P+ \displaystyle\ \frac{a}{V^{2}}\right)$ $(V-b)$ = $RT$
$P = \displaystyle\ \frac{RT} {V-b} -\displaystyle\ \frac{a}{V^{2}}$
At the critical point, 
$P = P _{C}, V= V _{C}$ and $T= T _{C}$ 
$\therefore P _{C} = \displaystyle\ \frac{RT _{C}}{V _{C}-b} - \frac{a}{V _{C}^{2}}$ .......(i)
At the critical point on the isothermal,
$\displaystyle\ \frac{dP _{C}}{dV _{C}} = 0$
$\therefore 0 = \displaystyle\ \frac{-RT _{C}}{(V _{C}-b)^{2}} + \frac{1a}{V _{C}^{3}}$
$\displaystyle\ \frac{RT _{C}}{(V _{C}-b)^{2}} = \displaystyle\ \frac{2a}{V _{C}^{3}}$ .... (ii)
Also at critical point,
$\displaystyle\ \frac{d^{2}P _{C}}{dV _{C}^{2}} = 0$
$0 = \displaystyle\ \frac{2RT}{(V _{C}-b)^{3}}- \frac{6a}{V _{C}^{4}}$
$\displaystyle\ \frac{2RT _{C}}{(V _{C-b})^{3}}$ = $\displaystyle\ \frac{5a}{V _{C}^{4}}$ ....(iii)
Dividing (ii) by (iii) we get
$\displaystyle\ \frac{1}{2}(V _{C}-b)$ = $\displaystyle\ \frac{1}{3} V _{C}$
$V _{C} = 3b$ ..... (iv)
Putting this value in (ii), we get 
$\displaystyle\ \frac{RT _{C}}{4b^{2}} = \displaystyle\ \frac{2a}{27{b}^{3}}$
$T _{C} = \displaystyle\ \frac{8a}{27bR}$ .......(v)
Putting the value of $V _{C}$ and $T _{C}$ in (i), we get 
$P _{C}$ = $\displaystyle\ \frac{R}{2b}\left(\frac{8a}{27bR}\right)$ = $\displaystyle\ \frac{a}{9b^{2}}$
$= \displaystyle\ \frac{a}{27b^{2}}$

The temperature of an ideal gas at atmospheric pressure is 300K and volume $lm^3$.If temperature and volume become double, then pressure will be

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. $10^5 N/m^2$

  2. $2\times 10^5 N/m^2$

  3. $0.5\times 10^5 N/m^2$

  4. $4\times 10^5 N/m^2$

Show answer
Correct Option: A
Explanation:

$\begin{array}{l} \dfrac { { { P _{ 1 } }{ V _{ 1 } } } }{ { { T _{ 1 } } } } =\dfrac { { { P _{ 2 } }{ V _{ 2 } } } }{ { { T _{ 2 } } } }  \ \Rightarrow \dfrac { { { { 10 }^{ 5 } }\times \left( { 1{ m^{ 3 } } } \right)  } }{ { 300K } } =\dfrac { { P\left( 2 \right)  } }{ { 600 } }  \ \Rightarrow P={ 10^{ 5 } }N/{ m^{ 2 } } \ Hence, \ option\, \, A\, \, is\, correct\, \, answer. \end{array}$

Assertion: Real gases do not obey the ideal gas equation.

Reason: In the ideal gas equation, the volume occupied by the molecules as well as the inter molecular forces are ignored.

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. Both assertion (A) and reason (R) are correct and R gives the correct explanation

  2. Both assertion (A) and reason (R) are correct but R doesnt give the correct explanation

  3. A is true but R is false

  4. A is false but R is true

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Correct Option: A
Explanation:

The ideal gas law treats the molecules of a gas as point particles with  perfectly elastic collisions. This works well for dilute gases in many experimental circumstances. But gas molecules are not point masses, and there are circumstances where the properties of the molecules have an experimentally measurable effect.

In the year 1984, the Bhopal gas tragedy was caused by the leakage of

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. Carbon monoxide

  2. Methyl isocyanate

  3. Nitrogen oxide

  4. Sulphur oxide

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Correct Option: B
Explanation:

The Bhopal gas tragedy was an industrial catastrophe that occurred in 1984 at the Methyl isocynate gas ($CH _3NCO$) was leaked from the plant. Union Carbide India Limited (UCIL) pesticide plant in Bhopal, Madhya Pradesh.

A real gas can be approximated to an ideal gas at

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. Low density

  2. High pressure

  3. High density

  4. Low temperature

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Correct Option: A
Explanation:

Real gas can be approximated as ideal gas when the pressure is low and the temperature is high
This means that per unit volume, there are less number of gas molecules because there is less force (pressure) and there is more energy (temperature), so the molecules will tend to move apart
So, density will be low.

If N be the Avogardo's number and R be the gas constant , then Boltzmann constant id given by 

real gases van der-waal equation: equation of state for real gas kinetic theory of gases thermal physics physics

  1. RN

  2. R/N

  3. N/R

  4. I/RN

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Correct Option: B