Tag: kinetic theory of gases

Questions Related to kinetic theory of gases

Real gases approaches ideal gas at high temperature and low pressure because

$A$.   Inter atomic separation is large 

$B$.   Size of the molecule is negligible when compared to inter atomic separation 

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

  1. a & b are true

  2. only a is true

  3. only b is true

  4. a & b are false

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Correct Option: A
Explanation:

Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure as the forces against intermolecular forces becomes less significant compared to the particles' kinetic energy, and the size of the molecules becomes less significant compared to the empty space between them.

A sample of an ideal gas occupies a volume V at a pressure P and absolute temperature T, the mass of each molecule is m. The expression for the density of gas is (k= Boltzmann's constant)

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

  1. $mkT$

  2. $P/kT$

  3. $P/kTV$

  4. $Pm/kT$

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Correct Option: D

The equation of state of n moles of a non-ideal gas can be approximated by the equation 
$ (P + \dfrac{an^2}{V^2})(V -nb) = nRT $ 
where a and b are constants characteristics of the gas. Which of the following can represent the equation of a quasistatic adiabat for this gas (Assume that $C _V$ , the molar heat capacity at constant volume, is independent of temperature) ?

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

  1. $T(V-nb)^{R/C _v}=$ constant

  2. $T(V-nb)^{C _v/R}=$ constant

  3. $ \begin {pmatrix} T + \frac {ab}{V^2R} \end{pmatrix} (V-nb)^{R/C _v} = $ constant

  4. $ \begin {pmatrix} T + \frac {n^2 ab}{V^2R} \end{pmatrix} (V-nb)^{C _v/R} = $ constant

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Correct Option: A
Explanation:

For  a reversible adiabatic process, we have $dS = 0$ (Entropy change = 0)


The entropy equation is $TdS = nC _VdT+T(\frac{\partial P}{\partial T}) _VdV$

From the non-ideal gas equation, $(P+\frac{an^2}{V^2})(V-nb)=nRT$
$(\frac{\partial P}{\partial T}) _V=\frac{nR}{V-nb}$

for $dS = 0$, we have
$nC _VdT = -T(\frac{\partial P}{\partial T}) _VdV=-nRT\frac{dV}{V-nb}$
$\Rightarrow \frac{dT}{T} = -\frac{nR}{C _V}\frac{dV}{V-nb}$
$\Rightarrow ln(\frac{T}{T _0})=-\frac{R}{C _V} ln(\frac{V-nb}{V _0-nb})$

$\Rightarrow T(V-nb)^{\frac{R}{C _V}}=T _0(V _0-nb)^{\frac{R}{C _V}}$

i.e., $T(V-nb)^{\frac{R}{C _V}} = \textrm{constant}$

The size of container B is double that of A and gas in B is at double the temperature and pressure than that in A. The ratio of molecules in the two containers will then be -

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

  1. $\frac{N _B}{N _A} = \frac{1}{1}$

  2. $\frac{N _B}{N _A} = \frac{2}{1}$

  3. $\frac{N _B}{N _A} = \frac{4}{1}$

  4. $\frac{N _B}{N _A} = \frac{1}{2}$

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Correct Option: A

Two vertical parallel glass plates are partially submerged in water. The distance between the plates is $d = 0.10 mm$, and their width is $l  = 12 cm$. Assuming that the water between the plates does not reach the upper edges of the plates and that the wetting is complete, find the force of their mutual attraction.

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

  1. $17N$

  2. $13N$

  3. $10$

  4. $19N$

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Correct Option: B

For gaseous decomposition of ${PCI} _{5}$ in a closed vessel the degree of dissociation '$\alpha $', equilibrium pressure 'P' & ${'K} _{p}'$ are related as

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

  1. $\ \alpha =\sqrt { \frac { { K } _{ p } }{ P } } $

  2. $\ \alpha =\frac { 1 }{ \sqrt { { K } _{ p }+P } } $

  3. $\ \alpha =\sqrt { \frac { { K } _{ p }+P }{ { K } _{ p } } } $

  4. $\alpha =\sqrt { { K } _{ p }+P } $

Show answer
Correct Option: A
Explanation:

${  \quad \quad \quad \quad \quad \quad \quad PCl } _{ 5 }\rightleftharpoons { PCl } _{ 3(9) }+{ Cl } _{ 2(9) }\\ Initial\quad mole\quad \quad \quad 1\quad  \quad 0\quad\quad\quad 0\\ After\quad mole\quad \quad 1-\alpha \quad \quad  \alpha \quad\quad  \alpha \\ decomposition$

Total mole$=1-\alpha+\alpha+\alpha\\=1+\alpha$

Total pressure$=P$

Partial pressure of $PCl _5=P(\cfrac{1-\alpha}{1+\alpha})$

PArtial pressure of $PCl _3=P(\cfrac{\alpha}{1+\alpha})$

Partial pressure of $PCl _2=P(\cfrac{\alpha}{1+\alpha})$

Then $K _p=\cfrac{(PCl _3)(Cl _2)}{(PCl _5)}\\ \quad=\cfrac{P(\cfrac{\alpha}{1+\alpha})P(\cfrac{\alpha}{1+\alpha})}{P(\cfrac{1-\alpha}{1+\alpha})}\\ \quad=\cfrac{P^2\alpha^2}{(1+\alpha)^2}\times\cfrac{(1+\alpha)}{P(1-\alpha)}\\K _p=\cfrac{P\alpha^2}{1-\alpha^2}$

now, $1-\alpha^2<<1$

so that $K _p=P\alpha^2\\ \alpha^2=\cfrac{K _p}{P}\\ \alpha=\sqrt{\cfrac{K _p}{P}}$

 

If pressure of ${CO} _{2}$ (real gas) in a container is given by $P=\cfrac { RT }{ 2V-b } -\cfrac { a }{ 4{ b }^{ 2 } } $, then mass of the gas in container is:

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

  1. $11g$

  2. $22g$

  3. $33g$

  4. $44g$

Show answer
Correct Option: B
Explanation:

According to Van Der waal's equation for $n$ mole of real gas 


$\bigg( P +\dfrac{n^2 a}{V^2}\bigg)(V- nb)=nRT\implies P=\dfrac{nRT}{V-nb}-\dfrac{n^2a}{V^2}$

Given that Pressure of $CO _2$ gas in a contaner is given by:
$P= \dfrac{RT}{2V-b}-\dfrac{a}{4b^2}$

Compairing it with the standard Van der waal's equation we get :
$n=\dfrac12$

Therefore, Number of moles in a container , $n=\dfrac12$
Molar mass of $CO _2= 44\ gm$
Mass of gas in the container, $m= \dfrac12\times 44 =22 gm$