JEE Advanced PYQ Kinetic Theory of an Ideal Gas and Gas Laws

Q 1 :

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 [2013]

1 : 4
1 : 2
6 : 9
8 : 9

1

2

3

4

(4)

$P_{1} M_{1} = P_{1} R T and P_{2} M_{2} = P_{2} R T$

$∴$ $\frac{P_{1}}{P_{2}} \times \frac{M_{1}}{M_{2}} = \frac{P_{1}}{P_{2}}$

$\frac{4}{3} \times \frac{2}{3} = \frac{P_{1}}{P_{2}}$

$∴$ $\frac{P_{1}}{P_{2}} = \frac{8}{9}$

Q 2 :

A real gas behaves like an ideal gas if its [2010]

pressure and temperature are both high
pressure and temperature are both low
pressure is high and temperature is low
pressure is low and temperature is high

1

2

3

4

(4)

A real gas behaves as an ideal gas at low pressure and high temperature.

Q 3 :

Which of the following graphs correctly represents the variation of $β = - \frac{d V / d P}{V}$ with $P$ for an ideal gas at constant temperature? [2002]

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1

2

3

4

(1)

$According to Boyle's law, P V = constant.$

$∴$ $P d V + V d P = 0$

$\Rightarrow \frac{P d V}{d P} = - V$ $; β = - (\frac{1}{V}) (\frac{d V}{d P}) = (\frac{1}{P})$

$\Rightarrow β \times P = 1$

$Hence graph between β and P will be a rectangular hyperbola.$

Q 4 :

A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be ________ (take the acceleration due to gravity = 10 ${ms}^{- 2}$ and the universal gas constant = 8.3 J ${mol}^{- 1} K^{- 1}$ ). [2020]

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(6)

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$Initially partition is held at a distance of 4 m from the top.$

$Temperature, T = 300 K$

$P V = P A h = n R T$

$P_{0} 4 A = 0.1 R \times 300$ $(For both)$

$Let piston shifts by x meter$ $and final temperature is T, then$

$P_{2} A (4 - x) = 0.1 R T$

$∴$ $P_{2} A = \frac{0.1 R T}{4 - x} = \frac{R T}{10 (4 - x)}$ $and P_{1} A = \frac{R T}{10 (4 + x)}$

$Finally P_{2} A - P_{1} A = m g = 83 = \frac{R T}{10} (\frac{1}{4 - x} - \frac{1}{4 + x})$

$\Rightarrow \frac{R T}{10} (\frac{2 x}{16 - x^{2}}) = 83$

$0.2 C_{v} \times 300 + m g \cdot x = 0.2 C_{v} T$

$\Rightarrow \frac{3}{2} R \times \frac{2}{10} \times 300 + 83 x = \frac{2}{10} \times \frac{3}{2} R T$

$\Rightarrow (\frac{900 R}{10} + 83 x) = 3 (\frac{R T}{10})$

$\Rightarrow (\frac{300 R}{10} + \frac{83}{3} x) = \frac{R T}{10}$ $\Rightarrow (30 R + \frac{83 x}{3}) (\frac{2 x}{16 - x^{2}}) = 83$

$\Rightarrow 83 (3 + \frac{x}{3}) (\frac{2 x}{16 - x^{2}}) = 83$ $\Rightarrow \frac{(9 + x)}{3} (\frac{2 x}{16 - x^{2}}) = 1$

$\Rightarrow 18 x + 2 x^{2} = 48 - 3 x^{2}$ $\Rightarrow 5 x^{2} + 18 x - 48 = 0$

$∴$ $x = 1.78 \approx 2$

$Hence distance from top when reached equilibrium = 4 + 2 = 6 m$

Q 5 :

A cylindrical furnace has height (H) and diameter (D) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure $P_{a}$ and its temperature becomes T = 360 K. The hot air with density $ρ$ rises up a vertical chimney of diameter d = 0.1 m and height h = 9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $ρ_{a} = 1.2 {kg m}^{- 3}$ , pressure $P_{a}$ and temperature $T_{a} = 300 K$ enters the furnace. Assume air as an ideal gas, neglect the variations in $ρ$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.

[Given the acceleration due to gravity $g = 10 {m s}^{- 2}$ and $π = 3.14$ ]

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Q. Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is ________ gm $s^{- 1}$ . [2023]

(47.10)

$Since, pressure P = constant$ $ρ_{a} T_{a} = ρ T$

$\Rightarrow 1.2 \times 300 = ρ (360)$

$∴$ $ρ = 1 {kg/m}^{3}$

$Applying Bernoulli's theorem between upper and bottom point$

$Assuming velocity of hot air inside the furnace ≃ 0$

$P_{a} + 0 + 0 = P_{a} - ρ_{a} g (h) + ρ g (h) + \frac{1}{2} ρ V^{2}$

$∴$ $V = \sqrt{\frac{2 (ρ_{a} - ρ) g \times 9}{ρ}} = \sqrt{2 (0.2) 90} = 6$

$Therefore the steady mass flow rate of air existing the chimney$

$Q = ρ π (\frac{d^{2}}{4}) V$ $= 1 \times 3.14 \times \frac{{(0.1)}^{2}}{4} \times 6$

$= 0.0471 kg/s = 47.10 {gms}^{- 1}$

Q 6 :

A cylindrical furnace has height (H) and diameter (D) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure $P_{a}$ and its temperature becomes T = 360 K. The hot air with density $ρ$ rises up a vertical chimney of diameter d = 0.1 m and height h = 9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $ρ_{a} = 1.2 {kg m}^{- 3}$ , pressure $P_{a}$ and temperature $T_{a} = 300 K$ enters the furnace. Assume air as an ideal gas, neglect the variations in $ρ$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.

[Given the acceleration due to gravity $g = 10 {m s}^{- 2}$ and $π = 3.14$ ]

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Q. When the chimney is closed using a cap at the top, a pressure difference $ΔP$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $ΔP$ is ________ ${Nm}^{- 2}$ . [2023]

(18)

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$Pressure P_{a} = P_{in} + ρ g (h)$

$\Rightarrow P_{a} = P_{inside} + ρ g (9)$

$\Rightarrow P_{inside} = P_{a} - ρ g (9)$

$And, P_{outside} = P_{a} - ρ_{a} g (9)$

$∴$ $Δ P = P_{inside} - P_{outside}$

$= (ρ_{a} - ρ) g \times 9$

$= (1.2 - 1) \times 10 \times 9 = 18 {N/m}^{2}$

Topic Question Set

Q 1 :

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 [2013]

Q 2 :

A real gas behaves like an ideal gas if its [2010]

Q 3 :

Which of the following graphs correctly represents the variation of $β = - \frac{d V / d P}{V}$ with $P$ for an ideal gas at constant temperature? [2002]

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Topic Question Set

Q 1 : 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 [2013]

Q 2 : A real gas behaves like an ideal gas if its [2010]

Q 3 : Which of the following graphs correctly represents the variation of β=-dV/dPV with P for an ideal gas at constant temperature? [2002]

Want Us to Email you About Special Offers & Updates?

Q 1 :

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 [2013]

Q 2 :

A real gas behaves like an ideal gas if its [2010]

Q 3 :

Which of the following graphs correctly represents the variation of $β = - \frac{d V / d P}{V}$ with $P$ for an ideal gas at constant temperature? [2002]