Assuming that the gas inside the sun behaves very much like the perfect gas, the temperature at the centre of the sun is nearly (the number density of gas particles = 2 ρ M H ), Boltzmann's constant k B = 1 . 4 × 10 – 23 J K – 1 , mass of proton M H = 1 . 67 × 10 – 27 kg , M 0 = 2 × 10 30 kg , R 0 = 7 × 10 5 km and pressure at the center of the sun, P C = 5 × 10 14 N m - 2

As we know that the ideal gas law in relation with Boltzmann's constant can be written as, P V = n N A k B T ∴ P = n N A k B T V … 1 (where P is pressure, n is number of moles, V is volume, k B is Boltzmann's constant and T is temperature) Number density of gas particle can be written as, n N A V which is equal to 2 ρ M H Also, ρ = M 0 V 0 ⇒ ρ = M 0 4 3 π R 0 3 Putting value of n N A V in equation 1 , we will get, P = 2 ρ M H k B T ⇒ P = 2 M 0 4 3 π R 0 3 M H k B T ⇒ T = 2 π R 0 3 3 × M H P k B M 0 ⇒ T = 2 3 × 3 . 14 × 7 × 10 8 3 × 1 . 67 × 10 - 27 × 5 × 10 14 1 . 4 × 10 - 23 × 2 × 10 30 ⇒ T = 2 . 2 × 10 7 K

Two vessels A and B , thermally insulated, contain an ideal monoatomic gas. A small tube fitted with a valve connects these vessels. Initially the vessel A has 2 liters of gas at 300 K and 2 × 10 5 N m - 2 pressure while vessel B has 4 liters of gas at 350 K and 4 × 10 5 Nm - 2 pressure. The valve is now opened and the system reaches equilibrium in pressure and temperature. Calculate the new pressure and temperature. R = 25 3 J mol - 1 - K - 1

For ideal gas, U = 3 2 n R T = 3 2 P V U 1 + U 2 = U 3 2 P 1 V ⁡ 1 + 3 2 P ⁡ 2 V ⁡ 2 = 3 2 PV ∴ P ⁡ = P ⁡ 1 V ⁡ 1 + P ⁡ 2 V ⁡ 2 V ⁡ On putting the values we have P = 2 × 2 × 10 5 + 4 × 4 × 10 5 2 + 4 = 3 . 3 × 1 0 5 N m - 2 Number of moles will be conserved. n = n 1 + n 2 and therefore P 1 V 1 R T 1 + P 2 V 2 R T 2 = P V R T 2 × 1 0 5 × 2 3 0 0 + 4 × 1 0 5 × 4 3 5 0 = 1 0 × 1 0 5 × 6 3 T 4 3 0 0 + 1 6 3 5 0 = 2 0 T T = 338 . 71 K

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The temperature of an open room of volume $30 m^{3}$ increased from $17 º C to 27 º C$ due to the sunshine. The atmospheric pressure in the room remains $1 \times 10^{5} Pa$ . If $n_{i}$ and $n_{f}$ are the number of molecules in the room before and after heating, then $n_{f} - n_{i}$ will be

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Important Questions on Kinetic Theory of Gases

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 1

Assuming that the gas inside the sun behaves very much like the perfect gas, the temperature at the centre of the sun is nearly (the number density of gas particles

= \frac{2 ρ}{M_{H}}

), Boltzmann's constant

k_{B} = 1.4 \times 10^{- 23} J K^{- 1}

, mass of proton

M_{H} = 1.67 \times 10^{- 27} kg

M_{0} = 2 \times 10^{30} kg

R_{0} = 7 \times 10^{5} km

and pressure at the center of the sun,

P_{C} = 5 \times 10^{14} N m^{- 2}

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 2

Find the pressure of air in a vessel being evacuated as a function of evacuation time $t$ . The vessel's volume is $V$ and initial pressure is $p_{0} .$ The process is assumed to be isothermal and the evacuation rate equal to $C$ , and independent of pressure.

Note: The evacuation rate is the gas volume being evacuated per unit time, with that volume being measured under the gas pressure attained by that moment.

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 3

Find the maximum attainable temperature of an ideal gas, in each of the following processes,

(a) $p = p_{0} - α V^{2}$ ,

(b) $p = p_{0} e^{- β V}$ ,

where, $p_{0}, α$ and $β$ are the positive constants and $V$ is the volume of one mole of gas.

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 4

Two vessels

A

and

B

, thermally insulated, contain an ideal monoatomic gas. A small tube fitted with a valve connects these vessels. Initially the vessel

A

has

2 liters

of gas at

300 K

and

2 \times 10^{5} N m^{- 2}

pressure while vessel

B

has

4 liters

of gas at

350 K

and

4 \times 10^{5} {Nm}^{- 2}

pressure. The valve is now opened and the system reaches equilibrium in pressure and temperature. Calculate the new pressure and temperature.

(R = \frac{25}{3} J {mol}^{- 1} - K^{- 1})

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 5

In the given figure a glass tube lies horizontally with the middle $20 cm$ containing mercury. The two ends of the tube contains air at $27 º C$ and at a pressure $76 cm$ of mercury. Now the air column on one side is maintained at $0 º C$ and the other side is maintained at $127 º C$ . Find the new length of the air column on the cooler side. Neglect the changes in the volume of mercury and of the glass.

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Alpha Question Bank for Engineering: Physics>Chapter 18 - Kinetic Theory of Gases>Exercise - 4>Q 6

A thermally insulated piston divides a nonconducting container in two compartments, right compartment of

2 V, T

and

2 P

, while in the left compartment the respective values are

V, T

and

P

. Total moles in total system of both compartments is

5 moles

(same molar mass). If the piston can slide freely, and in the final equilibrium position, volume of right compartment is

\frac{xV}{5}

then find the value of

x

The temperature of an open room of volume 30 m3 increased from 17ºC to 27ºC due to the sunshine. The atmospheric pressure in the room remains 1×105 Pa. If ni and nf are the number of molecules in the room before and after heating, then nf – ni will be

Important Questions on Kinetic Theory of Gases

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The temperature of an open room of volume $30 m^{3}$ increased from $17 º C to 27 º C$ due to the sunshine. The atmospheric pressure in the room remains $1 \times 10^{5} Pa$ . If $n_{i}$ and $n_{f}$ are the number of molecules in the room before and after heating, then $n_{f} - n_{i}$ will be