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Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

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Q. 13. A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free-fall acceleration is equal to g. The temperature of the nitrogen varies along the height h so that its density is the same throughout the volume. Find the temperature gradient dT/dh. 

Solution. 13. Consider a thin layer at a height h and thickness dh. Let p and dp+p be the pressure on the two sides of the layer. The mass of the layer is Sdhp. Equating vertical downward force to the upward force acting on the layer.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET    (1)

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

That means, temperature of air drops by 34°C at a height of 1 km above bottom.


Q. 14. Suppose the pressure p and the density p of air are related as p/pn = const regardless of height (n is a constant here). Find the corresponding temperature gradient.

Solution. 14.  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET     (1)

But, from    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET     (2)

We have from gas low   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Thus,    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

But,    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

so,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 15. Let us assume that air is under standard conditions close to the Earth's surface. Presuming that the temperature and the molar mass of air are independent of height, find the air pressure at the height 5.0 km over the surface and in a mine at the depth 5.0 km below the surface. 

Solution. 15. We have, dp = - p g dh and from gas law  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Integrating, we get

or,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

(where p0 is the pressure at the surface of the Earth.)

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

[Under standard condition, Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Pressure at a height of 5 atm   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Pressure in a mine at a depth of 5 km  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 16. Assuming the temperature and the molar mass of air, as well as the free-fall acceleration, to be independent of the height, find the difference in heights at which the air densities at the temperature 0°C differ 

(a) e times; (b) by η = 1.0%. 

Solution. 16. We have dp = - pg dh but from gas law  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Thus    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET at const, temperature 

So,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Integrating within limits   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Thus    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 17. An ideal gas of molar mass M is contained in a tall vertical cylindrical vessel whose base area is S and height h. The temperature of the gas is T, its pressure on the bottom base is P0. Assuming the temperature and the free-fall acceleration g to be independent of the height, find the mass of gas in the vessel. 

Solution. 17. From the Barometric formula, we have

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

and from gas law

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

So, at constant temperature from these two Eqs.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET     (1)

Eq. (1) shows that density varies with height in the same manner as pressure. Let us consider the mass element of the gas contained in the coltimn.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Hence the sought mass, 

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 18. An ideal gas of molar mass M is contained in a very tall vertical cylindrical vessel in the uniform gravitational field in which the free-fall acceleration equals g. Assuming the gas temperature to be the same and equal to T, find the height at which the centre of gravity of the gas is located.

Solution. 18. As the gravitational field is constant the centre of gravity and the centre of mass are same. The location of C.M.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

But from Barometric formula and gas law  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

So,    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 19. An ideal gas of molar mass /If is located in the uniform gravitational field in which the free-fall acceleration is equal to g. Find the gas pressure as a function of height h, if p = p0  at h = 0, and the temperature varies with height as 

(a) T = T0  (1 — ah); (b) T = T0  (1 + ah),

where a is a positive constant. 

Solution. 19. (a) We know that the variation of pressure with height of a fluid is given by :

dp = - p g dh

But from gas law   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

From these two Eqs.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Integrating,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Hence,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

(b) Proceed up to Eq. (1) of part (a), and then put  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET and proceed further in the same fashion to get

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 20. A horizontal cylinder closed from one end is rotated with a constant angular velocity ω about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to P0, the temperature to T, and the molar mass of air to M. Find the air pressure as a function of the distance r from the rotation axis. The molar mass is assumed to be independent of r. 

Solution. 20. Let us consider the mass element of the gas (thin layer) in the cylinder at a distance r from its open end as shown in the figure.

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Using Newton’s second law for the element 

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

So,     Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Thus,    Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 21. Under what pressure will carbon dioxide have the density p = 500 g/1 at the temperature T = 300 K? Carry out the calculations both for an ideal and for a Van der Waals gas. 

Solution. 21. For an ideal gas law

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

So,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

For Vander Waal gas Eq. 

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 22. One mole of nitrogen is contained in a vessel of volume V = 1.00 1. Find:
 (a) the temperature of the nitrogen at which the pressure can be calculated from an ideal gas law with an error η = 10% (as compared with the pressure calculated from the Van der Waals equation of state);
 (b) the gas pressure at this temperature. 

Solution. 22.  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

(The pressure is less for a Vander Waal gas than for an ideal gas)

or,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

(b) The corresponding pressure is

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 23. One mole of a certain gas is contained in a vessel of volume V = 0.250 1. At a temperature T1 = 300 K the gas pressure is p1 = 90 atm, and at a temperature T= 350 K the pressure is p2 = 110 atm. Find the Van der Waals parameters for this gas. 

Solution. 23. Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

So,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or.  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Also,  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Using T1 = 300 K, p1 = 90 atms, T2 = 350 K, p2 = 110 atm, V = 0.250 litre 

a = 1.87 atm. litre2/mole2, b = 0.045 litre/mole


Q. 24. Find the isothermal compressibility x of a Van der Waals gas as a function of volume V at temperature T.  

Note. By definition Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Solution. 24. Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

or,   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET


Q. 25. Making use of the result obtained in the foregoing problem, find at what temperature the isothermal compressibility x of a Van der Waals gas is greater than that of an ideal gas. Examine the case when the molar volume is much greater than the parameter b. 

Solution. 25. For an ideal gas  Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEETIrodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

Now   Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET

If a, b do not vary much with temperature, then the effect at high temperature is clearly determined by b and its effect is repulsive so compressibility is less.

The document Irodov Solutions: Equation of The Gas State Processes - 2 - Notes | Study Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
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