Kinetic Molecular Theory Of Gases

An ideal gas of density

%5Crho%3D02%20%5C%2Ckg%20%5C%2C%20m%20%5E%7B-3%7D

enters a chimney of height

h

at the rate of

%5Calpha%3D08%20%5C%2Ckg%20%5C%2Cs%20%5E%7B-1%7D

from its lower end, and escapes through the upper end as shown in the figure The cross-sectional area of the lower end is

A_1%3D01%20%5C%2C%20m%20%5E2

and the upper end is

A_2%3D04%20%5C%2C%20m%20%5E2

The pressure and the temperature of the gas at the lower end are

600%20%5C%2CPa

and

300%20%5C%2CK

, respectively, while its temperature at the upper end is

150%20%5C%2CK

The chimney is heat insulated so that the gas undergoes adiabatic expansion Take

g%3D10%20%5C%2Cms%20%5E%7B-2%7D

and the ratio of specific heats of the gas

%5Cgamma%3D2

Ignore atmospheric pressure Which of the following statement(s) is(are) correct?

$%5Cfrac%7Bdm%7D%7Bdt%7D%20%3D%20p_1%20A_1%20v_1%20%3D%200.8%20%5C%2C%20%5Ctext%7Bkg%2Fs%7D%20%5C%2C%20A$
$v_1%20%3D%20%5Cfrac%7B0.8%7D%7B0.2%20%5Ctimes%200.1%7D%20%3D%2040%20%5C%2C%20%5Ctext%7Bm%2Fs%7D$

g = 10 m/s²
$%5Cgamma%20%3D%202$

Gas undergoes adiabatic expansion,
$p_1%20-%20%5Cgamma%20T%5E%7B%5Cgamma%7D%20%3D%20%5Ctext%7BConstant%7D$
$%5Cfrac%7BP_2%7D%7BP_1%7D%3D(%5Cfrac%7BT_1%7D%7BT_2%7D)%5E%7B%5Cfrac%7Br%7D%7B1-%5Cgamma%7D%7D$

$P_2%3D%5Cfrac%7B600%7D%7B4%7D%3D150P_a$

Now $%5Crho%E2%88%9D%5Cfrac%7BP%7D%7BT%7D$

$%5Cfrac%7B%5Crho_1%7D%7B%5Crho_2%7D%3D(%5Cfrac%7BP_1%7D%7BP_2%7D)(%5Cfrac%7BT_1%7D%7BT_2%7D)$

$(%5Cfrac%7B150%7D%7B600%7D)(%5Cfrac%7B300%7D%7B150%7D)%3D%5Cfrac%7B1%7D%7B2%7D$

Now
$P_1%20A_1%20%5CDelta%20x_1%20-%20P_2%20A_2%20%5CDelta%20x_2%20%3D%202%20mV_1%20-%20mV_2%20%2B%20mgh_2%20-%20mgh_1%20%2B%20%5Cfrac%7Bf%7D%7B2%7D(P_2%20V_2%20-%20P_1%20V_1)$

Simplifying we get. $%5Cfrac%7BV_2%7D%7BV_1%7D%20-%20%5Cfrac%7BV_1%7D%7BV_2%7D%20%3D%20%5Cfrac%7B2P%7D%7Bgh%7D%20%5Cfrac%7Bm%7D%7Bm%7D$
$%E2%87%92%5Cfrac%7B2%5Ctimes600%7D%7B0.2%7D-%5Cfrac%7B2%5Ctimes150%7D%7B0.1%7D$

$%3D%5Cfrac%7B20%5E2-40%5E2%7D%7B2%7D%2B10h$
$h%20%3D%20360%20m$

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Kinetic Molecular Theory Of Gases

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs