An ideal gas is kept in a long cylindrical vessel fitted with a frictionless piston of cross-sectional area 10 cm2 and weight 1 kg. The length of the gas column in the vessel is 20 cm. The atmospheric pressure is 100 kPa. The vessel is now taken into a spaceship revolving round the earth as a satellite. The air pressure in the spaceship is maintained at 100 kPa. Find the length of the gas column in the cylinder.
Given
Area of cross section A= 10cm2=10
10-4m2
Mass of piston ‘m’ = 1kg
Weight of piston ‘mg’=1
9.8N
Length of the gas column l=20cm=0.20m
Atmospheric pressure Po=100kPa=105Pa
Air pressure in the spaceship =100kPa=Po
Let the length of the gas column in the spaceship be l’.
Pressure on gas before taking to spaceship= P1
P1= pressure due to weight of piston + atmospheric pressure
Now,
Volume of the gas column before taking to spaceship V1=area
length
V1=A
l
Volume of the gas after taking to spaceship V2=A
l’
The pressure of surroundings has been kept same as the atmospheric pressure. So, this means the temperature of surrounding, before and after taking to spaceship is same.
Therefore, we can apply boyle’s law we state that ‘PV=constant, when temperature is constant’.
We also know that in satellite or in spaceship the effect of gravity is negligible. So, pressure on gas due to weight of piston in spaceship will be zero.
So, pressure on the gas in spaceship P2=air pressure of spaceship=Po
Applying boyle’s law before and after taking vessel to spaceship, we get
P1V1=P2V2
(9.8
103+105)
0.2=105
l’
109.8
103
0.2=105
l’
The length of gas column in spaceship is 22cm.