A vertical cylinder of height 100 cm contains air at a constant temperature. The top is closed by a frictionless light piston. The atmospheric pressure is equal to 75 cm of mercury. Mercury is slowly poured over the piston. Find the maximum height of the mercury column that can be put on the piston.

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Solution

Given, $P_{I}$ = Atmospheric pressure

= $75 \times ρ g$ ,

$V_{1} = 100 \times A$

$P_{2}$ = Atmospheric pressure + Mercury pressure

= $75 ρ g + h ρ g$

(if h = height of mercury)

$(V_{2} = (100 - h)$

$P_{1} V_{1} = P_{2} V_{2}$

$\Rightarrow 75 ρ g (100 A)$

= $(75 + h) ρ g (100 - h) A$

$\Rightarrow 75 \times 100 = (75 + h) (100 - h)$

$\Rightarrow 7500 = 7500 - 75 h + 100 h - h^{2}$

$\Rightarrow h^{2} - 100 h + 75 h = 0$

$\Rightarrow h^{2} - 25 h = 0$

$\Rightarrow h^{2} = 25 h$

$\Rightarrow$ h = 25 cm

Height of Mercury that can be poured

= 25 cm

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