Question

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.

Answer 1

Answer:

25 cm​

Explanation:

P1 = Atmospheric pressure = 75 × ƒg

V1 = 100 × A

P2 = Atmospheric pressure + Mercury pressure = 75ƒg + hgƒg (if h = height of mercury)

V2 = (100 – h) A

P1V1 = P2V2

=>75ƒg(100A) = (75 + h)ƒg(100 – h)A

=> 75 × 100 = (74 + h) (100 – h)

=> 7500 = 7500 – 75 h + 100 h – h2

=> h2 – 25 h => 0

=> h2 = 25 h

=> h = 25 cm

Height of mercury that can be poured = 25 cm​

Answer 2

The maximum height of the mercury column is 25 cm

Explanation:

Height of the vertical cylinder = 1 m

Atmospheric pressure of mercury (Hg) = 0.75 m

                                                         Hg = 0.75 ρg Pa

$\text {Density of Hydrogen} = 13500 \frac{kg}{m^{3}}$

Let h be mercury height of the piston.

$P_{2}=P_{1}+h \rho g$

Let the CSA be A.

$\begin{aligned}&\mathrm{V}_{1}=\mathrm{Ah}=\mathrm{A}\\&\mathrm{V}_{2}=(1-\mathrm{h}) \mathrm{A}\end{aligned}$

Applying the law of Boyle, we get

$\begin{aligned}&\mathrm{P}_{1} \mathrm{V}_{1}=\mathrm{P}_{2} \mathrm{V}_{2}\\&0.75 \rho g A=P_{2}(1-h) A\end{aligned}$

0.75 ρg = ( 0.75 ρg + hρg) (1−h)

0.75 = (0.75 + h)(1−h)

H = 0.25 mm  

H = 25 cm

The maximum height of the mercury column that can be put on the piston is 25 cm  where the height of the cylinder is 100 cm and the atmospheric pressure of is equal to 75 cm of mercury.