A vertical cylinder of cross sectional area A is fitted with a tight-fitting, fr

Question

A vertical cylinder of cross sectional area A is fitted with a tight-fitting, frictionless piston of mass m. If n moles of an ideal gas are in the cylinder at a temperature of T, use Newton's second law for equilibrium to show that the height h at which the piston is in equilibrium under its own weight is given by h = nRT/mg + P_0 A where P0 is atmospheric pressure. Is the pressure inside the cylinder less than, equal to, or greater than atmospheric pressure? If the gas in the cylinder is warmed, how would the answer for be affected?

Explanation / Answer

Given,

Mass of piston = m ; No of moles = n ; Temp. = T ;

We need to prove: h = nRT/(mg + P0A)

We know for an ideal gas:

PV = n R T ( R is the Gas constant and V is the volume, P is the pressure)

P = n R T/V

The net force on the piston should sum to zero in equilibrium

(Pressure = Force/Area => Force = Pressure x Area)

P A = Patm A + mg

P A = P0 A + mg

(n R T/V) A = P0 A + mg

we know that, V = A x height => Height = h = V/A ; A/V = 1/h

n R T/h = P0 A + mg

h = n R T / (P0A + mg)

(b)Its greater than the atmospheric pressure, because it is the sum of

Pgas = Patm + P gauge = Patm + mg/A

c)From the expression:

h = n R T / (P0A + mg)

h is directly proportional to T, so with the increase of T height will also be increased.

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