Question

A gas is enclosed in a vessel of volume V
at a pressure P. It is being pumped out of
the vessel by means of a piston-pump with
a stroke volume v. What is the final
pressure in the vessel after 'n' strokes of
the pump ? Assume temperature remains
constant​

Answer 1

Answer:

A gas is enclosed in a vessel of volume V

at a pressure P. It is being pumped out of

the vessel by means of a piston-pump with

a stroke volume v. What is the final

pressure in the vessel after 'n' strokes of

the pump ? Assume temperature remains

constant

Answer 2

AnSwer :-

According to ideal gas equation PV = nRT, at constant temperature for a given mass,

PV = P¹V¹

Now as stroke volume is v during 1st stroke for constant mass (say m) volume changes from V to (V + v) and so if pressure changes from P to P₁, the above equation yields

PV = P₁(V + v), .i.e.,

$\sf P_1 = P\bigg[ \dfrac{V}{V + v} \bigg] \: \: \: \: .......(1)$

After the first stroke, the gas left in the vessel has again volume V but at Pressure P₁ (with mass m₁ < m).

Now the second stroke will take place from these initial conditions and if P₂ is the pressure of the gas in the cylinder at the end of 2nd stroke,

P₁V = P₂(V + v), .i.e.,

$\sf P_2 = P_1 \bigg[ \dfrac{V}{V + v} \bigg]$

Substituting the value of P₁ from equation (1) in the above.

$\sf P_2 = P \bigg[ \dfrac{V}{V + v} \bigg] ^{2}$

Repeating the same for n strokes, the pressure of the gas in the vessel after nth stroke will be,

$\sf P_n = P \bigg[\dfrac{V}{V +v} \bigg]^{2} = P \bigg[\dfrac{1}{1 + (v/V)}\bigg]^{2}$

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