Question

An ideal gas in a container of volume \[V\] at a pressure \[P\]. It is being pumped out of the container by using a pump with stroke volume \[V\] . What is the final pressure in the container after n-strokes of the pump? (Assume temperature remains same)
(A) \[P{\left( {\dfrac{V}{{V + v}}} \right)^n}\]
(B)$P{\left( {\dfrac{{1 - V}}{v}} \right)^n}$
(C)$P\dfrac{{{V^n}}}{{{v^n}}}$
(D)$P{\left( {\dfrac{V}{{V - V}}} \right)^n}$

Answer 1

Hint: The empirical relationship between temperature (\[T\]), volume (\[V\]), pressure (\[P\]) and no of moles of gas (\[n\]) is combining called as gas law or ideal gas equation.

Formula used:
Ideal gas equation
\[PV = nRT\],
 Where,
\[n\]is number of moles, \[P\]is pressure, \[V\] (constant) is volume, \[T\] is temperature and \[R\] is the universal gas constant.

Complete step by step answer:
 It is given that the volume of the container is \[V\], pressure is \[P\], stroke volume\[V\].
Let \[T\] be the temperature. Along with that we also know that,
\[\;v \ll V\]
We have to find the final pressure in the container after the n-strokes of the pump,
It is an isothermal process, thus \[T\]=constant,
Let in the first pump that is When\[n = 1\], volume \[V\] is reduced from the container then fractional change in volume is\[\dfrac{{\left( {V - v} \right)}}{V}\].
⟹\[\dfrac{{(V - v)}}{V}\]
Let us use the ideal gas equation which is as following,
\[PV = nRT\]
From the above equation we can write the following equation as,
\[P = n \times constant\] (as all other parameters are constant)
So from here we can conclude that the way number of moles will vary similarly pressure will vary,
Let us write the remaining number of moles as,
Number of moles = initial number of mole × fractional change in volume.
\[\;\;{N_1} = n \times \dfrac{{(V - v)}}{V}\] (1)
Similarly let us now write the pressure.
   ${P_1} = P \times \dfrac{{V - v}}{V}$
$ = P\left( {1 - \dfrac{v}{V}} \right)$ (2)
Now, after the second stroke the pressure is given below.
${P_2} = {P_1}\dfrac{{V - v}}{V}$
$ = P{\left( {1 - \dfrac{v}{V}} \right)^2}$ (3)
So in equation (2) and (3) we can observe a pattern. Hence let us write the pressure after n-number of strokes.
${P_n} = P{\left( {1 - \dfrac{v}{V}} \right)^n}$
Hence, pressure of the container after n strokes is$P{\left( {1 - \dfrac{v}{V}} \right)^n}$.
Additional information:
Apart from ideal gases, some gases don't obey the gas equation. Such gases are called real or non-ideal behaving gases.
For the real gases, the gas equation is modified by making changes in volume and pressure, which is given as follows,
$
  \left( {P + \dfrac{{a{n^2}}}{{{V^2}}}} \right)(V - nb) = RTP \\
    \\
 $ , this is called the vander Waals equation.
\[P\]=pressure,
\[V\]=volume of gas
\[R\]=gas constant
$T$=Absolute temperature
$a,b$=Van der Waals constant

Note:
An isothermal process is a process in which we keep the temperature of the system constant.
For volume,$\dfrac{V}{{100}}$ , the final pressure in the container after nth stroke is, $P{\left( {\dfrac{{99}}{{100}}} \right)^n}$.