Question

A cycle tube has volume $2000\,c{m^3}$. Initially the tube is filled to ${\left( {\dfrac{3}{4}} \right)^{th}}$ of its volume by air at pressure of $105\,N{m^{ - 2}}$. It is to be inflated to a pressure of $6 \times {10^5}\,N{m^{ - 2}}$ under isothermal conditions. The number of strokes of the pump, which gives $500\,c{m^3}$ air in each stroke, to inflate the tube is?
(A) $21$
(B) $12$
(C) $42$
(D) $11$

Answer 1

Hint: This problem is related to thermodynamics because it is given that the process is an isothermal process. So, the temperature is constant throughout the process. By using Boyle's formula, the number of strokes of the pump can be determined.

Formula Used:
Boyle’s formula for the isothermal process, the product of pressure and volume is equal to constant. Then,
${p_f}{v_f} = {p_i}{v_i} + n\left( {{p_p}{v_p}} \right)$
Where, ${p_f}$ is the final pressure of the tube, ${v_f}$ is the final volume of the tube, ${p_i}$ is the initial pressure of the tube, ${v_i}$ is the initial volume of the tube, ${p_p}$ is the pressure of the pump, ${v_p}$ is the volume of the pump and $n$ is the number of strokes of the pump.

Complete step by step answer:
Given that,
The final volume of the tube, ${v_f} = 2000\,c{m^3}$
The initial volume of the tube, ${v_i} = \dfrac{3}{4} \times 2000\,c{m^3}$
${v_i} = 1500\,c{m^3}$,
The initial pressure of the tube, ${p_i} = 105\,N{m^{ - 2}}$
The final pressure of the tube, ${p_f} = 6 \times {10^5}\,N{m^{ - 2}}$
The volume of the pump, \[{v_p} = 500\,c{m^3}\]
According to Boyle’s law,
${p_f}{v_f} = {p_i}{v_i} + n\left( {{p_p}{v_p}} \right)\,......................\left( 1 \right)$
Substituting the given values in the equation (1), then the equation (1) is written as,
$6 \times {10^5} \times 2000 = 105 \times 1500 + n\left( {{p_p} \times 500} \right)$
The pressure of the pump is assumed to be the pressure of the atmospheric air, and substitute in the above equation, then,
$6 \times {10^5} \times 2000 = 105 \times 1500 + n\left( {1.01 \times {{10}^5} \times 500} \right)$
On multiplying, then the above equation is written as,
$12 \times {10^8} = 157500 + n\left( {505 \times {{10}^5}} \right)$
On further steps for calculation,
$12 \times {10^8} - 157500 = n\left( {505 \times {{10}^5}} \right)$
On simplifying the LHS in the above equation, then
$1199842500 = n\left( {505 \times {{10}^5}} \right)$
By keeping the term $n$ in one side and the other terms in other side, then the above equation is written as,
$n = \dfrac{{1199842500}}{{505 \times {{10}^5}}}$
On dividing, then the above equation is written as,
$n = 23.7$
Thus, the number of strokes should not be in decimal.
Hence, the option (A) is the correct answer.

Note: The pressure of the pump is not mentioned, so we have to assume the pressure of the pump as atmospheric pressure. And by using Boyle's law isothermal process equation, the number of strokes can be determined.