Question

How many cylinders of hydrogen at atmospheric pressure are require to fill a balloon whose volume is \[500{\rm{ }}{{\rm{m}}^3}\], if hydrogen is stored in cylinder of volume \[0.05{\rm{ }}{{\rm{m}}^3}\] at an absolute pressure of \[15 \times {10^5}{\rm{ Pa}}\]?
A. \[700\]
B. \[675\]
C. \[605\]
D. \[710\]

Answer 1

Hint: We will be using Boyle’s law of a gas and by removing the constant of proportionality and rewriting the general form of Boyle’s law into the form a gas having two states to obtain the volume of hydrogen at atmospheric pressure.

Complete step by step answer:
It is given that the hydrogen is stored in a cylinder of volume \[0.05{\rm{ }}{{\rm{m}}^3}\] at an absolute pressure of \[15 \times {10^5}{\rm{ Pa}}\]. We have to calculate the number of cylinders of hydrogen required to fill a balloon having a volume of \[500{\rm{ }}{{\rm{m}}^3}\] at atmospheric pressure.
 We know that the value of atmospheric pressure is \[1.013 \times {10^5}{\rm{ Pa}}\].
Let us consider the state at which hydrogen is stored in cylinder as state 1 and atmospheric conditions as state 2:
\[\begin{array}{l}
{P_1} = 15 \times {10^5}{\rm{ Pa}}\\
{{\rm{V}}_1} = 0.05{\rm{ }}{{\rm{m}}^3}\\
{P_2} = 1.013 \times {10^5}{\rm{ Pa}}
\end{array}\]
Boyle’s law tells us about the relationship of pressure and volume at constant temperature. In other words we can say that according to Boyle’s law, pressure of a gas is inversely proportional to its volume at constant temperature.
\[P \propto \dfrac{1}{V}\]
As we remove the constant of proportionality the following expression will be obtained.
\[PV = {\rm{constant}}\]
Let us write the above expression for state 1 and state 2.
\[{P_1}{V_1} = {P_2}{V_2}\]
Substitute \[15 \times {10^5}{\rm{ Pa}}\] for \[{P_1}\], \[0.05{\rm{ }}{{\rm{m}}^3}\] for \[{V_1}\] and \[1.013 \times {10^5}{\rm{ Pa}}\] for \[{P_2}\] in the above expression to find out the value of \[{V_2}\].
\[\left( {15 \times {{10}^5}{\rm{ Pa}}} \right)\left( {0.05{\rm{ }}{{\rm{m}}^3}} \right) = \left( {1.013 \times {{10}^5}{\rm{ Pa}}} \right){V_2}\]
Rearranging the above expression,
\[\begin{array}{l}
 \Rightarrow {V_2} = \dfrac{{\left( {15 \times {{10}^5}{\rm{ Pa}}} \right)\left( {0.05{\rm{ }}{{\rm{m}}^3}} \right)}}{{\left( {1.013 \times {{10}^5}{\rm{ Pa}}} \right)}}\\
 \Rightarrow {V_2} = 0.7403{\rm{ }}{{\rm{m}}^3}
\end{array}\]
Here \[{V_2}\] is the volume of the given hydrogen at atmospheric pressure \[{10^5}{\rm{ Pa}}\].
It is given that we have to fill a balloon whose volume is \[500{\rm{ }}{{\rm{m}}^3}\] at atmospheric conditions. Therefore, the volume of the given balloon has to be equated to the product of the number of cylinders required and \[{V_2}\].
\[N \times {V_2} = 500{\rm{ }}{{\rm{m}}^3}\]
Let us substitute \[0.7403{\rm{ }}{{\rm{m}}^3}\] for \[{V_2}\] in the above equation.
\[\begin{array}{c}
N\left( {0.7403{\rm{ }}{{\rm{m}}^3}} \right) = 500{\rm{ }}{{\rm{m}}^3}\\
N = \dfrac{{500{\rm{ }}{{\rm{m}}^3}}}{{0.7403{\rm{ }}{{\rm{m}}^3}}}\\
 = 675.4
\end{array}\]
Taking round-off of the value of N.
\[N = 675\]
Therefore, \[675\] is the number of cylinders of hydrogen at atmospheric pressure are require to fill a balloon whose volume is \[500{\rm{ }}{{\rm{m}}^3}\] and option (B) is correct.

Note:Alternate method: We can also solve this question by using the ideal gas equation which is expressed in terms of pressure, temperature and volume. By keeping the temperature as a constant value, we will get the expression in terms of pressure and temperature which will come out equal to the expression obtained by Boyle’s law. Also, do not round-off the value of atmospheric pressure because this will lead to incorrect answers.