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

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:

{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.