Question

A balloon contains $500{{m}^{3}}$ of helium at $27{}^\circ C$ and 1 atmosphere pressure. The volume of helium at $-3{}^\circ C$ temperature and 0.5 atmosphere pressure will be
A. $500{{m}^{3}}$
B. $700{{m}^{3}}$
C. $900{{m}^{{{3}^{{}}}}}$
D. $1000{{m}^{3}}$

Answer 1

As a very first step, one could note down all the given values from the question. You could then recall the ideal gas equation. Assume that the number of moles of the given helium gas remains constant and then you will get a proportionality relation for pressure, volume and temperature and hence find the answer.
Formula used:
Ideal Gas equation,
$PV=nRT$

Complete step by step answer:
In the question we are given a balloon containing helium. The initial conditions are given as,
${{P}_{1}}=1atm$
${{V}_{1}}=500{{m}^{3}}$
${{T}_{1}}=27{}^\circ C=300K$
We are given another set of conditions where,
${{P}_{2}}=0.5atm$
${{T}_{2}}=-3{}^\circ C=270K$
We are supposed to find the volume of helium contained now.
Let recall the ideal gas equation given by,
$PV=nRT$
We know that R is the universal gas constant and then for a given number of moles of a gas, we could say that,
$PV\alpha T$
$\Rightarrow \dfrac{{{P}_{1}}{{V}_{1}}}{{{T}_{1}}}=\dfrac{{{P}_{2}}{{V}_{2}}}{{{T}_{2}}}$
Substituting the given values we see that,
$\Rightarrow \dfrac{1\times 500}{300}=\dfrac{0.5\times {{V}_{2}}}{270}$
$\therefore {{V}_{2}}=900{{m}^{3}}$
Therefore, we found the volume of helium under the given second set of conditions to be,
${{V}_{2}}=900{{m}^{3}}$

So, the correct answer is “Option C”.

Note: A very important step that should be followed in questions related to this topic is that the temperature when given in degree Celsius should be converted to Kelvin scale by adding 273 to the given value in Celsius. Since, all the other quantities are given in SI units one shouldn’t worry about them.