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In 1992, a Danish study concluded that a standard toy balloon, made from latex and filled with helium, could rise to 10,000 m (where the pressure is 1/3 of that at sea level) in the atmosphere before bursting. In the study, a number of balloons were filled with helium, and then placed in a chamber maintained at −20°C. The pressure in the chamber was gradually reduced until the balloons exploded, and then the researchers determined the height above sea level corresponding to that pressure. Assume each balloon was filled with helium at +20°C and at about atmospheric pressure. Determine the balloon's volume just before it exploded, if its volume when it was first filled was 640 cm^3.

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Answer:

1658.01807 cm³

Explanation:

[tex]V_1[/tex] = Initial volume = 640 cm³

[tex]V_2[/tex] = Final volume

[tex]T_1[/tex] = Initial temperature = 20°C

[tex]T_2[/tex] = Final temperature = -20°C

[tex]P_1[/tex] = Initial pressure = 1 atm (sea level)

Final pressure

[tex]P_2=\dfrac{1}{3}P_1\\\Rightarrow P_2=\dfrac{1}{3}\ atm[/tex]

We have the relation

[tex]\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}\\\Rightarrow V_2=\dfrac{P_1V_1T_2}{T_1P_2}\\\Rightarrow V_2=\dfrac{1\times 640\times 253.15}{293.15\times \dfrac{1}{3}}\\\Rightarrow V_2=1658.01807\ cm^3[/tex]

The final volume is 1658.01807 cm³

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