Inflate a balloon inside a bottle

Classical Mechanics Level 2

A glass bottle and the air contained in it are heated in a bath of boiling water to a temperature of 100 degrees. The bottle is sealed airtight with a balloon in its neck. Then the bottle is placed in an ice bath, so that the trapped air is cooled to 0 degrees. Due to the resulting negative pressure, the balloon is inflated. What is the remaining volume of the trapped air in the bottle? Specify the result in milliliters and round to the nearest integer.

Details and Assumptions: The bottle has a volume of 1.5 liters. The balloon is stretchable arbitrarily and forms a perfect heat insulation to the outside air. The air is an ideal gas.


The answer is 1098.

Markus Michelmann by Markus Michelmann , 35, Germany

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1 solution

Arjen Vreugdenhil
Dec 21, 2017

Use the ideal gas law, p V = n R T pV = nRT . The pressure and number of particles remain constant. The volume of the air changes according to V 2 V 1 = n R T 2 / p n R T 1 / p = T 2 T 1 . \frac{V_2}{V_1} = \frac{nRT_2/p}{nRT_1/p} = \frac{T_2}{T_1}. Therefore V 2 = V 1 ⋅ T 2 T 1 = 1 500 mL ⋅ 273 K 373 K = 1 098 mL . V_2 = V_1\cdot \frac{T_2}{T_1} = \SI{1500}{mL}\cdot \frac{\SI{273}{K}}{\SI{373}{K}} = \boxed{\SI{1098}{mL}}.

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