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The simple answer is because the pressure remains the same. When pressure remains constant, the change in volume will be directly proportional to the change in temperature.

The air molecules release energy as they travel around inside the balloon. What we call pressure is technically the force of the air molecules colliding with the inside of the balloon and each other. The kinetic energy is related to the speed of the molecules. The faster the molecules are moving, the greater the kinetic energy. The greater the kinetic energy, the greater the pressure in the inside of the balloon. The outside air is doing the same thing to outside of the balloon. Therefore, the pressure pushing out is equal to the pressure pushing in. Temperature decreases because the molecules inside the balloon are releasing their energy in the form of heat as they interact with each other. In fact, they heat energy passes through the balloon and is absorbed by the air on the outside. Over time, more energy is released, and the temperature drops. The volume of the balloon decreases in order to maintain a constant pressure. As the surface area decreases, the total pressure pushing in also decreases so that it always equals the pressure pushing out.

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Q: Why does a balloon get smaller in cold weather?
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