Which statement best explains why a hot-air balloon rises when the air in the balloon is heated? a. According to Charles’s law, the temperature of a gas is directly related to its volume. Thus the volume of the balloon increases, making the density smaller. This lifts the balloon. b. Hot air rises inside the balloon, and this lifts the balloon. c. The temperature of a gas is directly related to its pressure. The pressure therefore increases, and this lifts the balloon. d. Some of the gas escapes from the bottom of the balloon, thus decreasing the mass of gas in the balloon. This decreases the density of the gas in the balloon, which lifts the balloon. e. Temperature is related to the root mean square velocity of the gas molecules. Thus the molecules are moving faster, hitting the balloon more, and thus lifting the balloon. Justify your choice, and for the choices you did not pick, explain what is wrong with them.

Step by step solution

01

Statement a:

According to Charles's Law, the temperature of a gas is directly related to its volume. When the air inside a hot-air balloon is heated, its temperature increases, resulting in an increase in volume. As the volume of the gas in the balloon increases, its density decreases because density is inversely proportional to volume. This decreased density causes the buoyant force to be greater than the force of gravity, making the hot-air balloon rise.

02

Statement b:

Hot air does rise due to its lower density compared to the cooler air surrounding it. However, this statement doesn't fully explain the mechanism behind the balloon's ascent. It doesn't take into account the relationship between temperature, volume, and density, which is crucial to understanding how a hot-air balloon rising.

03

Statement c:

The temperature of a gas is directly related to its pressure according to Gay-Lussac's Law. But the statement implies that the increased pressure causes the balloon to rise, which is incorrect. An increase in pressure alone does not cause the balloon to lift; it is the change in density due to the increase in volume as per Charles's Law which helps it rise.

04

Statement d:

Although the mass of gas inside the balloon may be slightly reduced during the heating process, it does not significantly affect the overall density. More importantly, the main mechanism behind a hot-air balloon's ascent is the change in density due to the increase in volume as a result of the increased temperature according to Charles's Law.

05

Statement e:

While the temperature is related to the root mean square velocity of gas molecules, stating that molecules move faster and hit the balloon more, lifting it, is incorrect. The primary force behind the ascent of a hot-air balloon comes from the change in density due to the increased volume caused by increased temperature, not simply the effect of faster-moving molecules. In conclusion, the best explanation for the ascent of a hot-air balloon when the air inside is heated is given by the

06

Statement a:

According to Charles’s law, the temperature of a gas is directly related to its volume. Thus the volume of the balloon increases, making the density smaller. This lifts the balloon. Statements b, c, d, and e either provide incomplete reasoning or incorrect statements for the associated mechanism.

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