Question

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, decreasing the density. b. Hot air rises inside the balloon, which lifts the balloon. c. The temperature of a gas is directly related to its pressure. The pressure therefore increases, which 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 velocity of the gas molecules. Thus the molecules are moving faster, hitting the balloon more, and lifting the balloon. For choices you did not pick, explain what you feel is wrong with them, and justify the choice you did pick.

Short answer

The best statement that explains why a hot-air balloon rises when the air inside the balloon is heated is option (a). As temperature increases, the volume of the gas inside the balloon also increases according to Charles's Law, ultimately decreasing its density. This decrease in density makes the hot-air balloon less dense than the cooler surrounding air, causing it to rise.

Step-by-step solution

Charles's Law

Charles's Law states that for an ideal gas at a constant pressure, the volume of the gas is directly proportional to its temperature. As the temperature of the air inside the hot-air balloon increases, the volume of the air also increases. This is because the gas molecules spread out as they are heated, occupying more space. #Step 2: Analyze each option based on scientific accuracy#

Option (a)

This statement says that Charles's Law is the reason for balloons to rise as the volume of the balloon increases, decreasing the density. As we mentioned earlier, Charles's Law correctly explains the relationship between the temperature and volume of the gas inside the balloon. As a result, this option seems like a reasonable explanation.

Option (b)

This statement suggests that the hot air rises inside the balloon, lifting it. While it's true that hot air is less dense and tends to rise, this statement does not provide a clear explanation of why the balloon itself rises in comparison to the colder air surrounding it. It doesn't address the key idea that the overall density of the air inside the balloon decreases, making it less dense than the surrounding air.

Option (c)

This statement attributes the lift of the balloon to the direct relationship between the temperature and pressure of a gas. While in some cases, the temperature and pressure may be related, Charles's Law talks specifically about the relationship between temperature and volume, with pressure held constant. Therefore, this statement does not capture the essence of the phenomenon.

Option (d)

This statement focuses on gas molecules escaping from the bottom of the balloon, reducing its mass, and subsequently lowering the density. This explanation may have some merit, but it does not address the main phenomenon of the gas expanding as it heats up and its effect on density. Thus, this option is not as comprehensive as option (a).

Option (e)

This statement relates temperature to the velocity of gas molecules, claiming that fast-moving molecules hit the balloon more and cause it to rise. While hotter molecules do move faster, the increase in collisions doesn't necessarily lift the balloon. The main factor that causes the balloon to rise is the decrease in density due to the increase in volume. #Step 3: Select and justify the best option#

Best answer

Option (a) is the best choice because it explains the relationship between the temperature and volume of the gas inside the hot-air balloon according to Charles's Law. As the temperature increases, the volume of the gas increases as well, decreasing its overall density. This decrease in density is what causes the hot-air balloon to rise, as it becomes less dense than the cooler air surrounding it. The other options do not provide clear explanations or address the central concept of the volume's role in the density of the gas inside the balloon.

Key concepts

Gas Laws

Gas laws provide us with a foundation for understanding the behavior of gases under various conditions. These laws describe how pressure, volume, temperature, and the quantity of gas are interrelated.

- **Boyle's Law** tells us that for a constant temperature, the pressure of a gas is inversely proportional to its volume. This means when volume decreases, pressure increases, and vice versa.- **Charles's Law** states that for a gas at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. So, when the temperature rises, the volume increases.- **Avogadro's Law** explains that volume is directly proportional to the number of gas molecules, assuming temperature and pressure are constant.- **Ideal Gas Law** combines all the above, giving the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is moles of gas, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.

These gas laws are integral in explaining real-life phenomena such as a hot-air balloon's flight, as they describe how gas behavior changes with varying external conditions.

Temperature and Volume Relationship

The temperature and volume relationship in gases is neatly captured by Charles's Law. This law shows how gases expand when heated and condense when cooled.

When we talk about gases, molecules are always in constant motion. When we heat a gas, the kinetic energy of these molecules increases. As a result:

• Molecules move faster and spread out more.
• Increased movement requires more space, causing the gas to expand.
• Therefore, the volume increases with rising temperature.

So, in the context of a hot-air balloon, when the air inside is heated, it expands because of this increased volume. This expansion plays a crucial role in reducing the density of the gas inside the balloon, allowing it to rise.

Density and Buoyancy

Density and buoyancy are key concepts in understanding why hot-air balloons rise. Density is defined as mass per unit volume, while buoyancy relates to an object's ability to float in a fluid.

- **Density**: When air inside the balloon is heated, its volume increases which results in a lower density because density is mass divided by volume. - **Buoyancy**: An object will float if it's less dense than the fluid it displaces. In the case of the hot-air balloon, the warmer, less dense air inside the balloon makes it buoyant enough to rise through cooler, denser air surrounding it.

In simple terms, heating the air reduces its density, and when the air inside the balloon becomes lighter than the outside air, the balloon lifts. This buoyant force is similar to the reason ships float on water.

Ideal Gases

An ideal gas is a theoretical concept that simplifies the behavior of gases to make them easier to study and predict.

Ideal gases are comprised of:

• Molecules that do not interact aside from elastic collisions.
• Molecules that are in constant, random motion.
• Behaviors that are highly predictable using the Ideal Gas Law \(PV = nRT\).

Real gases approximate ideal gas behavior under many conditions, making these concepts incredibly valuable. They allow chemists and physicists to predict how gases will behave when subjected to changes in pressure, temperature, and volume.

In the case of the hot-air balloon, treating the air inside as an ideal gas helps us apply Charles's Law effectively for simplicity and predictability.