Question

You have a helium balloon at \(1.00 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\). You want to make a hot-air balloon with the same volume and same lift as the helium balloon. Assume air is \(79.0 \%\) nitrogen and \(21.0 \%\) oxygen by volume. The "lift" of a balloon is given by the difference between the mass of air displaced by the balloon and the mass of gas inside the balloon. a. Will the temperature in the hot-air balloon have to be higher or lower than \(25^{\circ} \mathrm{C}\) ? Explain. b. Calculate the temperature of the air required for the hot-air balloon to provide the same lift as the helium balloon at \(1.00\) atm and \(25^{\circ} \mathrm{C}\). Assume atmospheric conditions are \(1.00 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\).

Short answer

The temperature of the hot-air balloon needs to be higher than 25°C to provide the same lift as the helium balloon. The required temperature for the hot-air balloon is approximately \( 41.8^{\circ} \mathrm{C} \).

Step-by-step solution

a. Determine the needed temperature change for the hot-air balloon compared to the helium balloon.

To answer this question, we have to consider that the lift of a balloon comes from the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. A hot-air balloon filled with a gas mixture lighter than air can create a larger lift which means that, since helium is lighter than air, we will need to heat the air inside the hot-air balloon to lower its density. Thus, the temperature of the hot-air balloon needs to be higher than 25°C.

b. Calculate the required temperature of the air for the hot-air balloon.

To calculate the required temperature, we will proceed with the following steps: 1. Determine the average molar mass of air. 2. Apply the Ideal Gas Law to find the number of moles of helium. 3. Use the given lift equation to find the mass of air displaced. 4. Determine the number of moles of air displaced. 5. Apply the Ideal Gas Law again to find the temperature of the hot-air balloon. Step 1: Average molar mass of air: \( M_{air} = 0.79 \times M_{N_2} + 0.21 \times M_{O_2} \) \( M_{air} = 0.79 \times 28.02 + 0.21 \times 32.00 \) \( M_{air} = 28.96 \; g/mol \) Step 2: Using Ideal Gas Law for helium balloon: \( PV = n_{He} RT_{He}\) \( n_{He} = \frac{PV}{RT_{He}} \) Step 3: Calculate the mass of air displaced: \( m_{air\; displaced} = \rho_{He} V - m_{He} \) \( m_{air\; displaced} = \frac{n_{He}M_{He}}{V} - \frac{n_{He}M_{air}}{V} \) Step 4: Number of moles of air displaced: \(n_{air\; displaced} = \frac{m_{air\; displaced}}{M_{air}} \) Step 5: Apply Ideal Gas Law for hot-air balloon: \( PV = n_{air\; displaced} RT_{hot\; air} \) \( T_{hot\; air} = \frac{n_{air\; displaced} R}{\frac{PV}{M_{air}}} \) After calculating all these steps, you will find that the required temperature for the hot-air balloon is approximately \( 41.8^{\circ} \mathrm{C} \).

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is essential when discussing the principles behind the lift of a hot-air balloon. The law combines Boyle's, Charles's, and Avogadro's laws into one comprehensive equation, which is often expressed as PV = nRT. Here, P stands for the pressure of the gas, V is the volume it occupies, n is the number of moles of the gas, R is the universal gas constant, and T is the absolute temperature in Kelvin.

This formula allows us to calculate any one of the gas properties if the others are known. For instance, when calculating the lift of a balloon, we can use the Ideal Gas Law to find out the number of moles of gas inside the balloon, which will help us determine its temperature or volume for a given pressure.

In the context of our problem, to determine the lift provided by a hot-air balloon, we must use the Ideal Gas Law to find the temperature required inside the balloon that would give the same lift as a helium balloon.

Molar Mass of Air

The molar mass of air is a key factor in calculating the lift of a balloon because it allows us to quantify the mass of the gas inside the balloon compared to the mass of the air it displaces. Air is not a single gas but a mixture, primarily composed of nitrogen (N₂) and oxygen (O₂).

To calculate the average molar mass of air, we consider it as 79% nitrogen and 21% oxygen by volume. By using the individual molar masses of nitrogen and oxygen, 28.02 g/mol and 32.00 g/mol respectively, we obtain the average molar mass of air as a weighted sum, which yields approximately 28.96 g/mol.

This value is an integral part of the calculations involving the Ideal Gas Law, as it directly affects the moles of gas and, consequently, the temperature and lift calculations for the hot-air balloon.

Density and Buoyancy

Density, the mass per unit volume of a substance, and buoyancy, the force exerted by a fluid on an object immersed in it, are fundamental to understanding hot-air balloon lift. The lift of a balloon, as defined, is the difference in mass between the air displaced by the balloon and the mass of gas inside the balloon. This principle relies on Archimedes' principle, which states that the buoyant force exerted on an object is equal to the weight of the fluid it displaces.

When the air inside the balloon is heated, it expands and its density decreases. Since the balloon displaces the same volume of air whether it is filled with hot air or cold air, the weight of the air displaced remains constant. However, the weight of the hot air inside the balloon is now less than the weight of the air displaced causing an upward buoyant force, which is the lift.

Achieving the right balance between the density of the air inside the balloon and the air outside is crucial for the balloon to rise. If the density inside is too high, the balloon will not lift, while if it is too low, the lift might be excessive, making it difficult to control.

Temperature and Gas Volume

Temperature and gas volume are intrinsically linked through Charles's Law, which is part of the Ideal Gas Law. This law states that at constant pressure, the volume of a gas is directly proportional to its temperature measured in Kelvin.

In the case of the hot-air balloon, heating the air inside the balloon increases its temperature and causes the air to expand, thereby increasing its volume assuming the pressure remains constant. Since the balloon fabric confines the air, the balloon maintains the same shape and lifts as volume increases, and that can only happen if the mass of the gas inside becomes less dense than the air outside.

For our exercise, the hot-air balloon must provide the same lift as a helium balloon. Helium, being lighter than air, provides significant lift at room temperature; for the hot-air balloon to match this, we must increase the temperature of the air inside until the lift is equivalent to that provided by the helium. To conclude, the temperature inside the hot-air balloon needs to be carefully calibrated to achieve the desired lift and controlled flight.