Problem 91 - Chemistry: Principles and Reactions

Question

The buoyant force on a balloon is equal to the mass of air it displaces. The gravitational force on the balloon is equal to the sum of the masses of the balloon, the gas it contains, and the balloonist. If the balloon and the balloonist together weigh \(168 \mathrm{~kg}\), what would the diameter of a spherical hydrogen-filled balloon have to be in meters if the rig is to get off the ground at \(22^{\circ} \mathrm{C}\) and \(758 \mathrm{~mm} \mathrm{Hg}\) ? (Take \(\mathrm{MM}_{\text {air }}=29.0 \mathrm{~g} /\) mol.)

Step-by-Step Solution

Answer

Answer: The diameter of the spherical hydrogen-filled balloon required to lift the rig off the ground is approximately 6.354 meters.

Step 1: Write down the given information

The combined weight of the balloon and the balloonist is given as 168 kg. The temperature is given as 22°C (which needs to be converted to Kelvin) and the pressure as 758 mmHg (which needs to be converted to Pascals). The molar mass of air is given as 29.0 g/mol.

Step 2: Convert the temperature and pressure to SI units

To convert from Celsius to Kelvin, add 273.15: $$T = 22 + 273.15 = 295.15 \mathrm{~K}$$ To convert from mmHg to Pascals, use the conversion factor 1 mmHg = 133.322 Pa: $$P = 758 \times 133.322 = 101100 \mathrm{~Pa}$$

Step 3: Calculate the buoyant force required to lift the rig

The buoyant force required to lift off the ground is equal to the gravitational force acting on the balloon and balloonist, which is $$F = m \times g$$ Using the given mass (168 kg) and acceleration due to gravity (9.81 m/s²), we have: $$F = 168 \times 9.81 = 1648.08 \mathrm{~N}$$

Step 4: Find the mass of the displaced air using Archimedes' principle

According to Archimedes' principle, the buoyant force is equal to the mass of air displaced multiplied by gravity: $$F = m_{\text{air}} \times g$$ Solve for the mass of the displaced air: $$m_{\text{air}} = \frac{F}{g} = \frac{1648.08}{9.81} = 168 \mathrm{~kg}$$

Step 5: Determine the density of air at the given conditions using the ideal gas law

The ideal gas law in terms of density is: $$PV = nRT$$ Rearranging to find the density of air: $$\rho_{\text{air}} = \frac{m_{\text{air}}}{V} = \frac{n \times MM_{\text{air}}}{V} = \frac{P \times MM_{\text{air}}}{RT}$$ Using the given pressure, molar mass of air, temperature, and the gas constant R (8.314 J/mol·K), we can calculate the density of air: $$\rho_{\text{air}} = \frac{101100 \times 0.029}{8.314 \times 295.15} = 1.185 \mathrm{~kg/m}^3$$

Step 6: Calculate the volume of the displaced air

Now, we can find the volume of the air displaced by the balloon: $$V = \frac{m_{\text{air}}}{\rho_{\text{air}}} = \frac{168}{1.185} = 141.776 \mathrm{~m}^3$$

Step 7: Calculate the diameter of the spherical balloon

The volume of a sphere can be determined using the formula: $$V = \frac{4}{3}\pi r^3$$ Rearranging to solve for the radius: $$r = \sqrt[3]{\frac{3V}{4\pi}}$$ Inserting the calculated volume: $$r = \sqrt[3]{\frac{3 \times 141.776}{4\pi}} = 3.177 \mathrm{~m}$$ The radius is 3.177 m, so the diameter of the balloon is $$D = 2r = 2 \times 3.177 = 6.354 \mathrm{~m}$$ The diameter of the spherical hydrogen-filled balloon required to lift the rig off the ground is approximately 6.354 meters.

Key Concepts

Archimedes' Principle

Archimedes' principle is a fundamental law in physics that describes how the buoyant force on an object submerged in a fluid relates to the weight of the fluid displaced by the object. The principle can be applied to both liquids and gases and is named after the ancient Greek mathematician and inventor Archimedes, who first postulated this relationship.

The principle states that the buoyant force experienced by an object in a fluid is equal to the weight of the fluid that the object displaces. This concept allows us to calculate whether an object will sink or float when placed in a fluid. In the case of the balloon in the exercise, the balloon will float if the buoyant force - which is equal to the weight of the air displaced by the balloon - is greater than or equal to the gravitational force on the balloon and its payload, including the balloonist.

To determine whether the balloon rises or not, we need to calculate the volume of the balloon that leads to a mass of air displacement equal to the weight of the balloon system (balloon, gas, and balloonist). Archimedes' principle serves as the crucial starting point for this calculation, guiding us in understanding the relationship between buoyancy and displaced volume.

Ideal Gas Law

The ideal gas law is a powerful equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and quantity (n in moles) of an ideal gas. The law is generally expressed as \( PV = nRT \), where R is the universal gas constant. This equation is based on the assumption that particles of an ideal gas do not interact with each other and that they occupy no volume; although real gases do not perfectly meet these criteria, the ideal gas law serves as a good approximation under many conditions.

In this case, we utilize the ideal gas law to determine the density of air under specific conditions. Density is mass per unit volume, which is represented as \( \rho = \frac{m}{V} \). By manipulating the ideal gas law, we can express it in terms of density: \( \rho = \frac{PM}{RT} \), where M is the molar mass. This equation allows us to calculate the density of air with known pressure, temperature, and molar mass, providing a crucial link between the physical states of the air and the ability to calculate the buoyant force necessary for the balloon to float.

Density of Air

The density of air, denoted as \( \rho \), is the mass per unit volume of Earth's atmosphere. Air density is a crucial factor in various scientific calculations and technological applications because it affects the movement of objects through the atmosphere, including balloons, airplanes, and weather patterns.

Air density can vary depending on temperature, pressure, and humidity. Warmer air is less dense than cooler air, and air at higher altitudes is less dense due to lower air pressure. In the context of the exercise provided, the density of air at the given temperature \(22^{\text{\tiny{\circ}}} \text{C}\) and pressure (758 mm Hg) is calculated using a modified form of the ideal gas law. The calculated density is then used to ascertain the volume of air that the balloon must displace to generate a buoyant force sufficient to lift the balloon and the balloonist off the ground.

Gravitational Force Calculation

Gravitational force calculation, often known as weight calculation, concerns the force exerted on an object due to the gravity of a celestial body, usually Earth. It is determined using the formula \( F = mg \), where 'm' is the mass of the object and 'g' is the acceleration due to gravity, approximately \(9.81 \text{m/s}^2\) on Earth's surface.

In the context of our balloon exercise, we calculate the gravitational force on the entire system comprising the balloon, gas, and human payload, which must be counteracted by the buoyant force for the balloon to ascend. Once we have the total weight, we can then use Archimedes' principle to find out how much air needs to be displaced by the balloon (which, in turn, relates to the volume of the balloon) to achieve lift-off. These calculations integrate the laws of physics to resolve practical questions in aviation and fluid dynamics.