Question

The densities of air, helium, and hydrogen (at \(p\) \(=\) 1.0 atm and \(T\) \(=\) 20\(^\circ\)C) are 1.20 kg/m\(^3\), 0.166 kg/m\(^3\), and 0.0899 kg/m\(^3\), respectively. (a) What is the volume in cubic meters displaced by a hydrogen- filled airship that has a total "lift" of 90.0 kN? (The "lift" is the amount by which the buoyant force exceeds the weight of the gas that fills the airship.) (b) What would be the "lift" if helium were used instead of hydrogen? In view of your answer, why is helium used in modern airships like advertising blimps?

Short answer

Airship must displace ~8244 m³ of air for 90kN lift with hydrogen; helium produces ~74.9kN lift. Helium is used for safety.

Step-by-step solution

Understand the Concept of Lift

The 'lift' of an airship is the net upward force, which is the buoyant force minus the weight of the gas inside the airship. This is due to the difference in density between the air and the gas. The buoyant force is equal to the weight of the air displaced by the volume of the airship.

Calculate the Volume of Displaced Air for Hydrogen

To find the volume of displaced air, we use the formula for lift: \( F_{lift} = \rho_{air} V g - \rho_{gas} V g \), where \( \rho_{air} \) and \( \rho_{gas} \) are the densities of air and the gas (hydrogen in this case), \( V \) is the volume, and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Since \( F_{lift} = 90.0 \) kN = 90,000 N, solve for \( V \):\[90,000 = (1.20 - 0.0899) V (9.8)\]\[V = \frac{90,000}{(1.20 - 0.0899) \times 9.8}\]\[V \approx 8243.71 \text{ m}^3\]

Calculate the Lift with Helium

Repeat the lift calculation using helium by substituting helium's density into the formula:\[ F_{lift, helium} = \rho_{air} V g - \rho_{helium} V g\]\[ F_{lift, helium} = (1.20 - 0.166) \times 8243.71 \times 9.8\]\[ F_{lift, helium} \approx 74,854.58 \text{ N} \]

Compare and Conclude

Hydrogen provides greater lift than helium because it is less dense, but helium is used for safety reasons since helium is non-flammable compared to hydrogen. Despite the lower lift of helium (74.9 kN compared to 90.0 kN with hydrogen), the safety advantages make helium preferable.

Key concepts

Density Comparison

Density is a measure of how much mass is contained in a given volume. It plays a crucial role in determining buoyancy. For airships, the relevant densities are:

• Air: 1.20 kg/m³
• Helium: 0.166 kg/m³
• Hydrogen: 0.0899 kg/m³

Air, being denser than helium and hydrogen, contributes to the buoyant force experienced by the airship. The larger the difference between the density of the air and the gas used in the airship, the greater the lift that can be generated. This is why lighter gases like helium and hydrogen are preferred for filling airships.

Buoyant Force

Buoyant force is the upward force that keeps objects afloat and is caused by the displacement of fluid (in this case, air) by a volume of another substance (the gas in the airship). The buoyant force is calculated by considering the weight of the air displaced by the airship's volume.The formula to calculate buoyancy is:\[ F_{buoyant} = \rho_{air} \times V \times g \]Where:

• \( \rho_{air} \) is the density of air.
• \( V \) is the volume of the airship.
• \( g \) is the acceleration due to gravity \( \approx 9.8 \text{ m/s}^2 \).

To achieve lift, the buoyant force must exceed the weight of the gas inside the airship. Therefore, a balance between the buoyant force and the gas weight results in the net force known as 'lift'.

Helium versus Hydrogen

Both helium and hydrogen are lighter than air, making them suitable for use in airships. However, they have different densities which affects the lift they provide:

• Hydrogen has a density of 0.0899 kg/m³, which allows for more lift because it displaces more air relative to its own weight.
• Helium, with a density of 0.166 kg/m³, provides slightly less lift than hydrogen but is considered due to its safety characteristics.

When comparing lift capabilities, hydrogen's lower density results in a greater buoyant force and thus, more lift. Despite this, helium is often chosen because it is non-flammable, which makes it safer in applications like modern airships.

Safety in Modern Airships

Safety is a primary concern in the design and operation of modern airships. Although hydrogen provides more lift due to its lower density, it is highly flammable and poses significant risks, as evidenced by historical accidents. Helium, on the other hand, is an inert gas meaning it does not react and is not flammable. This makes helium the preferred choice for modern airships despite its slightly lower lifting capability. Therefore:

• Helium's safety attributes outweigh its lower performance in terms of lift.
• Modern airships and blimps often prioritize safety over maximizing lift.

The use of helium in modern airships exemplifies a commitment to ensuring the highest standards of safety in airship design and operation.