Question

A balloon weighing \(50 \mathrm{~kg}\) is filled with \(685 \mathrm{~kg}\) of helium at 1 atm pressure and \(25^{\circ} \mathrm{C}\). What will be its pay load if it displaced \(5108 \mathrm{~kg}\) of air? (a) \(4373 \mathrm{~kg}\) (b) \(4423 \mathrm{~kg}\) (c) \(5793 \mathrm{~kg}\) (d) none of these

Short answer

The payload of the balloon is 4373 kg, making option (a) the correct answer.

Step-by-step solution

Understand the concept of payload

The payload of a balloon is the difference between the weight of the air displaced by the helium in the balloon and the total weight of the balloon including the helium inside. It can be calculated by subtracting the weight of the balloon and the helium from the weight of the air displaced.

Calculate the total weight of the balloon and the helium

Add the weight of the balloon to the weight of the helium to get the total weight. This can be calculated as follows: Total weight = weight of balloon + weight of helium = 50 kg + 685 kg = 735 kg.

Calculate the payload

Subtract the total weight of the balloon and the helium from the weight of the air displaced to find the payload. This can be calculated as follows: Payload = weight of air displaced - total weight = 5108 kg - 735 kg = 4373 kg.

Key concepts

Payload Calculation in Aerial Vehicles

When attempting to understand the payload capacity of aerial vehicles such as balloons or blimps, one must start with the basic definition of payload. In the field of aeronautics, payload refers to the carrying capacity of an aircraft, minus the weight of the pilot, crew, and any fuel or necessary gear. It essentially represents the additional weight the vehicle can carry, which could range from passengers, cargo, or scientific instruments.

For the example given in the exercise, an understanding of the payload is crucial to calculate how much additional weight the balloon can lift. The payload is determined by the buoyant force, which is the lifting force a balloon experiences when it displaces air. To calculate the payload of the balloon, one needs to subtract the total weight of the balloon and its contained helium from the weight of the air it displaces. This payload computation, when applied to our problem, involves simple arithmetic: the air displacement weight is given as 5108 kg, and the total weight of the balloon and helium together is 735 kg, thus the payload capacity is computed to be 4373 kg.

Buoyancy in Chemistry

Buoyancy, a key concept in both physics and chemistry, refers to the ability of an object to float in a fluid (liquid or gas). This principle is governed by Archimedes' Principle, which states that the buoyant force - the upward force that keeps things afloat - is equal to the weight of the fluid that the object displaces. In chemical terms, this is often related to the density of the materials involved.

In the context of our helium-filled balloon, the balloon rises because helium is less dense than air. Thus, when the balloon displaces a certain amount of air, as per Archimedes’ principle, it experiences an upward, buoyant force equal to the weight of the air displaced. This is why knowing the weight of the displaced air (5108 kg in the exercise) is so pivotal for payload calculations. Furthermore, the presumption that the balloon is floating implies that the buoyant force is at least equal to the gravitational force on the balloon and helium combined.

Gas Laws in Physical Chemistry

The behavior of gases is an important aspect of physical chemistry, especially related to buoyancy and payload calculations. Gas laws help us understand and predict how gases will react to changes in pressure, temperature, and volume. Key laws that inform these calculations include Boyle's Law, which shows the inverse relationship between pressure and volume, and Charles's Law, which illustrates the direct proportionality between volume and temperature.

In the provided exercise, knowing that helium is at 1 atm pressure and 25°C is significant because it allows us to approximate the volume and therefore the displacement of air based on standard gas laws. While the payload calculation primarily relies on the difference in weight (a force), gas laws indirectly inform this calculation. They are necessary for determining how much gas can be filled in the balloon (without mentioning the exact volume of helium, which is decided by the gas laws based on given pressure and temperature). Correctly understanding gas laws is imperative in ensuring the safe and effective operation of balloons for transportation or research purposes.