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Chapter 4: Problem 24

In "The Unparalleled Adventure of One Hans Pfaall" by Edgar Allen Poe, the hero discovers a gas whose density is " \(37.4\) times" less than that of hydrogen. How much better at lifting would a balloon filled with the new gas be compared to one filled with hydrogen?

Short Answer

Expert verified
Answer: Yes, the new gas is better at lifting compared to hydrogen.

Step by step solution

01

Understand buoyancy and lifting force

The lifting force of a balloon is due to buoyancy, which is explained by Archimedes' principle. Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In the case of a balloon floating in the air, the lifting force is determined by the difference in density between the gas inside the balloon and the surrounding air, multiplied by the volume of the balloon and the acceleration due to gravity.
02

Calculate the density ratios

Let the density of hydrogen be \(d_H\) and the density of the new gas be \(d_{NG}\). We are given that the new gas has a density of 37.4 times less than that of hydrogen, so we can write the relationship between the two densities as: \(d_{NG} = \frac{d_H}{37.4}\)
03

Calculate the difference in densities

Let the density of air be \(d_A\). The difference in densities between the hydrogen-filled balloon and the air is: \(\Delta d_H = d_A - d_H\) Similarly, the difference in densities between the new gas-filled balloon and the air is: \(\Delta d_{NG} = d_A - d_{NG}\)
04

Calculate the lifting force ratios

Now, we will calculate the ratio of the lifting forces for the two balloons. The lifting force of a balloon is directly proportional to the difference in densities, so we can write the relationship between the two lifting forces as: \(\frac{F_{NG}}{F_H} = \frac{\Delta d_{NG}}{\Delta d_H}\) Substituting the expression for density difference, we get: \(\frac{F_{NG}}{F_H} = \frac{d_A - \frac{d_H}{37.4}}{d_A - d_H}\)
05

Simplify the expression

Let's simplify the expression by multiplying both the numerator and the denominator by \(37.4\): \(\frac{F_{NG}}{F_H} = \frac{37.4\cdot d_A - d_H}{37.4\cdot d_A - 37.4 \cdot d_H}\)
06

Calculate how much better the new gas is at lifting

The ratio \(\frac{F_{NG}}{F_H}\) represents how much better the new gas is at lifting compared to hydrogen. If the ratio is greater than 1, the new gas is better at lifting, and if it is less than 1, hydrogen is better. Since the ratio is independent of the actual densities, we can see that: \(\frac{F_{NG}}{F_H} = \frac{37.4\cdot d_A - d_H}{37.4\cdot d_A - 37.4 \cdot d_H} > 1\) This shows that the new gas would be better at lifting compared to hydrogen, and our calculation gives the difference in the lifting capabilities depending on the densities of the gases involved.

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