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Chapter 17: Problem 9

Amount of Helium in a Blimp The Goodyear blimp Spirit of Alcron is \(62.6 \mathrm{m}\) long and contains \(7023 \mathrm{m}^{3}\) of helium. When the temperature of the helium is \(285 \mathrm{K}\), its absolute pressure is \(112 \mathrm{kPa} .\) Find the mass of the helium in the blimp.

Short Answer

Expert verified
The mass of the helium is approximately 133 kg.

Step by step solution

01

Understand the Ideal Gas Law

The ideal gas law is given by the formula \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.
02

Identify Given Values

From the problem, \( P = 112 \) kPa, \( V = 7023 \) m\(^{3}\), and \( T = 285 \) K. The ideal gas constant \( R \) for helium is 8.314 J/(mol路K).
03

Convert Pressure to Proper Units

The standard unit of pressure is Pascals (Pa), so we convert kPa to Pa by multiplying by 1,000. Thus, \( P = 112,000 \) Pa.
04

Solve for Number of Moles \(n\)

Rearrange the ideal gas law to \( n = \frac{PV}{RT} \). Substitute the known values: \( n = \frac{112,000 \times 7023}{8.314 \times 285} \).
05

Calculate the Number of Moles

Compute the expression to find \( n \). After calculation, \( n = 33258 \) moles approximately.
06

Determine Mass from Moles

The molar mass of helium is 4 g/mol. To find the mass, use \( \,mass = n \times \,molar\, mass \), thus \( mass = 33258 \times 4 \,g/mol = 133032 \, g \) which is approximately 133 kg.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Helium in Context
Helium is a fascinating element, renowned for being incredibly light and inert. It is a noble gas, meaning it is rarely reactive with other elements. Helium's low density makes it perfect for use in blimps, allowing them to float effortlessly due to helium being lighter than the surrounding air. Besides aeronautical applications, helium is also used in cryogenics and as a protective gas for arc welding.

Given its nature, helium plays a crucial role in applications where weight and inertness are essential. Its low molar mass also allows for it to be used in calculations related to gas laws, which is particularly useful in understanding various scientific phenomena and engineering solutions.
Understanding Pressure Conversion
Pressure conversion is a vital step in solving problems that involve gases, as pressure is commonly recorded in various units like kPa, atm, or Pa. Understanding how to convert between these units is crucial for accurate calculations. For instance, if pressure is given in kilopascals (kPa), converting it to pascals (Pa) is essential when using the ideal gas law, which requires pressure to be in standard units.

One kilopascal (kPa) is equivalent to 1,000 pascals (Pa). This means to convert kilopascals to pascals, simply multiply the pressure in kPa by 1,000. This standardization ensures the units are consistent across all variables within the ideal gas equation, making calculations straightforward and reliable.
The Role of Molar Mass
Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and is crucial for converting between the number of moles and the mass of a substance. For helium, the molar mass is 4 g/mol, which is relatively low compared to other elements, making it an ideal candidate for applications where low density is beneficial.

When calculating the mass of a gas from its number of moles, use the relationship:
  • mass = number of moles 脳 molar mass
This formula allows you to determine how much a specified number of moles would weigh, aiding in practical applications like the weight calculation of helium in a blimp, as shown in the exercise.
Calculating the Number of Moles
One of the core components of utilizing the ideal gas law is determining the number of moles of gas present in a given volume, pressure, and temperature. The formula
  • \( n = \frac{PV}{RT} \)
allows you to calculate the moles from known quantities:
  • \( P \) - Pressure in Pascals
  • \( V \) - Volume in cubic meters (m鲁)
  • \( R \) - Ideal gas constant (8.314 J/(mol路K))
  • \( T \) - Temperature in Kelvin
This calculation is essential in determining how the amount of gas in a blimp influences its buoyancy and mass. By rearranging the ideal gas law and substituting the provided values, you can easily find the number of moles in the helium-filled blimp. This process ultimately aids in calculating the gas's mass, validating the blimp's lift capacity.

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Most popular questions from this chapter

Standard temperature and pressure (STP) is defined as a temperature of \(0^{\circ} \mathrm{C}\) and a pressure of \(101.3 \mathrm{kPa}\). What is the volume occupied by one mole of an ideal gas at STP?

A gas has a temperature of \(310 \mathrm{K}\) and a pressure of \(101 \mathrm{kPa}\). (a) Find the volume occupied by 1.25 mol of this gas, assuming it is ideal. (b) Assuming the gas molecules can be approximated as small spheres of diameter \(2.5 \times 10^{-10} \mathrm{m},\) determine the fraction of the volume found in part (a) that is occupied by the molecules. (c) In determining the properties of an ideal gas, we assume that molecules are points of zero volume. Discuss the validity of this assumption for the case considered here.

Is the number of atoms in one mole of helium greater than, less than, or equal to the number of atoms in one mole of oxygen? Helium consists of individual atoms, He, and oxygen is a diatomic gas, \(\mathrm{O}_{2}\)

An ideal gas is kept in a container of constant volume. The pressure of the gas is also kept constant. (a) If the number of molecules in the gas is doubled, does the rms speed increase, decrease, or stay the same? Explain. (b) If the initial rms speed is \(1300 \mathrm{m} / \mathrm{s},\) what is the final rms speed?

A \(35-g\) ice cube at \(0.0 ~^{\circ} \mathrm{C}\) is added to \(110 \mathrm{g}\) of water in a \(62-g\) aluminum cup. The cup and the water have an initial temperature of \(23^{\circ} \mathrm{C} .\) (a) Find the equilibrium temperature of the cup and its contents. (b) Suppose the aluminum cup is replaced with one of equal mass made from silver. Is the equilibrium temperature with the silver cup greater than, less than, or the same as with the aluminum cup? Explain.
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