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Chapter 14: Problem 16

A Goodyear blimp typically contains \(5400 \mathrm{m}^{3}\) of helium (He) at an absolute pressure of \(1.1 \times 10^{5}\) Pa. The temperature of the helium is \(280 \mathrm{K}\) What is the mass (in \(\mathrm{kg}\) ) of the helium in the blimp?

Short Answer

Expert verified
The mass of helium in the blimp is approximately 102.36 kg.

Step by step solution

01

Identify the given values

We are given the following data: Volume of helium, \( V = 5400 \, \mathrm{m}^3 \); Pressure of helium, \( P = 1.1 \times 10^5 \, \mathrm{Pa} \); Temperature of helium, \( T = 280 \, \mathrm{K} \). We also know the molar mass of helium, \( M_{\mathrm{He}} = 4.00 \, \mathrm{g/mol} \), but we need to use it in kilograms per mole: \( 4.00 \times 10^{-3} \, \mathrm{kg/mol} \). The gas constant \( R \) is \( 8.314 \, \mathrm{J/(mol \cdot K)} \).
02

Use the ideal gas law to find the number of moles

Apply the ideal gas law, \( PV = nRT \), where \( n \) is the number of moles of helium. Rearrange to find \( n \): \[n = \frac{PV}{RT}\]. Substitute the given values into the equation: \[n = \frac{(1.1 \times 10^5 \, \mathrm{Pa}) \times (5400 \, \mathrm{m}^3)}{8.314 \, \mathrm{J/(mol \cdot K)} \times 280 \, \mathrm{K}}\].
03

Calculate the number of moles

Calculate \( n \) using the formula derived in Step 2: \[n = \frac{1.1 \times 10^5 \, \times 5400}{8.314 \times 280}\]. This simplifies to \[n \approx 25589.7 \, \mathrm{mol}\].
04

Convert moles to mass

Use the molar mass of helium to convert moles to mass. The mass \( m \) is given by \( m = n \times M_{\mathrm{He}} \). So, \( m = 25589.7 \, \mathrm{mol} \times 4.00 \times 10^{-3} \, \mathrm{kg/mol} \).
05

Calculate the mass

Substitute the values to find the mass: \[m = 25589.7 \, \times 4.00 \times 10^{-3} = 102.359 \, \mathrm{kg}\]. Therefore, the mass of the helium in the blimp is approximately \(102.36 \, \mathrm{kg}\).

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Key Concepts

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

Helium
Helium is an extremely light and inert gas that is often used for inflating blimps and balloons. Its chemical symbol is He, and it is the second lightest element after hydrogen. Helium is part of the noble gas group, which means it does not react easily with other elements due to its stable electronic configuration. This makes it ideal for applications where a non-reactive atmosphere is needed, such as in blimps.
  • It has a low molecular weight, allowing it to lift heavy objects when contained in large volumes.
  • Helium is non-flammable, adding an element of safety over gases like hydrogen.
  • It is colorless, odorless, and tasteless, making its use unobtrusive in various scenarios.
Its lightness and non-reactivity make it a popular choice not just for aviation, but also in scientific, medical, and industrial applications.
Molar Mass
Molar mass is an essential concept when dealing with gases in chemistry. It represents the mass of one mole of a substance, often measured in grams per mole (g/mol) or kilograms per mole (kg/mol) for larger calculations. Helium has a very low molar mass of 4.00 g/mol, or converted to the standard unit for the ideal gas law calculations, 4.00 x 10-3 kg/mol.
  • Molar mass allows us to connect the macroscopic mass of a substance to the microscopic scale of atoms or molecules.
  • It is used to convert the number of moles to actual physical mass, which is particularly useful in experiments and chemical reactions.
  • Understanding molar mass is crucial for applying the ideal gas law, where it helps in determining the relationship between moles and mass of a gas.
When dealing with calculations in thermodynamics or stoichiometry, knowing the molar mass enables conversion from moles to grams or kilograms, which is often required to solve problems involving gases.
Temperature
Temperature is fundamental in the study of gases, as it directly affects their energy and state. It is measured in Kelvin (K) for scientific calculations involving gases. The Kelvin scale is used because it starts at absolute zero, where theoretically, particles have minimal motion and energy.
  • Temperature influences the speed and vibration of gas particles, impacting the pressure and volume of the gas.
  • The Ideal Gas Law, which relates pressure, volume, temperature, and moles, uses Kelvin to maintain accuracy across various conditions.
  • In the case of helium in a blimp, the specified temperature of 280 K helps determine the behavior and properties of the gas, influencing its calculation of moles and mass.
Understanding the temperature in Kelvin is critical for calculations like those involving the Ideal Gas Law, facilitating accurate results regardless of external conditions.
Gas Constant
The gas constant, denoted as \( R \), is vital for calculations involving the ideal gas law. It bridges the relationship between pressure, volume, temperature, and number of moles of a gas. The value of the gas constant is \( 8.314 \) J/(mol路K), and it serves to standardize the units used in these equations.
  • It provides a constant proportionality factor required to equate the macroscopic properties of gases.
  • Using \( R \) ensures that calculations remain consistent across different conditions and units.
  • In the provided exercise, the gas constant allows us to compute the number of moles from the known conditions of helium.
Without \( R \), the relationship between the variables in the Ideal Gas Law wouldn't be as clearly defined, making it challenging to relate physical quantities like pressure and temperature practically.

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Most popular questions from this chapter

A tank contains \(11.0 \mathrm{g}\) of chlorine gas \(\left(\mathrm{Cl}_{2}\right)\) at a temperature of \(82^{\circ} \mathrm{C}\) and an absolute pressure of \(5.60 \times 10^{5} \mathrm{Pa} .\) The mass per mole of \(\mathrm{Cl}_{2}\) is \(70.9 \mathrm{g} / \mathrm{mol}\) (a) Determine the volume of the tank. (b) Later, the temperature of the tank has dropped to \(31^{\circ} \mathrm{C}\) and, due to a leak, the pressure has dropped to \(3.80 \times 10^{5}\) Pa. How many grams of chlorine gas have leaked out of the tank?

A certain element has a mass per mole of \(196.967 \mathrm{g} / \mathrm{mol}\). What is the mass of a single atom in (a) atomic mass units and (b) kilograms? (c) How many moles of atoms are in a 285 -g sample?

A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains \(0.034 \mathrm{m}^{3}\) of helium at an absolute pressure of \(1.2 \times 10^{5} \mathrm{Pa} .\) The cylinder contains helium at an absolute pressure of \(1.6 \times 10^{7} \mathrm{Pa}\) and has a volume of \(0.0031 \mathrm{m}^{3} .\) The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled?

A tube has a length of \(0.015 \mathrm{m}\) and a cross-sectional area of \(7.0 \times \mathrm{x}\) \(10^{-4} \mathrm{m}^{2} .\) The tube is filled with a solution of sucrose in water. The diffusion constant of sucrose in water is \(5.0 \times 10^{-10} \mathrm{m}^{2} / \mathrm{s} .\) A difference in concentration of \(3.0 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3}\) is maintained between the ends of the tube. How much time is required for \(8.0 \times 10^{-13} \mathrm{kg}\) of sucrose to be transported through the tube?

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is \(A=2.50 \times 10^{-3} \mathrm{m}^{2}\) The initial pressure, volume, and temperature of the gas are, respectively, \(P_{0}, V_{0}=6.00 \times 10^{-4} \mathrm{m}^{3},\) and \(T_{0}=273 \mathrm{K}\) and the spring is initially stretched by an amount \(x_{0}=\) \(0.0800 \mathrm{m}\) with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are \(P_{\mathrm{f}}, V_{\mathrm{f}},\) and \(T_{\mathrm{f}},\) and the spring is stretched by an amount \(x_{\mathrm{f}}=0.1000 \mathrm{m}\) with respect to its unstrained length. What is the final temperature of the gas?
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