Question

What mass of helium, in grams, is required to fill a \(5.0-\mathrm{I}\) balloon to a pressure of 1.1 atm at \(25^{\circ} \mathrm{C} ?\)

Short answer

The mass of helium required is approximately 0.90 grams.

Step-by-step solution

Understanding the Ideal Gas Law

The Ideal Gas Law is given by the equation \( PV = nRT \), 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 in Kelvin.

Converting Temperature to Kelvin

The given temperature is \( 25^{\circ} \mathrm{C} \). To convert this to Kelvin, use the formula \( T(K) = T(\degree C) + 273.15 \). So, \( T = 25 + 273.15 = 298.15 \) K.

Identifying Given Values

You are given \( P = 1.1 \) atm, \( V = 5.0 \) L, and \( T = 298.15 \) K. The ideal gas constant, \( R \), is \( 0.0821 \) L atm/mol K.

Solving for Moles of Gas

Rearrange the ideal gas law to solve for \( n \): \( n = \frac{PV}{RT} \). Substitute the known values: \( n = \frac{(1.1 \text{ atm})(5.0 \text{ L})}{(0.0821 \text{ L atm/mol K})(298.15 \text{ K})} \).

Calculating the Number of Moles

Calculate \( n \) using the rearranged equation: \( n = \frac{5.5}{24.466} \approx 0.225 \) moles. Thus, the balloon contains approximately 0.225 moles of helium gas.

Converting Moles to Grams

The molar mass of helium is approximately 4.00 g/mol. To find the mass in grams, use the formula \( \text{mass} = n \cdot \text{molar mass} \): \( \text{mass} = 0.225 \times 4.00 \approx 0.90 \) grams.

Key concepts

Helium Gas

Helium is a noble gas with the chemical symbol He. It is both lightweight and non-reactive, making it ideal for applications like filling balloons. When dealing with helium gas, it's essential to consider its atomic and molecular properties.

• Helium is the second lightest element, with an atomic number of 2.
• It exists as a monoatomic gas, meaning it is composed of single atoms rather than molecules.
• This gas is colorless, odorless, and tasteless.

Helium does not easily form compounds with other elements due to its stable electron configuration. This inertness makes helium a safe choice for many scientific and industrial purposes. When calculating properties like moles or mass, remember that the molar mass of helium is approximately 4.00 g/mol.

Mole Calculations

Moles are a central concept in chemistry, representing a specific number of particles, typically atoms or molecules. One mole is defined as exactly 6.022 x 10²³ particles, a value known as Avogadro's number. To perform mole calculations:

• First, identify the relevant chemical species and their weights or molar masses.
• Use the ideal gas law to determine the number of moles when dealing with gases, especially under specific conditions of temperature and pressure.

For instance, you can calculate the number of moles using the formula:\[ n = \frac{PV}{RT}\]Here, \( P \) is the pressure, \( V \) is the volume, \( R \) is the ideal gas constant \( (0.0821 \text{ L atm/mol K}) \), and \( T \) is the temperature in Kelvin. Once you have the moles, converting this to mass involves multiplying by the substance's molar mass.

Temperature Conversion

Converting temperatures between Celsius and Kelvin is a routine task in scientific calculations. The Kelvin scale is preferred in chemistry for its absolute nature, which begins at absolute zero, the theoretical point where particles have minimum thermal energy. The conversion formula is straightforward:\[ T(K) = T(°C) + 273.15\]Using this formula, a temperature of 25°C converts to 298.15 K. It's critical to perform this conversion when using the ideal gas law or any other gas-related calculations, as these calculations require absolute temperature values in Kelvin. Remember, you should never perform gas law calculations using Celsius.

Gas Laws

Gas laws describe the behavior of gases in different conditions. The ideal gas law, in particular, is a valuable tool for predicting how gases behave under varying pressures, volumes, and temperatures. The ideal gas law is expressed as:\[ PV = nRT \]In this equation:

• \( P \) is the pressure in atmospheres.
• \( V \) is the volume in liters.
• \( n \) is the number of moles.
• \( R \) is the ideal gas constant \((0.0821 \, \text{L atm/mol K})\).
• \( T \) is the temperature in Kelvin.

This law assumes ideal conditions where gas molecules do not interact with each other and occupy no volume themselves. While real gases may deviate slightly due to intermolecular forces and the volume of gas particles, the ideal gas law provides a close approximation that is useful for solving many practical problems in chemistry.