Question

A balloon holds \(30.0 \mathrm{kg}\) of helium. What is the volume of the balloon if its pressure is 1.20 atm and the temperature is \(22^{-} \mathrm{C} ?\)

Short answer

The volume of the balloon is approximately 152.0 m³.

Step-by-step solution

Convert Temperature

First, convert the temperature from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \]Given temperature is \( 22^{\circ}C \), thus:\[ T = 22 + 273.15 = 295.15 K \]

Identify Known Values

We are given:- Mass of helium, \( m = 30.0\, \text{kg} \)- Pressure, \( P = 1.20\, \text{atm} \)- Temperature, \( T = 295.15\, K \)We want to find the volume \( V \). The ideal gas law will be used here.

Convert Pressure to Pascals

Convert the pressure from atm to Pascals (SI unit) using the conversion:\[ 1 \text{ atm} = 101325 \text{ Pa} \]So, \[ P = 1.20 \times 101325 = 121590 \text{ Pa} \]

Use the Ideal Gas Law

The ideal gas law is: \[ PV = nRT \]We need to find the number of moles \( n \) of helium first.

Calculate Moles of Helium

Helium has a molar mass of approximately \( 4.00 \text{ g/mol} \). Convert the mass from kilograms to grams:\[ m = 30.0\, \text{kg} = 30000\, \text{g} \]Calculate moles:\[ n = \frac{30000}{4.00} = 7500 \text{ mol} \]

Solve for Volume

Substitute the values into the ideal gas law and solve for \( V \):\[ V = \frac{nRT}{P} \]Using the gas constant \( R = 8.314 \text{ J/mol} \cdot \text{K} \):\[ V = \frac{7500 \text{ mol} \times 8.314 \text{ J/mol} \cdot \text{K} \times 295.15 \text{ K}}{121590 \text{ Pa}} \]Calculate to find \( V \):\[ V \approx 152.0 \text{ m}^3 \]

Key concepts

Gas Volume Calculation

Calculating the volume of a gas involves understanding the relationship between pressure, volume, temperature, and the amount of gas. The Ideal Gas Law is central to this calculation. The law is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the amount of substance (in moles)
• \( R \) is the ideal gas constant, 8.314 J/mol·K
• \( T \) is the temperature in Kelvin

The problem we have involves a balloon filled with helium under specific conditions of pressure and temperature, and we must use the ideal gas law to find the volume.
Firstly, identify known quantities like the amount of helium and conversion of units into those compatible with the Ideal Gas Law, such as converting temperature to Kelvin and pressure to Pascals.
Then, solve for \( V \) using the equation to understand how these factors impact the gas's volume.

Helium Properties

Helium is a unique gas with distinct properties that make it interesting and useful in various applications. Here are some notable properties:

• Helium is the second lightest element and is non-toxic, non-reactive, and inert.
• It has a molar mass of approximately 4.00 g/mol.
• Helium has low density, making it ideal for balloons and airships. This explains why the mass of helium is converted to moles in the calculation.

When dealing with gases like helium, understanding its properties helps in predicting how it behaves under different conditions.
For instance, despite a large mass of 30.0 kg in the given exercise, when converted to grams and then to moles, it indicates a significant amount of substance to be considered in the gas law calculation.

Temperature Conversion

Temperature conversion is a crucial step in many calculations involving gases because the Ideal Gas Law requires temperature in Kelvin. Kelvin is the absolute temperature scale used in scientific calculations.
To convert Celsius to Kelvin, we use:\[ T(K) = T(°C) + 273.15 \]Given that the temperature in our problem was 22°C, simply add 273.15 to convert it:\[ 22°C = 295.15K \]This conversion ensures that calculations involving energy, kinetic movement of particles, and gas behavior remain consistent.
Using Kelvin allows for a more predictable and mathematically manageable way of understanding thermal energy since Kelvin doesn’t dip below zero, avoiding negative temperature implications in physics.

Pressure Conversion

In gas calculations, pressure is often provided in one unit but required in another for the Ideal Gas Law. The SI unit for pressure is the Pascal (Pa).
To convert from atmospheres (atm) to Pascals, use the conversion:\[ 1 ext{ atm} = 101325 ext{ Pa} \]If we're given a pressure of 1.20 atm, the conversion is straightforward:\[ P = 1.20 imes 101325 = 121590 ext{ Pa} \]These conversions are necessary for consistent measurements and applying physical laws accurately. Using the SI unit ensures the calculation adheres to standard international practice, yielding reliable and uniform results applicable in scientific and engineering contexts.