Question

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

Short answer

The volume of the balloon is approximately 151368.75 L.

Step-by-step solution

Understand the Problem

We need to find the volume of a helium-filled balloon given its mass, the pressure, and the temperature. For this, we'll use the Ideal Gas Law: \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Convert Given Units

First, let's convert the temperature from Celsius to Kelvin by using the formula \(T_{K} = T_{C} + 273.15\). So, \(22^{\circ}C + 273.15 = 295.15 \, K\).

Calculate Number of Moles

We need to calculate the number of moles using the mass of helium. The molar mass of helium is \(4.00 \, \text{g/mol}\). Convert the mass from kilograms to grams: \(30.0 \, \text{kg} = 30000 \, \text{g}\). Then, calculate the moles \( n = \frac{30000}{4.00} = 7500 \, \text{moles} \).

Use Ideal Gas Law

Insert the known values into the Ideal Gas Law. \(P = 1.20 \, \text{atm}\), \(n = 7500\), \(R = 0.0821 \, \text{L} \, \text{atm} \, \text{K}^{-1} \, \text{mol}^{-1}\), and \(T = 295.15 \, \text{K}\). Rearrange the equation to solve for \(V\), resulting in \(V = \frac{nRT}{P}\).

Calculate Volume

Substitute the values: \[V = \frac{7500 \times 0.0821 \times 295.15}{1.20}\]. Performing the multiplication and division gives us the volume \(V \approx 151368.75 \, \text{L}\).

Key concepts

Gas Constant

The gas constant, often denoted as \( R \), is a crucial part of the Ideal Gas Law, \( PV = nRT \). This constant relates the energy scale to the temperature scale and the amount of substance. In most problems concerning gases, particularly when using the Ideal Gas Law, the value of \( R \) is expressed as \( 0.0821 \, \text{L} \, \text{atm} \, \text{K}^{-1} \, \text{mol}^{-1} \).

Here’s how \( R \) fits into equations:

• It is a universal constant that links pressure, volume, and temperature to the amount of gas in moles.
• Always use the same units for consistency. For this value of \( R \), make sure pressure is in atm, volume in liters, and temperature in Kelvin.

Consistency and unit conversion are key when working with \( R \). Keeping this in mind will help you use the ideal gas law efficiently.

Temperature Conversion

Temperature affects gas behavior significantly, and using the wrong units can lead to errors. In scientific calculations, it's essential to convert Celsius to Kelvin. The Kelvin scale is an absolute temperature scale starting at absolute zero, where 0 Kelvin means no thermal energy.

The conversion from Celsius to Kelvin is straightforward: add 273.15 to the Celsius temperature. So for converting \(22^{\circ}C\), simply calculate:

• 22° C + 273.15 = 295.15 K

Why Kelvin? Because the Ideal Gas Law requires temperature in Kelvin to accurately describe the behavior of gases. Kelvin gives a linear measurement of temperature aligned with the energy within the systems.

Molar Mass

Molar mass is the weight of one mole of a substance, typically expressed in grams per mole (\(\text{g/mol}\)). It's imperative to know when dealing with gas calculations. For example, helium has a molar mass of \(4.00 \, \text{g/mol}\). This provides the connection between the mass of the gas and the number of moles, crucial for using the Ideal Gas Law.

Steps for using molar mass:

• First, ensure you have the mass of the substance in the correct unit, usually grams.
• Divide the mass by the molar mass to get the number of moles.

In this problem, \(30.0 \, \text{kg}\) of helium is converted to grams and divided by the molar mass, \(4.00 \, \text{g/mol}\), to calculate the moles. Knowing this step is fundamental in obtaining quantities needed for subsequent calculations using the Ideal Gas Law.

Pressure

Pressure, indicated by \( P \) in the Ideal Gas Law, describes how force is applied by the gas particles colliding with the walls of their container per unit area. It is typically measured in atmospheres (atm), which is the pressure exerted by the Earth's atmosphere at sea level.

Understanding pressure in the context of gas laws:

• Pressure balances with factors like temperature and volume to maintain a certain gas behavior.
• In the Ideal Gas Law \(PV = nRT\), pressure distinctly influences the volume if temperature and moles are constant.

When solving for the volume of a balloon in this exercise, pressure is given as \(1.20 \, \text{atm}\). Knowing how pressure works helps in correctly applying it in volume calculations involving gases. Correct unit usage here ensures accurate and practical results.