Question

Calculate the density of helium in a helium balloon at \(25.0^{\circ} \mathrm{C}\). (Assume that the pressure inside the balloon is \(1.10 \mathrm{~atm} .)\)

Short answer

The density of helium in the balloon is approximately 0.178 g/L.

Step-by-step solution

Understand the problem

We need to find the density of helium gas inside a balloon at a temperature of \( 25.0^{\circ} \mathrm{C} \) and a pressure of \( 1.10 \, \mathrm{atm} \). We will use the ideal gas law to find the volume and calculate the density using the relation \( \text{Density} = \frac{\text{mass}}{\text{volume}} \).

Convert temperature to Kelvin

To use the ideal gas law, we need to convert the temperature from Celsius to Kelvin. The conversion formula is \( T(K) = T(^{\circ}C) + 273.15 \). So, \( T = 25.0 + 273.15 = 298.15 \, \mathrm{K} \).

Use the Ideal Gas Law

The ideal gas law is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is temperature in Kelvin. To find the molar density \( \frac{n}{V} \), we rearrange the equation to \( \frac{n}{V} = \frac{P}{RT} \).

Calculate molar density

Plug the values into the rearranged ideal gas law: \( \frac{n}{V} = \frac{1.10}{0.0821 \times 298.15} \approx 0.0446 \, \mathrm{mol/L} \).

Find density in g/L

Multiply the molar density by the molar mass of helium. The molar mass of helium is approximately 4.00 g/mol. Thus, density \( = 0.0446 \, \, \mathrm{mol/L} \times 4.00 \, \, \mathrm{g/mol} \approx 0.178 \, \mathrm{g/L} \).

Key concepts

Density Calculation

To calculate the density of a gas, like helium in a balloon, we must understand that density is defined as mass per unit volume. In the context of gases, this can often involve indirect measurement, especially in a classroom setting. We rely on the density formula:

• \( \text{Density} = \frac{\text{mass}}{\text{volume}} \)

The challenge in such problems is that the volume might not be directly measured but inferred through other known variables like temperature, pressure, and amount of substance (moles). By using the ideal gas law, we can find these relationships effectively and solve for density. The density of a gas can change with variations in temperature and pressure, which is why understanding these relations is crucial.

Molar Density

Molar density refers to the number of moles of a substance present per unit volume. This concept is pivotal when working with gases and the ideal gas law, as it directly allows us to link macroscopic properties like pressure and temperature, to the microscopic amount of substance, or moles.The ideal gas law equation:

• \( PV = nRT \)

helps us form the expression for molar density \( \frac{n}{V} \) by rearranging it:

• \( \frac{n}{V} = \frac{P}{RT} \)

This gives a straightforward way to calculate molar density when pressure \( P \), temperature \( T \), and the ideal gas constant \( R \) are known. In our helium balloon example, plugging in pressure in atm, temperature in Kelvin, and using the constant 0.0821 L atm/mol K results in calculating the molar density in moles per litre.

Temperature Conversion

Converting temperature from Celsius to Kelvin is a simple yet essential step when dealing with gas laws. The Kelvin scale is used in scientific calculations because it's based on absolute zero, providing a direct measure of thermal energy.The conversion formula is:

• \( T(K) = T(^{\circ}C) + 273.15 \)

For example, converting 25.0°C into Kelvin:

• \( 25.0 + 273.15 = 298.15 \, \mathrm{K} \)

This conversion is crucial because the ideal gas law, \( PV = nRT \), requires temperature to be in Kelvin to ensure units are consistent and calculations correct. Always remember, no matter the initial temperature unit, Kelvin is key for any gas calculation.