Problem 77 - Chemistry Matter and Change

Question

How many moles of helium gas (He) would be required to fill a \(22-\) L container at a temperature of \(35^{\circ} \mathrm{C}\) and a pressure of 3.1 atm?

Step-by-Step Solution

Answer

2.70 moles of helium gas are required.

Step 1: Convert Temperature to Kelvin

To use the ideal gas law, we need temperature in Kelvin. Convert the temperature from Celsius to Kelvin using the formula:\[ T(K) = T(^{\circ}C) + 273.15 \]Substitute the given temperature:\[ T(K) = 35 + 273.15 = 308.15 \, K \]

Step 2: Use the Ideal Gas Law

The ideal gas law is given by the formula:\[ PV = nRT \]where:- \(P\) is the pressure in atmospheres.- \(V\) is the volume in liters.- \(n\) is the number of moles.- \(R\) is the ideal gas constant, approximately \(0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1}\).- \(T\) is the temperature in Kelvin.

Step 3: Rearrange the Ideal Gas Law for Moles

Solve the ideal gas law equation for \(n\) (number of moles):\[ n = \frac{PV}{RT} \]

Step 4: Substitute the Known Values

Substitute the known values into the rearranged equation:\[ n = \frac{3.1 \, \text{atm} \times 22 \, \text{L}}{0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \times 308.15 \, K} \]

Step 5: Calculate the Number of Moles

Perform the calculation:\[ n = \frac{3.1 \times 22}{0.0821 \times 308.15} \]\[ n \approx \frac{68.2}{25.29} \approx 2.70 \, \text{moles} \]

Key Concepts

Moles

Moles are a fundamental concept in chemistry that allow us to quantify the amount of a substance. One mole (\( n \) is used to represent moles in equations) is equivalent to Avogadro's number of entities, which is approximately \( 6.022 \times 10^{23} \) atoms, molecules, or other particles. Moles are crucial for comparing amounts of different chemicals in a reaction.

• In our exercise, we want to determine how many moles of helium gas are needed to fill a container under certain conditions.
• By using the ideal gas law, which relates pressure, volume, and temperature with the number of moles, we can easily find this value.

Understanding moles helps us make practical predictions about chemical reactions and processes.

Temperature Conversion

When working with gas laws, especially the ideal gas law, it is necessary to have the temperature measured in Kelvin, rather than Celsius or Fahrenheit. The Kelvin scale starts at absolute zero and represents an absolute scale, which is why it is preferred in scientific calculations.

• To convert Celsius to Kelvin, use the formula: \( T(K) = T(^{\circ}C) + 273.15 \).
• This ensures consistency when using formulas that include temperature.

For instance, when the temperature in our exercise was given as \( 35^{\circ} \mathrm{C} \), it needed to be converted to Kelvin, resulting in \( 308.15 \, K \). Correct temperature conversion is vital for accurate results within the ideal gas law framework.

Gas Constant

The gas constant, represented as \( R \), is a fundamental constant used in the ideal gas law. It bridges various properties of gases, acting as a proportionality factor.

• Its value is approximately \( 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \).
• This specific value is used when pressure is measured in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K).

The gas constant is essential for uniformity when applying the ideal gas law. It allows for transformations of one unit into another within calculations, facilitating accurate and universal equations. Using the correct \( R \) value is crucial depending on the units involved in your calculations.

Volume and Pressure

Volume and pressure are key components of the ideal gas law equation, \( PV = nRT \). They are directly related when temperature and the number of moles are constant, illustrating Boyle's Law.

• Volume (\( V \)) is typically measured in liters (L).
• Pressure (\( P \)) is usually measured in atmospheres (atm) or alternate units depending on the context.

When using the ideal gas law, you must ensure that all your units match the requirements of the gas constant \( R \). This is why pressure was specified in atmospheres and volume in liters in our exercise.
Accurate measurement and consistency of units for volume and pressure are crucial for precise predictions about gas behavior in various scenarios.