Question

A 55 L cylinder contains \(A r\) at 145 atm and \(26^{\circ}\) C. What minimum volume of air at STP must have been liquefied and distilled to produce this Ar? Air contains \(0.934 \%\) Ar, by volume.

Short answer

Using the aforementioned steps, determine the required volume of air at STP that was needed. The final computation will provide the specific volume in liters.

Step-by-step solution

Convert the temperature to Kelvin

The given temperature is \(26^{\circ}\) C. Converting it to Kelvin, we have \(T = 26 + 273.15 = 299.15\) K.

Use the Ideal Gas Law

Using the Ideal Gas Law equation \(PV = nRT\), and knowing that the molecular weight of Argon (Ar) is approximately 40 g/mol, the number of moles of Argon can be calculated.

Calculate the moles of Argon

First, solve the ideal gas law, \(PV = nRT\), for \(n\), where \(n\) is the number of moles, \(P\) is the pressure, \(V\) is the volume, \(R\) is the ideal gas constant, and \(T\) is the temperature. So, \(n = PV/RT\). Substitute the known values: \(P = 145\) atm, \(V = 55\) L, \(R = 0.0821\) L atm/mol K (value of gas constant), and \(T = 299.15\) K. This yields \(n = 145 \times 55 / (0.0821 \times 299.15)\). Calculate the value of \(n\).

Convert the volume percent to volume fraction

The volume percent of Argon in air is given as 0.934%. Converting this to a volume fraction, we have \(0.934 / 100 = 0.00934\). This is the volume fraction of Ar in the air.

Calculate the volume of air at STP

Finally, to find the volume of air that was needed to produce this Argon, divide the number of Argon moles by the volume fraction of Argon in air. As STP conditions define 1 mole of any gas to occupy 22.4 liters, multiply the obtained value by 22.4 to convert moles into liters at STP. Calculate the result and that gives the required volume of air at STP.

Key concepts

Gas Laws

Gas Laws are the scientific principles that describe the behavior of gases in terms of their pressure, volume, temperature, and number of particles. The Ideal Gas Law is a popular equation under these laws, expressed as \( PV = nRT \). Here, \( P \) stands for pressure, \( V \) for volume, \( n \) represents the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature measured in Kelvin.

Understanding the Ideal Gas Law is crucial because it provides a good approximation of the behavior of gases under many conditions. However, it's "ideal" because it makes assumptions that simplify real-world scenarios, assuming gases consist of point particles with no volume and no intermolecular forces.

• 37: The Ideal Gas Law provides a powerful tool to calculate missing variables when three of the four variables are known.


• The temperature always needs to be converted to Kelvin because the law requires absolute temperature.


• This law particularly helps in predicting how changes in conditions like an increase in temperature or pressure will affect the gas.

Argon

Argon (symbol Ar) is a noble gas found in the periodic table. It is colorless, odorless, and very stable due to its complete valence electron shell. This stability makes it inert and non-reactive under most conditions, which is why it is used in various industries.

Argon is about \( 0.934 \% \) of the Earth's atmosphere and is obtained mainly as a by-product of the liquid air separation process. Its inertness makes it useful in environments where materials might otherwise react with oxygen or other substances. For example:

• In light bulbs, Argon is used to prevent oxygen from corroding the filament.


• Argon is utilized in double-glazed windows as an insulating material due to its poor thermal conductivity.


• In welding, Argon prevents metal oxidation by creating an inert atmosphere.

Understanding Argon's role in air and its usage helps comprehend why we calculate how much argon can be extracted from liquefied air.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It uses balanced chemical equations to derive the relationships between the quantities of substances consumed and produced.

In our exercise, stoichiometry bridges the Ideal Gas Law with real-world applications. By calculating the moles of Argon using the Ideal Gas Law, we employ stoichiometric methods to find the equivalent volume of air necessary to produce that amount of Argon.

Here is a simplified flow of stoichiometric calculations related to gases:

• Determine the number of moles using \( PV = nRT \).


• Convert given percentages to fractions for calculation precision, such as finding volume fractions.


• Relate moles to volume using standard molar volume at STP (Standard Temperature and Pressure), typically 22.4 liters per mole.

This approach solidifies the connection between calculations and understanding how much air is needed to obtain specific elements or compounds, as seen with Argon.