Problem 18 - Chemistry: Principles and Reactions

Question

A 2.00-L tank, evacuated and empty, has a mass of \(725.6 \mathrm{~g}\). It is filled with butane gas \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\) at \(22^{\circ} \mathrm{C}\) to a pressure of \(1.78 \mathrm{~atm} .\) What is the mass of the tank after it is filled?

Step-by-Step Solution

Answer

Answer: The mass of the tank after it is filled with butane gas is 734.079 g.

Step 1: Find the number of moles of butane gas in the tank

Use the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant (in this case, \(0.0821 \frac{L \cdot atm}{K \cdot mol} \)), and T is temperature in Kelvin. We are given P, V, and T. First, convert the given temperature from Celsius to Kelvin: T = 22 + 273.15 = 295.15 K. Now, plug the given values to find the number of moles of butane gas in the tank: \((1.78 \mathrm{~atm}) \times (2.00 \mathrm{~L}) = n \times (0.0821 \frac{L\cdot atm}{K \cdot mol}) \times (295.15 \mathrm{~K})\) Solving for n, we get: \(n = \frac{(1.78 \mathrm{~atm}) \times (2.00 \mathrm{~L})}{(0.0821 \frac{L\cdot atm}{K \cdot mol}) \times (295.15 \mathrm{~K})}= 0.146 \mathrm{~mol}\)

Step 2: Calculate the mass of butane gas in the tank

Now, we need to find the mass of the butane gas in the tank using the number of moles found in the previous step. The molecular formula of butane is given as \(\mathrm{C}_{4} \mathrm{H}_{10}\), which means it has 4 carbon atoms and 10 hydrogen atoms. The molar mass of butane is calculated as follows: Molar mass of \(\mathrm{C} = 12.01 \mathrm{~g/mol}\) Molar mass of \(\mathrm{H} = 1.008 \mathrm{~g/mol}\) Molar mass of butane = \(4 \times 12.01 \mathrm{~g/mol} + 10 \times 1.008 \mathrm{~g/mol} = 58.12 \mathrm{~g/mol}\) Now, we can calculate the mass of butane gas in the tank: Mass = Moles × Molar mass Mass of butane = \(0.146 \mathrm{~mol} \times 58.12 \mathrm{~g/mol} = 8.479 \mathrm{~g}\)

Step 3: Calculate the mass of the tank after it is filled

Lastly, we need to add the mass of the butane gas found in the previous step to the mass of the empty tank given in the problem. The mass of the empty tank is \(725.6 \mathrm{~g}\). Mass of the filled tank = Mass of the empty tank + Mass of the butane gas Mass of the filled tank = \(725.6 \mathrm{~g} + 8.479 \mathrm{~g} = 734.079 \mathrm{~g}\) Thus, the mass of the tank after it is filled with butane gas is \(734.079 \mathrm{~g}\).

Key Concepts

Molar Mass Calculation

The concept of molar mass is essential in chemistry as it helps us determine the mass of a given substance based on the number of moles. Molar mass is the weight of one mole of a substance, measured in grams per mole (g/mol). For any chemical compound, it can be calculated by summing the atomic masses of its constituent atoms.
For butane \( \mathrm{C_4H_{10}} \), we have:

• Carbon (C): 4 atoms
• Hydrogen (H): 10 atoms

The atomic mass of Carbon is approximately \( 12.01 \mathrm{~g/mol} \), and for Hydrogen, it is \( 1.008 \mathrm{~g/mol} \).
Hence, the molar mass of butane is calculated as:
\[ 4 \times 12.01 + 10 \times 1.008 = 48.04 + 10.08 = 58.12 \mathrm{~g/mol} \]
This calculation is crucial when determining the mass of butane gas in a container using the mole concept.

Butane Gas Properties

Butane is a gaseous hydrocarbon belonging to the alkane series with the chemical formula \( \mathrm{C_4H_{10}} \). Its properties play a significant role in understanding its behavior under different conditions, which is crucial in tasks involving calculations with gases.
Some key properties of butane are:

• It is colorless and highly flammable.
• Its boiling point is around \( -0.5^{ ext{o}} \mathrm{C} \), allowing it to be easily liquified.
• Butane is commonly used in fuel for lighters and portable stoves.
• Under standard conditions, it exists as a gas.

Understanding these properties helps us predict how butane will respond to changes in pressure and temperature, which are critical aspects used in the ideal gas law calculations.

Mole Concept

The mole concept is a fundamental principle in chemistry that denotes the amount of a substance. One mole contains exactly \( 6.022 \times 10^{23} \) entities, which could be atoms, molecules, ions, etc., and is represented by Avogadro's number.
When dealing with gases, the mole concept is pivotal in calculations involving the ideal gas law. The ideal gas law is expressed as:
\[ PV = nRT \]
where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume.
• \( n \) is the number of moles.
• \( R \) is the universal gas constant, \( 0.0821 \frac{L \cdot atm}{K \cdot mol} \).
• \( T \) is the temperature in Kelvin.

The concept is vital for converting between the volume of a gas and the number of moles, enabling us to predict how gases will react when subject to various conditions. In our example, knowing the number of moles of butane allowed us to determine its mass and, ultimately, the mass of the filled tank.