Problem 6 - Chemistry in 30 days

Question

A \(10 \mathrm{~L}\) cylinder of nitrogen at \(4.0\) atm pressure and \(27^{\circ} \mathrm{C}\) developed a leak. When the leak was repaired \(2.36\) atm of nitrogen remained in the cylinder still at \(27^{\circ} \mathrm{C}\). How many grams of nitrogen escaped ? (a) \(18.7 \mathrm{~g}\) (b) \(0.67 \mathrm{~g}\) (c) \(52.6 \mathrm{~g}\) (d) \(10.0 \mathrm{~g}\)

Step-by-Step Solution

Answer

18.7 g of nitrogen escaped, option (a).

Step 1: Use Ideal Gas Law to Find Initial Moles

The ideal gas law is given by \( PV = nRT \). First, we calculate the initial amount of moles \( n_1 \) in the cylinder using the initial conditions: \( P_1 = 4.0 \) atm, \( V = 10 \) L, and \( T = 27^{\circ}C = 300 \) K. The ideal gas constant \( R \) is \( 0.0821 \) L·atm/(mol·K).\[ n_1 = \frac{P_1 V}{RT} = \frac{4.0 \times 10}{0.0821 \times 300} \approx 1.63 \]

Step 2: Use Ideal Gas Law for Final Moles

Next, find the moles of nitrogen \( n_2 \) remaining after the leak using the final pressure \( P_2 = 2.36 \) atm but with the same volume and temperature.\[ n_2 = \frac{P_2 V}{RT} = \frac{2.36 \times 10}{0.0821 \times 300} \approx 0.961 \]

Step 3: Calculate Moles Escaped

Subtract the moles remaining \( n_2 \) from the initial moles \( n_1 \) to find the moles of nitrogen that escaped.\[ n_{\text{escaped}} = n_1 - n_2 = 1.63 - 0.961 = 0.669 \]

Step 4: Convert Moles Escaped to Grams

Using the molar mass of nitrogen gas (\( N_2 \)), which is \( 28 \text{ g/mol} \), convert the moles of nitrogen that escaped into grams.\[ ext{grams escaped} = n_{\text{escaped}} \times 28 = 0.669 \times 28 \approx 18.732 \]

Step 5: Determine Correct Answer

The calculated mass of nitrogen that escaped is approximately \( 18.7 \) g. This matches with option (a).

Key Concepts

Moles Calculation

Understanding moles calculation is crucial in solving chemical problems. The number of moles is a measure of how many molecules are in a given amount of substance. It is represented by the formula from the ideal gas law:

• \( PV = nRT \), where:
  • \( P \) is pressure in atmospheres
  • \( V \) is volume in liters
  • \( n \) is the number of moles
  • \( R \) is the ideal gas constant (\ 0.0821 L·atm/(mol·K)
  • \( T \) is temperature in Kelvin

By rearranging the formula to \( n = \frac{PV}{RT} \), you can calculate the moles of a gas if you know its pressure, volume, and temperature. This calculation was crucial in determining how much nitrogen gas escaped during the leak. Initially, we calculated the moles present using the provided pressure, volume, and temperature. After the leak, we recalculated using the reduced pressure to find the remaining moles. This step allowed us to identify the quantity that escaped.

Nitrogen Gas

Nitrogen gas, symbolized as \( N_2 \), is a diatomic molecule that is colorless, odorless, and makes up about 78% of the Earth's atmosphere. Here are some key points about nitrogen gas:

• It is essential for life, as it is a major component of amino acids and proteins.
• Nitrogen gas has a molar mass of 28 g/mol, which is useful for converting between moles and grams when solving chemical problems.
• It behaves as an ideal gas under many conditions commonly found in textbook problems.

In the context of our exercise, nitrogen gas was used to demonstrate how an ideal gas behaves when its pressure changes due to a leak. By using the ideal gas law, students can predict the amount of gas that escaped based on the initial and final states of the gas in terms of pressure, volume, and temperature.

Gas Leaks

Gas leaks may seem like a challenging concept, but understanding them is vital for safety and calculations in chemistry. When a gas leaks from a container:

• The number of molecules within the container decreases, leading to changes in pressure.
• The volume and temperature may remain constant, which simplifies the use of the ideal gas law in calculations.

In our example with nitrogen gas, the leak resulted in reduced pressure. By comparing the initial and final pressures, we determined the number of moles that escaped. This concept is crucial not only in academic exercises but also in real-world applications, where monitoring and controlling environmental gas levels are necessary.

Chemical Problem-Solving

Chemical problem-solving involves applying scientific principles to find solutions. In the exercise, multiple skills were combined to solve the problem of the escaping nitrogen gas. Here's a breakdown of the approach:

• Understanding the properties of the involved substances, like nitrogen gas.
• Applying the ideal gas law to establish relationships between pressure, volume, and temperature to calculate moles.
• Utilizing stoichiometry to convert between different units, such as moles to grams.
• Critical analysis of the problem to ensure each step accurately reflects the chemical principles involved.

Such systematic approaches not only help in solving specific problems but also foster a deeper comprehension and ability to tackle a variety of chemical scenarios. In this instance, students learn not only to execute calculations but also to interpret and apply their findings in a practical context.