Question

A cylinder with a movable piston records a volume of \(12.6 \mathrm{~L}\) when \(3.0 \mathrm{~mol}\) of oxygen is added. The gas in the cylinder has a pressure of \(5.83\) atm. The cylinder develops a leak and the volume of the gas is now recorded to be \(12.1 \mathrm{~L}\) at the same pressure. How many moles of oxygen are lost?

Short answer

Answer: About 0.13 mol of oxygen was lost.

Step-by-step solution

Write down the initial state of the cylinder

The initial state of the cylinder is described by a volume of \(12.6 \mathrm{~L}\), a pressure of \(5.83\,\mathrm{atm}\), and \(3.0\,\mathrm{mol}\) of oxygen.

Write down the final state of the cylinder

The final state of the cylinder is described by a volume of \(12.1\,\mathrm{L}\), a pressure of \(5.83\,\mathrm{atm}\), and an unknown number of moles of oxygen (which we will call \(n_2\)).

Set up the proportional relationship equation

Because the pressure and temperature remain constant, we can use the proportional relationships between volume and the number of moles to find the number of moles lost. The relationship can be written as: \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\) Where \(V_1\) and \(V_2\) are the initial and final volumes, and \(n_1\) and \(n_2\) are the initial and final number of moles, respectively.

Solve the equation for \(n_2\)

Insert the given values of \(V_1\), \(V_2\), and \(n_1\) into the equation: \(\frac{12.6\,\mathrm{L}}{3.0\,\mathrm{mol}} = \frac{12.1\,\mathrm{L}}{n_2}\) Solve the equation for \(n_2\), which represents the final number of moles of oxygen: \(n_2 = \frac{12.1\,\mathrm{L} \times 3.0\,\mathrm{mol}}{12.6\,\mathrm{L}}\) \(n_2 \approx 2.87\,\mathrm{mol}\)

Calculate the number of moles lost

To find the number of moles of oxygen lost, simply subtract the final number of moles \(n_2\) from the initial number of moles \(n_1\): Moles lost = \(n_1 - n_2\) Moles lost = \(3.0\,\mathrm{mol} - 2.87\,\mathrm{mol}\) Moles lost = \(0.13\,\mathrm{mol}\)

Write down the final answer

Therefore, during the leak, about \(0.13\,\mathrm{mol}\) of oxygen was lost.

Key concepts

Ideal Gas Equation

The Ideal Gas Equation is crucial in understanding how gases behave under different conditions. It is given by the formula \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This equation helps us relate these different properties of a gas and predict changes. In the context of the problem, since the temperature and pressure were constant, the Ideal Gas Equation simplifies our calculations by ensuring changes in volume directly relate to changes in moles of gas.

Moles

Moles are a fundamental concept for understanding quantities in chemistry. A mole represents \( 6.022 \times 10^{23} \) entities, such as atoms or molecules, and helps chemists quantify substances at the molecular level.
For the given problem, calculating the number of moles was key to determining how much oxygen was lost from the cylinder. Initially, the cylinder had \( 3.0 \) moles of oxygen. When the volume changed due to the leak, it became necessary to calculate the new number of moles using the proportional volume-mole relationship. Ultimately, by understanding moles, you can quantify the exact amount of a substance gained or lost in a chemical setting.

Volume-Pressure Relationship

The relationship between volume and pressure is central in understanding gas behavior, particularly in systems where the temperature is held constant, following Boyle's Law. \( PV = constant \)In our given scenario, although the exercise kept the pressure constant while the volume changed, understanding this relationship helps explain why volume changes even with constant pressure.
Since the temperature and pressure were constant, the volume's change was solely due to the change in the number of moles of gas. While Boyle's Law often details the inverse relationship in varying pressure situations, here it simplifies the understanding that less gas would occupy less volume but under unchanged pressure conditions.

Chemical Calculations

Chemical Calculations in the context of gas laws often involve using proportions and relationships like the Ideal Gas Law for detailed insights. In this exercise, you applied these calculations to the proportionate relationship between volume and the number of moles, as both pressure and temperature remained constant.

• Identified the initial and final states of the gas system.
• Utilized the proportional equation \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \) to find unknown quantities.
• Solved for the unknown final moles \( n_2 \).
• Calculated the moles of oxygen lost by comparing the initial and final quantities.

Understanding these calculation techniques allows for precise predictions and explanations of real-world chemical phenomena, like gas leaks, where a change in one aspect affects others predictably.