Problem 96 - Chemistry Matter and Change

Question

A fixed amount of oxygen gas is held in a \(1.00-\mathrm{L}\) tank at a pressure of 3.50 atm. The tank is connected to an empty 2.00-L tank by a tube with a valve. After this valve has been opened and the oxygen is allowed to flow freely between the two tanks at a constant temperature, what is the final pressure in the system?

Step-by-Step Solution

Answer

The final pressure in the system is 1.17 atm.

Step 1: Understand the Initial Conditions

We have two tanks. The first tank contains oxygen gas with a volume of \(1.00\, \mathrm{L}\) and a pressure of \(3.50\, \mathrm{atm}\). The second tank is initially empty and has a volume of \(2.00\, \mathrm{L}\). The total volume of the system when the valve is opened is \(V = 1.00\, \mathrm{L} + 2.00\, \mathrm{L} = 3.00\, \mathrm{L}\).

Step 2: Apply Boyle's Law to Find Final Pressure

Since the temperature is constant and the system is closed, we can apply Boyle's Law. Boyle's Law states that \(P_1 \cdot V_1 = P_2 \cdot V_2\). For the initial conditions, \(P_1 = 3.50\, \mathrm{atm}\) and \(V_1 = 1.00\, \mathrm{L}\). The final pressure is \(P_2\), and the final volume is \(V_2 = 3.00\, \mathrm{L}\).

Step 3: Solve for the Final Pressure

We set up the equation based on Boyle's Law: \(3.50\, \mathrm{atm} \times 1.00\, \mathrm{L} = P_2 \times 3.00\, \mathrm{L}\). Solving for \(P_2\), we have \(P_2 = \frac{3.50\, \mathrm{atm} \times 1.00\, \mathrm{L}}{3.00\, \mathrm{L}} = 1.17\, \mathrm{atm}\).

Key Concepts

Gas Pressure

Gas pressure is an essential concept to understand when dealing with gases and their properties. It is defined as the force exerted by gas molecules when they collide with the walls of their container. Think of it like small invisible balls constantly bouncing against the sides of a box. This pressure is what you measure with units such as atmospheres (atm), pascals (Pa), or inches of mercury (Hg). In our example, the oxygen gas in the first tank has an initial pressure of 3.50 atm before the valve is opened. Once the valve is opened and the gases are allowed to distribute themselves in a larger volume, the pressure will decrease due to a redistribution of gas molecules.
Several factors affect gas pressure:

• Number of Gas Molecules: More molecules typically mean higher pressure.
• Volume of Container: Larger volumes usually result in lower pressure if the amount of gas is constant.
• Temperature: Higher temperatures increase energy and collision speed, raising the pressure.

Understanding these factors is crucial for applying Boyle's Law and predicting changes in pressure under different conditions.

Volume of Gas

The volume of gas refers to the space that the gas occupies. Gas molecules are free to move around in the container they are placed in, filling up the available space. In our exercise, the initial oxygen gas is held in a 1.00 L tank. When the valve is opened, the gas expands into an additional empty space of 2.00 L, making the total volume 3.00 L. This expansion impacts the gas pressure, and Boyle's Law helps us predict the outcome.
Important points to remember about gases:

• Gases are compressible and take the shape of their container.
• Volume changes lead to pressure changes if the amount of gas and temperature are constant.
• The relationship between volume and pressure is inverse, which means as volume increases, pressure decreases if temperature stays constant.

Visualize the gas molecules zooming around and filling all nooks and crannies when thinking about the volume of gases. This helps to understand why changes in volume can cause significant shifts in pressure.

Constant Temperature

The exercise scenario highlights the condition of a constant temperature. This term indicates that as the gas is allowed to move between different volumes, its temperature remains unchanged. Temperature is a measure of the average kinetic energy of gas particles; hence, maintaining constant temperature means gas particles have consistent speeds and energy throughout the process.
Key aspects of constant temperature in gas behavior:

• Since temperature is constant, any adjustments in volume will inversely affect pressure (Boyle's Law).
• Changing temperature can separately affect both pressure and volume if not controlled.
• This condition is crucial for simplifying calculations and understanding gas laws, like Boyle's Law.

A constant temperature allows us to focus purely on the volume and pressure relationship, simplifying our understanding of the system dynamics.

Oxygen Gas Chemistry

Oxygen gas, represented as \({\text{O}_2}\), is a common yet crucial element in chemistry and environmental science. It plays vital roles in various biological and chemical processes. In chemistry, we often deal with oxygen in gaseous form when discussing reactions like combustion, respiration, or oxidation. In the example problem, understanding \({\text{O}_2}\) behavior when distributed between different volumes is essential in predicting pressure changes.
Quick points about \({\text{O}_2}\):

• It is a diatomic molecule, meaning it consists of two oxygen atoms bonded together.
• Oxygen is necessary for the process of respiration in living organisms.
• It reacts with other elements to form oxides and is a significant component of the Earth's atmosphere.

Understanding the basic properties and role of oxygen gas in chemistry helps connect theoretical scenarios to real-world applications, such as air pressure measurements and environmental balancing.