Problem 49 - Chemistry The Science in Context

Question

What is the final pressure of 1.00 mol of ammonia gas, initially at 1.00 atm, if the volume is a. gradually decreased from \(78.0 \mathrm{mL}\) to \(39.0 \mathrm{mL}\) at constant temperature? b. increased from \(43.5 \mathrm{mL}\) to \(65.5 \mathrm{mL}\) at constant temperature? c. decreased by \(40 \%\) at constant temperature?

Step-by-Step Solution

Answer

Question: Calculate the final pressure of 1.00 mol of ammonia gas for each of the following scenarios, assuming a constant temperature: (a) the volume is decreased to half of its initial value, (b) the volume is increased from 43.5 mL to 65.5 mL, and (c) the volume is decreased by 40%. Answer: (a) The final pressure in scenario (a) is 2.00 atm. (b) The final pressure in scenario (b) is approximately 0.666 atm. (c) The final pressure in scenario (c) is approximately 1.67 atm.

Step 1: Write down Boyle's Law formula

Write down Boyle's Law formula, which is P1 * V1 = P2 * V2. This will help us calculate the final pressure (P2) for each scenario.

Step 2: Calculate final pressure for scenario (a) decreasing the volume

Given the initial pressure P1 (1.00 atm), initial volume V1 (78.0 mL) and final volume V2 (39.0 mL), plug the values into the formula: P2 = (P1 * V1) / V2 P2 = (1.00 atm * 78.0 mL) / 39.0 mL P2 = 2.00 atm The final pressure in scenario (a) is 2.00 atm.

Step 3: Calculate final pressure for scenario (b) increasing the volume

Given the initial pressure P1 (1.00 atm), initial volume V1 (43.5 mL) and final volume V2 (65.5 mL), plug the values into the formula: P2 = (P1 * V1) / V2 P2 = (1.00 atm * 43.5 mL) / 65.5 mL P2 ≈ 0.666 atm The final pressure in scenario (b) is approximately 0.666 atm.

Step 4: Calculate final pressure for scenario (c) decreasing the volume by a percent

First, you need to find the final volume after a 40% decrease. We don't have the value for V1 in this case, so let's assume V1 = x mL. The final volume V2 will be 0.6 * V1 (since it's being decreased by 40%). Now, use the Boyle's Law formula: P2 = (P1 * V1) / V2 P2 = (1.00 atm * x mL) / (0.6x mL) After canceling out the x terms: P2 = 1.00 atm / 0.6 P2 ≈ 1.67 atm The final pressure in scenario (c) is approximately 1.67 atm.

Key Concepts

Gas Pressure Calculations

Gas pressure calculations often rely on understanding Boyle's Law. This principle states that for a given amount of gas at constant temperature, the product of its pressure and volume is constant. Hence, if you know the initial pressure and volume, you can find the final pressure when the volume changes, as per the relation: \[ P_1 \times V_1 = P_2 \times V_2 \] Here, \(P_1\) and \(V_1\) represent the initial pressure and volume, while \(P_2\) and \(V_2\) are the final pressure and volume, respectively. Using Boyle's Law allows us to calculate how changing the volume of a gas influences its pressure—provided the temperature does not vary. This relationship is inversely proportional, meaning if the volume decreases, the pressure increases, and vice versa. To solve practical problems:

• Identify initial conditions: initial pressure and volume.
• Determine the new volume condition.
• Use the formula above to solve for the unknown pressure.

Understanding these calculations is key to many real-world applications, including adjusting conditions in laboratory experiments, industrial processes, and even understanding natural phenomena.

Volume Change in Gases

Volume changes in gases are pivotal in many scientific and industrial applications. By compressing or expanding a gas, we can significantly alter its properties and behavior. Boyle's Law is central to understanding these changes. It tells us that when the volume of a gas decreases, its pressure increases, assuming the temperature and the amount of gas remain constant. For example:

• If a gas's volume is halved, its pressure doubles.
• If the volume increases, the pressure drops.

Understanding the interplay between volume and pressure helps us optimize conditions in tasks like gas storage, chemical reactions involving gases, or even the behavior of gases in our atmosphere. A clear grasp of how volume alterations affect gas pressure can also assist in predicting and managing fluctuations in environments where gases play a crucial role, like in air conditioning systems or scuba diving.

Ammonia Gas Properties

Ammonia is a colorless gas with a distinctive pungent smell. It is composed of nitrogen and hydrogen, having the chemical formula \(\text{NH}_3\). Its properties make it suitable for numerous applications, ranging from fertilizers to industrial refrigerants. Key characteristics of ammonia include:

• Light molecular weight, making it suitable for uses such as a cleansing agent.
• Ability to be easily compressed, which facilitates its storage and transport.
• High solubility in water, leading to its widespread use in aqueous solutions for cleaning and sanitation.

In terms of gas behavior, ammonia follows the same physical laws, such as Boyle's Law, which governs the relationship between its pressure, volume, and temperature. Understanding these properties allows industries to design processes that safely and effectively capitalize on ammonia's versatility. Proper handling and knowledge of its characteristics are crucial for ensuring safe usage and for achieving desired outcomes in various applications.