Problem 38 - Chemistry for Allied Health

Question

A nitrogen sample has a pressure of \(0.56\) atm with a volume of \(2.0 \mathrm{~L}\). What is the final pressure if the volume is compressed to a volume of \(0.75 \mathrm{~L}\) ? Assume constant moles and temperature.

Step-by-Step Solution

Answer

The final pressure is approximately 1.49 atm.

Step 1: Understanding the Given Information

We are given that the initial pressure of the nitrogen sample is \(0.56 \text{ atm}\) and the initial volume is \(2.0 \text{ L}\). The final volume is stated to be \(0.75 \text{ L}\). The problem asks for the final pressure, assuming that moles and temperature are constant.

Step 2: Identifying the Gas Law

The problem involves pressure and volume changes at constant temperature and moles. Recognize that Boyle's Law, \( P_1 V_1 = P_2 V_2 \), is the appropriate relationship to use, where \(P_1\) and \(V_1\) are initial conditions, and \(P_2\) and \(V_2\) are final conditions.

Step 3: Setting Up the Equation

Use the equation from Boyle's Law: \( P_1 V_1 = P_2 V_2 \). Substitute the known values into the equation: \(0.56 \times 2.0 = P_2 \times 0.75\).

Step 4: Solving for the Final Pressure \(P_2\)

Rearrange the equation to solve for \(P_2\):\[ P_2 = \frac{0.56 \times 2.0}{0.75} \]Calculate \(P_2\):\[ P_2 = \frac{1.12}{0.75} \approx 1.49 \text{ atm} \]

Step 5: Verifying the Solution

Check that the units make sense and that the math was done correctly. The resulting pressure is in atm, which is consistent with the units given for initial pressure. Also, because the volume decreased, it's logical for the pressure to increase, which it does.

Key Concepts

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in chemistry, particularly when dealing with gases. This concept is beautifully encapsulated by Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume, as long as the temperature and amount of the gas remain constant. Simply put:\[ P_1V_1 = P_2V_2 \] This means that if the volume of a gas decreases, its pressure increases, provided the temperature and the number of molecules stay the same. A real-world analogy is using a syringe: when you push the plunger, reducing the volume, the pressure inside increases. Understanding this relationship is crucial for predicting how gases will behave under varying conditions. It's a foundational principle used across many scientific fields, especially when working with gases in confined spaces.

Ideal Gas Equation

The ideal gas equation offers a more generalized form for understanding gas behavior, bringing together various individual gas laws. The equation is given as:\[ PV = nRT \] Where:

• \( P \) = pressure of the gas
• \( V \) = volume
• \( n \) = moles of gas
• \( R \) = ideal gas constant
• \( T \) = temperature in Kelvin

This relationship allows chemists to calculate one of these variables if the others are known. It embodies the principles of Boyle's Law (pressure and volume), Charles's Law (volume and temperature), and Avogadro's Law (volume and moles), making it an invaluable tool in scientific calculations. While it applies ideally to 'perfect' gases, it provides an essential approximation for real gases under a wide range of conditions.

Gas Laws

Gas laws are a set of fundamental principles that describe how gases behave and interact under different conditions. They simplify the complexity of gas behavior into manageable, predictable patterns. Key gas laws include:

• **Boyle’s Law**: showing the inverse relationship between pressure and volume at constant temperature.
• **Charles’s Law**: demonstrating that the volume of a gas is directly proportional to its absolute temperature at constant pressure.
• **Avogadro’s Law**: stating that equal volumes of gases, at the same temperature and pressure, contain an equal number of moles.

These laws lay the groundwork for understanding more complex interactions and are pivotal in a variety of scientific applications—from chemistry to physics and engineering. They assist scientists in making accurate predictions and adjustments during experiments and industrial processes involving gases.

Problem-Solving in Chemistry

Solving chemistry problems, like the one involving Boyle's Law, requires a step-by-step approach to ensure accuracy and understanding. Here are some effective steps to tackle such problems:

• **Identify what is known and unknown**: Outline the given information, such as initial conditions and the variable you need to find.
• **Select the appropriate formula**: Choose the right gas law or combination of laws that suits the situation.
• **Substitute values into the formula**: Insert the known values and rearrange the equation if needed to isolate the unknown variable.
• **Solve and check**: Perform the calculations, then verify that the solution is reasonable by considering the physical meaning and units involved.

This method ensures a systematic approach to problem-solving in chemistry, reducing errors and helping to deepen comprehension of the underlying principles.