Problem 52 - Chemistry A Molecular Approach

Question

A sample of nitrogen gas in a 1.75-L container exerts a pressure of 1.35 atm at \(25^{\circ} \mathrm{C}\). What is the pressure if the volume of the container is maintained constant and the temperature is raised to \(355^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The pressure of the nitrogen gas at 355°C is 2.85 atm.

Step 1: Identify the Known and Unknown Variables

Identify the given variables from the initial conditions and the final condition for which we need to solve. The initial pressure (P1) is 1.35 atm, initial temperature (T1) is 25°C which needs to be converted to Kelvin, the final temperature (T2) is 355°C also to be converted to Kelvin, and final pressure (P2) is the unknown we need to find. The volume remains constant.

Step 2: Convert Temperatures to Kelvin

Convert the Celsius temperatures to Kelvin using the formula K = °C + 273.15. For T1: 25°C + 273.15 = 298.15 K. For T2: 355°C + 273.15 = 628.15 K.

Step 3: Apply Gay-Lussac's Law

Since the volume is constant, use Gay-Lussac's Law, which states that the pressure of a gas is directly proportional to its temperature in Kelvin. The formula is given by P1/T1 = P2/T2.

Step 4: Solve for the Unknown Pressure P2

Rearrange the Gay-Lussac's Law equation to solve for P2: P2 = P1 * (T2/T1). Substitute in the known values to find P2: P2 = 1.35 atm * (628.15 K / 298.15 K).

Step 5: Calculate the Final Pressure

Perform the calculation from the previous step to find the final pressure: P2 = 1.35 atm * (628.15 K / 298.15 K) = 2.85 atm.

Key Concepts

Pressure-Temperature Relationship

Understanding the pressure-temperature relationship is critical in studying the behavior of gases. According to Gay-Lussac’s Law, there is a direct relationship between the pressure and temperature of a gas, provided the volume does not change. This means if you increase the temperature of a gas, its pressure increases as well, if the volume remains constant.

It's a key concept in thermodynamics, shaping how we interpret the workings of engines, refrigerators, and even the weather. In our exercise, we leveraged this law to determine how a substantial temperature increase affects the gas pressure within a fixed volume.

Converting Celsius to Kelvin

Temperature measurements in gas laws are always done in Kelvin, as it is an absolute temperature scale. The Kelvin scale is crucial for gas law calculations because it starts at absolute zero, where all molecular motion ceases.

To convert Celsius to Kelvin, we use the formula: \[ K = \text{\textdegree C} + 273.15 \]. For example, room temperature, usually around 25\text{\textdegree C}, converts to 298.15 K. This conversion is necessary in our problem to use Gay-Lussac’s Law correctly.

Gas Laws

Gas laws are a set of rules that describe how gases behave under various conditions of pressure, volume, and temperature. They are fundamental in understanding the physical properties of gases and their interactions. The main gas laws include Boyle's Law (pressure-volume relationship), Charles's Law (volume-temperature relationship), Avogadro's Law (volume-mole relationship), and Gay-Lussac's Law, which we’ve discussed in relation to pressure-temperature relationship.

These laws are often combined into the ideal gas law, encapsulated in the equation \[ PV = nRT \], where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin.

Solving for Unknown Pressure

Finding an unknown gas pressure when other conditions are given often requires using one of the gas laws. In the example provided, we applied Gay-Lussac’s Law which is represented by the formula \[ P1/T1 = P2/T2 \]. By rearranging, the formula to solve for the unknown pressure becomes \[ P2 = P1 \times (T2/T1) \].

This step encapsulates the essence of gas law problems - substituting known values into a formula derived from a fundamental principle, and algebraically solving for the unknown. In practical scenarios, solving for unknown pressure can be critical, from calculating the required pressure in a car's tires based on temperature changes to managing the pressurization systems in aircraft.