Question

A sample of nitrogen gas in a \(1.75-\mathrm{L}\) 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 \(355^{\circ} \mathrm{C}\) ?

Short answer

The pressure of the nitrogen gas after being raised to 355°C is approximately 2.84 atm.

Step-by-step solution

State the Given Information

Initially, the nitrogen gas is at a pressure of 1.35 atm, a volume of 1.75 L, and a temperature of 25°C which is 298 K (since 25°C + 273 = 298 K). The final temperature is 355°C which corresponds to 628 K (since 355°C + 273 = 628 K). The volume remains constant.

Apply Gay-Lussac's Law

Since the volume remains constant and the number of moles of gas doesn't change, Gay-Lussac's Law can be used, which states that the pressure of a gas is directly proportional to its temperature in Kelvin. The equation is expressed as: \( P_1/T_1 = P_2/T_2 \), where \( P_1 \) and \( T_1 \) are the initial pressure and temperature, and \( P_2 \) and \( T_2 \) are the final pressure and temperature.

Calculate the Final Pressure

We can rearrange Gay-Lussac's Law to solve for the final pressure, \( P_2 \): \( P_2 = P_1 \times (T_2/T_1) \). Substituting the given values results in: \( P_2 = 1.35 \, \text{atm} \times (628 \, \text{K} / 298 \, \text{K}) \).

Perform the Calculation

By carrying out the calculation, we get \( P_2 = 1.35 \, \text{atm} \times 2.108 \), which equals to approximately 2.84 atm. This is the final pressure of the nitrogen gas after the temperature is raised.

Key concepts

Gas Laws

Understanding gas laws is fundamental in comprehending how gases behave under varying conditions of temperature, pressure, and volume. These laws are not only crucial for chemistry and physics but also have practical applications in industries like meteorology, engineering, and even medicine.

One of the primary gas laws is Gay-Lussac's Law, which details the relationship between the pressure and temperature of a gas. When handling problems involving Gay-Lussac's Law, like in the exercise, it's important to ensure that the volume and the amount of gas remain constant. Simplifying complex scenarios into conditions where Gay-Lussac’s Law applies is a key skill in understanding gas behavior.

To solve problems effectively, always start by listing given information and verifying units are consistent—like converting Celsius to Kelvin for temperature. This step-by-step organization aids in clarity and reduces errors during calculations.

Pressure-Temperature Relationship

Gay-Lussac's Law showcases the direct pressure-temperature relationship in gases, stating that, for a given mass at constant volume, the pressure of a gas is directly proportional to its absolute temperature. In the context of the provided exercise, this relationship is used to predict how the pressure of nitrogen gas changes as its temperature increases.

To comprehend this concept, visualize a fixed amount of gas particles bouncing inside a container. As the temperature rises, the particles move faster and collide more forcefully with the container's walls, resulting in increased pressure. When approaching such problems, a thorough understanding of this proportional relationship allows students to confidently assess the outcome of temperature changes on pressure.

Remember, temperatures must always be in Kelvin when using gas laws. This is because Kelvin is an absolute scale with zero representing the absence of kinetic energy in particles, and it ensures the proportionality remains true.

Ideal Gas Behavior

The concept of an ideal gas is a simplified model that helps in the understanding of real gas behavior under different conditions. In ideal gas behavior, the particles are considered to have perfectly elastic collisions and no intermolecular forces. While no gas is truly ideal, many gases at high temperature and low pressure behave similarly enough to be approximated as such.

In the exercise involving nitrogen gas, we assume ideal behavior to apply Gay-Lussac's Law. It's important to recognize conditions under which real gases deviate from this ideal model, such as at very low temperatures or high pressures where intermolecular forces are significant.

By focusing on the ideal behavior, students can apply simple equations to predict gas behavior under controlled conditions. Yet, it's crucial to contextualize the ideal gas law within its limitations and encourage students to consider real-world complexities when applying it beyond theoretical exercises.