A 775-mL sample of \(\mathrm{NO}_{2}\) gas is at STP. If the volume changes to \(615 \mathrm{~mL}\) and the temperature changes to \(25^{\circ} \mathrm{C}\), what will be the new pressure?

Gas laws are fundamental principles that describe how gases behave under various conditions of pressure, volume, and temperature. These laws are essential for understanding and predicting gas behavior in fields like chemistry and physics. One key gas law is the combined gas law, which integrates three simpler gas laws: Boyle's law, Charles's law, and Gay-Lussac's law. Understanding this combination helps in solving more complex gas-related problems.
Boyle’s law states that the volume of a gas is inversely proportional to its pressure when the temperature is held constant: \[ P_1 V_1 = P_2 V_2 \]
Charles’s law states that the volume of a gas is directly proportional to its temperature when the pressure is kept constant: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Gay-Lussac's law states that the pressure of a gas is directly proportional to its temperature when volume is kept constant: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
Combining these three laws, we get the combined gas law, which is used to solve problems where pressure, volume, and temperature all change: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

In chemistry, accurate calculations are vital for understanding reactions, predicting outcomes, and scaling processes. Solving gas law problems often involves multiple steps, including converting units and rearranging equations. These skills are necessary for both academic and practical applications.
Let's break down the steps with our given problem:

• First, identify and list all given values such as initial and final volumes, initial and final temperatures, and initial pressure.
• Next, ensure all temperatures are in Kelvin, as this is the standard unit when working with gas laws.
• Finally, apply the combined gas law. Substitute the known values, rearrange to solve for the unknown variable, and compute the final answer.

In our specific example, we converted the temperature from Celsius to Kelvin. Then, we used given volumes and pressures to determine the unknown pressure using the combined gas law. Step-by-step calculations clarify how these values come together to predict the new pressure.

Understanding the relationship between pressure and volume is crucial in comprehending gas behaviors. Boyle's law directly addresses this relationship. When the temperature is constant, an increase in pressure will decrease the volume and vice versa.
In our problem, the initial and final volumes of a gas changed under constant mass and changing temperatures and pressures. By using the combined gas law equation, we linked changes in pressure (P), volume (V), and temperature (T) together.
Rearranging and solving the equation showed us that decreasing the volume of the gas while maintaining and adjusting for temperature leads to an increase in pressure. This phenomenon is predictable using Boyle's law principles within the broader scope of the combined gas law. Such calculations underscore the interdependence of pressure, volume, and temperature in real-world scenarios, from engine mechanics to atmospheric science.
Understanding these principles enables students to predict how gases will behave under different conditions, essential knowledge for any aspiring scientist.