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Boyle's Law is a fundamental principle in physics and chemistry that describes how the pressure of a gas tends to increase as the volume of the container decreases, and vice versa, provided the temperature remains constant. This relationship can be expressed mathematically as: \( P_1V_1 = P_2V_2 \) Where:

• \( P_1 \) and \( P_2 \) are the initial and final pressures of the gas respectively.
• \( V_1 \) and \( V_2 \) are the initial and final volumes respectively.

In the given exercise, Boyle's Law is utilized to find the new partial pressures of nitrogen and oxygen gases after mixing them in a larger volume. For example, starting with nitrogen gas at 825 Torr in 650 mL, we calculate its new pressure in the combined 1150 mL volume by rearranging the formula to find \( P_2 \). This step demonstrates the inverse relationship between pressure and volume, key to Boyle's Law. The same approach is taken for oxygen gas with its initial conditions. Boyle's Law is quite handy in scenarios involving gas mixtures and understanding gas behavior under varying volumes.

In a mixture of gases, each gas exerts a pressure independently, called its partial pressure. It is the pressure that the gas would exert if it were the only gas present in the entire volume. This concept is crucial to solving problems involving gas mixtures like in the exercise. To calculate partial pressures using Boyle's Law, we're essentially seeing how each individual gas contributes to the total pressure when volumes are combined. For nitrogen, the calculation involves adjusting its initial pressure from 825 Torr, found in its initial volume of 650 mL, to fit into the total volume of 1150 mL when the valve is opened. This yields a partial pressure of 466.304 Torr for nitrogen after mixing. Similarly, for oxygen gas starting at 732 Torr in 500 mL, Boyle's Law helps us find it's new pressure, 318.261 Torr, in the combined volume. These partial pressures play a significant role in predicting how gases behave in a mixed system, making it easier to calculate total pressure.

The total pressure of a gas mixture is the sum of the partial pressures of all the gases present. This idea comes from Dalton's Law of Partial Pressures, which states that each gas in a mixture exerts pressure independent of the other gases. In our exercise, once we have found the partial pressures for nitrogen and oxygen, which are 466.304 Torr and 318.261 Torr respectively, we simply add them to get the total pressure. \[ P_{\text{total}} = P_{\text{N}_2\text{ final}} + P_{\text{O}_2\text{ final}} \] \[ P_{\text{total}} = 466.304 + 318.261 = 784.565 \text{ Torr} \] This calculation shows how the pressures combine to give the overall pressure in the container after the gases have mixed. This total pressure is crucial in understanding how gases will behave in the system, and it is often used to predict outcomes in chemical reactions involving gases. Understanding both partial and total pressures helps clarify how gases interact when mixed.