the volume of the gas when the absolute temperature is 375 K.
(1) Boyle's law states that for a given mass of an ideal gas at a constant temperature, its pressure ([tex]$p N / m ^2$[/tex]) is inversely proportional to its volume ([tex]$\left(V m^3\right)$[/tex]). At a certain temperature, [tex]$3 m^3$[/tex] of a gas exerts a pressure of [tex]$10^5 N / m ^2$[/tex]. If the gas expands to [tex]$12 m^3$[/tex], calculate its pressure, giving your answer in standard form.
Answer (1)
Apply Boyle's Law: P 1 V 1 = P 2 V 2 .
Substitute the given values: P 1 = 1 0 5 N / m 2 , V 1 = 3 m 3 , and V 2 = 12 m 3 into the formula.
Calculate the final pressure: P 2 = 12 m 3 ( 1 0 5 N / m 2 ) ( 3 m 3 ) = 2.5 × 1 0 4 N / m 2 .
The final pressure of the gas is 2.5 × 1 0 4 N / m 2 .
Explanation
Understanding the Problem We are given a problem involving Boyle's Law, which states that for a fixed amount of gas at constant temperature, the pressure and volume are inversely proportional. This can be expressed as P 1 V 1 = P 2 V 2 , where P 1 and V 1 are the initial pressure and volume, and P 2 and V 2 are the final pressure and volume. We are given the initial volume V 1 = 3 m 3 and initial pressure P 1 = 1 0 5 N / m 2 . The gas expands to a final volume of V 2 = 12 m 3 . We need to find the final pressure P 2 .
Applying Boyle's Law We can use Boyle's Law to find the final pressure P 2 . The formula is P 1 V 1 = P 2 V 2 . We can rearrange this formula to solve for P 2 : P 2 = V 2 P 1 V 1
Substituting the Values Now, we substitute the given values into the formula: P 2 = 12 m 3 ( 1 0 5 N / m 2 ) ( 3 m 3 )
Calculating the Final Pressure Calculating the final pressure: P 2 = 12 3 × 1 0 5 N / m 2 = 4 1 × 1 0 5 N / m 2 = 0.25 × 1 0 5 N / m 2 = 2.5 × 1 0 4 N / m 2
Final Answer The final pressure is 2.5 × 1 0 4 N / m 2 .
Examples
Boyle's Law is useful in understanding how gases behave in various real-world scenarios. For example, it helps explain how a syringe works: when you increase the volume inside the syringe by pulling the plunger, the pressure decreases, allowing fluid to be drawn in. Similarly, in scuba diving, Boyle's Law helps divers understand how the volume of air in their lungs changes with depth due to pressure variations, which is crucial for safe diving practices. This principle is also applied in engines and compressors to control gas pressure and volume for efficient operation.
