Hikers noticed that a sealed bag of potato chips puffs up when taken to the top of a mountain. Suppose that at the valley floor below, the air pressure is 1.0 atm, the temperature is [tex]$25^{\circ} C$[/tex], and the volume of the bag is 0.985 L. At the top of the mountain, the temperature is [tex]$22^{\circ} C$[/tex] and the bag has puffed up to 1.030 L. What is the air pressure on top of the mountain? Type in your answer using the correct number of significant figures.
Formula: [tex]$\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}$[/tex]
[$\square$] atm
Answer (2)
The air pressure at the top of the mountain is approximately 0.947 atm. This was calculated using the combined gas law after converting the given temperatures into Kelvin. Significant figures were considered in rounding the final answer.
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Convert temperatures from Celsius to Kelvin: T 1 = 298.15 K and T 2 = 295.15 K .
Rearrange the combined gas law to solve for P 2 : P 2 = T 1 V 2 P 1 V 1 T 2 .
Substitute the given values into the equation: P 2 = ( 298.15 K ) ( 1.030 L ) ( 1.0 atm ) ( 0.985 L ) ( 295.15 K ) .
Calculate the final pressure and round to three significant figures: P 2 ≈ 0.947 atm .
Explanation
Problem Analysis We are given the initial pressure, volume, and temperature of a bag of potato chips at the valley floor, and the volume and temperature at the top of the mountain. We are asked to find the air pressure at the top of the mountain. We can use the combined gas law to solve this problem.
Combined Gas Law The combined gas law is given by the formula: T 1 P 1 V 1 = T 2 P 2 V 2 where: P 1 = initial pressure V 1 = initial volume T 1 = initial temperature P 2 = final pressure V 2 = final volume T 2 = final temperature
Convert Celsius to Kelvin First, we need to convert the temperatures from Celsius to Kelvin: T ( K ) = T ( ∘ C ) + 273.15 So, T 1 = 25 + 273.15 = 298.15 K T 2 = 22 + 273.15 = 295.15 K
Rearrange the Formula Now, we can rearrange the combined gas law to solve for P 2 :
P 2 = T 1 V 2 P 1 V 1 T 2
Substitute the Values Plug in the given values: P 2 = ( 298.15 K ) ( 1.030 L ) ( 1.0 atm ) ( 0.985 L ) ( 295.15 K )
Calculate the Final Pressure Calculate P 2 :
P 2 = 298.15 × 1.030 1.0 × 0.985 × 295.15 = 307.0945 290.62275 ≈ 0.946688 atm Since the given values have three significant figures, we should round our answer to three significant figures.
Final Answer Therefore, the air pressure at the top of the mountain is approximately 0.947 atm.
Examples
Understanding the relationship between pressure, volume, and temperature is crucial in many real-world applications. For example, when designing scuba diving equipment, engineers must account for the changing pressure at different depths to ensure the equipment functions correctly and safely. Similarly, meteorologists use these principles to predict weather patterns, as atmospheric pressure and temperature changes can indicate approaching storms or shifts in weather conditions. In the kitchen, a pressure cooker utilizes the relationship between pressure and temperature to cook food faster by raising the boiling point of water.
