Math Problem Statement
The gauge pressure in your car tires is 2.30 ✕ 105 N/m2 at a temperature of 35.0°C when you drive it onto a ferry boat to Alaska. What is their gauge pressure (in atm) later, when their temperature has dropped to −37.0°C? (Assume that their volume has not changed.)
Solution
To solve this problem, we can use the ideal gas law in the form of the combined gas law, which relates pressure and temperature for a fixed volume:
where:
- is the initial pressure,
- is the initial temperature (in Kelvin),
- is the final pressure,
- is the final temperature (in Kelvin).
Step 1: Convert Temperatures to Kelvin
The initial and final temperatures in Celsius need to be converted to Kelvin:
Step 2: Set Up the Combined Gas Law
We know the initial gauge pressure .
Now, we apply the combined gas law to solve for :
Substituting the values:
Step 3: Convert Pressure to Atmospheres
To convert from pascals (N/m²) to atmospheres, we use the conversion .
Final Answer:
The gauge pressure in the tires at is approximately 1.74 atm.
Would you like more details on any specific step?
Here are some related questions:
- How does a change in temperature affect tire pressure?
- Why do we need to convert temperatures to Kelvin in gas law calculations?
- What happens to the pressure if the volume of the tire changes as well?
- How would the answer change if we used absolute pressure instead of gauge pressure?
- What is the significance of gauge pressure versus absolute pressure in real-world applications?
Tip: Always convert temperatures to Kelvin when working with gas laws, as they rely on absolute temperature measurements.
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Math Problem Analysis
Mathematical Concepts
Gas Laws
Pressure-Temperature Relationship
Unit Conversion
Formulas
P1/T1 = P2/T2
P2 = P1 * (T2 / T1)
P (atm) = P (N/m²) / 1.013 × 10^5
Theorems
Combined Gas Law
Ideal Gas Law
Suitable Grade Level
Grades 9-11
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