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:

P1T1=P2T2\frac{P_1}{T_1} = \frac{P_2}{T_2}

where:

  • P1P_1 is the initial pressure,
  • T1T_1 is the initial temperature (in Kelvin),
  • P2P_2 is the final pressure,
  • T2T_2 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:

T1=35.0C+273.15=308.15 KT_1 = 35.0^\circ \text{C} + 273.15 = 308.15 \text{ K} T2=37.0C+273.15=236.15 KT_2 = -37.0^\circ \text{C} + 273.15 = 236.15 \text{ K}

Step 2: Set Up the Combined Gas Law

We know the initial gauge pressure P1=2.30×105N/m2P_1 = 2.30 \times 10^5 \, \text{N/m}^2.

Now, we apply the combined gas law to solve for P2P_2:

P2P1=T2T1\frac{P_2}{P_1} = \frac{T_2}{T_1} P2=P1T2T1P_2 = P_1 \cdot \frac{T_2}{T_1}

Substituting the values:

P2=(2.30×105)236.15308.15P_2 = (2.30 \times 10^5) \cdot \frac{236.15}{308.15} P21.76×105N/m2P_2 \approx 1.76 \times 10^5 \, \text{N/m}^2

Step 3: Convert Pressure to Atmospheres

To convert P2P_2 from pascals (N/m²) to atmospheres, we use the conversion 1atm=1.013×105N/m21 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2.

P2 (in atm)=1.76×1051.013×105P_2 \text{ (in atm)} = \frac{1.76 \times 10^5}{1.013 \times 10^5} P21.74atmP_2 \approx 1.74 \, \text{atm}

Final Answer:

The gauge pressure in the tires at 37.0C-37.0^\circ \text{C} is approximately 1.74 atm.

Would you like more details on any specific step?

Here are some related questions:

  1. How does a change in temperature affect tire pressure?
  2. Why do we need to convert temperatures to Kelvin in gas law calculations?
  3. What happens to the pressure if the volume of the tire changes as well?
  4. How would the answer change if we used absolute pressure instead of gauge pressure?
  5. 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