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Chapter 12: Problem 820

An automobile tire has a bursting pressure of \(60 \mathrm{lb} /\) in. \({ }^{2}\). The normal pressure inside the tire is \(32 \mathrm{lb} / \mathrm{in}^{2}\). When traveling at high speeds, a tire can get quite hot. Assuming that the temperature of the tire is \(25^{\circ} \mathrm{C}\) before running it, determine whether the tire will burst when the inside temperature gets to \(212^{\circ} \mathrm{F}\). Show your calculations.

Short Answer

Expert verified
The tire will not burst. The final pressure is 40.06 lb/in², less than the bursting pressure of 60 lb/in².

Step by step solution

01

- Convert Temperature Units

First, convert the temperature from Fahrenheit to Celsius. Use the formula \( T(^{\text{°F}}) = T(^{\text{°C}}) \times 9/5 + 32 \). Rearrange to solve for Celsius: \( T(^{\text{°C}}) = (T(^{\text{°F}}) - 32) \times 5/9 \). Substitute in the given temperature: \( T(^{\text{°C}}) = (212 - 32) \times 5/9 = 100 ^{\text{°C}} \).
02

- Apply Ideal Gas Law Relationship

Use the relationship of the ideal gas law, which states that \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Convert the temperatures to Kelvin: \( T_1 = 25 + 273 = 298 \text{ K} \) and \( T_2 = 100 + 273 = 373 \text{ K} \).
03

- Solve for Final Pressure

Rearrange the formula to solve for the final pressure: \( P_2 = P_1 \times \frac{T_2}{T_1} \). Substitute the known values: \( P_2 = 32 \text{ lb/in}^2 \times \frac{373}{298} \approx 40.06 \text{ lb/in}^2 \).
04

- Compare Final Pressure with Bursting Pressure

Compare the calculated pressure to the tire's bursting pressure. The calculated pressure \( 40.06 \text{ lb/in}^2 \) is less than the bursting pressure \( 60 \text{ lb/in}^2 \). Therefore, the tire will not burst.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

temperature conversion
Temperature conversion is a crucial step in many chemistry problems, especially when dealing with the ideal gas law. This exercise requires us to convert between Fahrenheit and Celsius. The formula for this conversion is \( T(^{\text{\degree F}}) = T(^{\text{\degree C}}) \times \frac{9}{5} + 32\).
Rearranging to solve for Celsius, we get \( T(^{\text{\degree C}}) = (T(^{\text{\degree F}}) - 32) \times \frac{5}{9}\).
  • For instance, converting 212°F to Celsius involves subtracting 32 and then multiplying by \(\frac{5}{9}\).
  • This yields 100°C, which is the temperature of the tire in Celsius.
Converting these temperatures accurately is key to applying the ideal gas law correctly.
pressure calculations
Pressure calculations are fundamental in determining how gases behave under different conditions. In the ideal gas law, pressure (P) is related to temperature (T). The formula we use is: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
  • First, convert temperatures to Kelvin. For 25°C, adding 273 gives 298 K. For 100°C, it becomes 373 K.
  • Rearrange the formula to solve for the final pressure: \ P_2 = P_1 \times \frac{T_2}{ T_1}\.
  • Substitute the known values: \( P_2 = 32 \text{ lb/in}^2 \times \frac{373}{298} \approx 40.06 \text{ lb/in}^2 \).
Comparing this final pressure with the tire's bursting pressure, we see that it is lower (\

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