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Chapter 12: Problem 77

The tires on an automobile were filled with air to 30. psi at \(71.0^{\circ} \mathrm{F}\). When driving at high speeds, the tires become hot. If the tires have a bursting pressure of 44 psi, at what temperature \(\left({ }^{\circ} \mathrm{F}\right)\) will the tires "blow out"?

Short Answer

Expert verified
The tires will blow out at approximately 316.8°F.

Step by step solution

01

Understand the problem

We need to find the temperature at which the pressure in the tires will increase to 44 psi from an initial pressure of 30 psi, given an initial temperature of 71.0°F.
02

Use ideal gas law relationship

The ideal gas law states that the pressure of a gas is directly proportional to its absolute temperature when the volume is constant: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\).
03

Convert initial temperatures to Kelvin

Convert the initial temperature from Fahrenheit to Kelvin: \(71.0^{\circ}F = 294.3^{\circ}K\). Use the formula: \(T(K) = (T(°F) − 32) × \frac{5}{9} + 273.15\).
04

Set up the proportion

Using the known pressures and the initial temperature in Kelvin, set up the equation: \(\frac{30}{294.3} = \frac{44}{T_2}\).
05

Solve for the unknown temperature in Kelvin

Rearrange the equation to solve for \(T_2\): \(T_2 = \frac{44 \times 294.3}{30} = 431.488K\).
06

Convert the temperature back to Fahrenheit

Convert the final temperature back to Fahrenheit: \(T(°F) = (T(K) - 273.15) × \frac{9}{5} + 32 = (431.488 - 273.15) × \frac{9}{5} + 32 \).
07

Calculate the final temperature in Fahrenheit

Perform the calculation: \(T(°F) = 316.78°F\).

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Key Concepts

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

Ideal Gas Law
The Ideal Gas Law is a fundamental principle in chemistry that describes the behavior of gases. It can be represented by the equation \[ PV = nRT \] where:
  • P stands for pressure
  • V represents volume
  • n is the number of moles of gas
  • R is the universal gas constant
  • T stands for temperature (in Kelvin)
In situations where the number of moles and the volume of the gas are constant, the Ideal Gas Law can be simplified to relate pressure and temperature directly: \[\frac{P1}{T1} = \frac{P2}{T2} \] This is the relationship we use when solving problems like the one provided.
Temperature Conversion
Temperature conversion is important when working with gas laws, as temperatures must be in Kelvin. Most chemistry problems provide temperature in Celsius or Fahrenheit. To convert from Fahrenheit to Kelvin, use the formula: \[ T(K) = (T(°F) − 32) \times \frac{5}{9} + 273.15 \]For the given exercise, the initial temperature is 71.0°F. Plugging in the values:\[ T(K) = (71.0 − 32) \times \frac{5}{9} + 273.15 = 294.3 \text{ K} \] Always remember to convert temperatures to Kelvin in gas law calculations!
Pressure Calculations
Pressure calculations often require a clear understanding of the initial and final states of the gas. For our problem, we start with an initial pressure of 30 psi and need to find the temperature at which the pressure reaches 44 psi using the proportion derived from the simplified Ideal Gas Law: \[ \frac{P_1}{T_1}=\frac{P_2}{T_2} \]After converting the initial temperature to Kelvin (294.3 K), the equation looks like this:\[\frac{30}{294.3} = \frac{44}{T_2} \]To find the unknown temperature (\[ T_2 \]), rearrange the equation:\[T_2 = \frac{44 \times 294.3}{30} = 431.488 \text{K } \]This gives us the temperature at which the tire pressure would reach 44 psi.
Chemical Principles
Several chemical principles underpin the steps we've taken to solve the problem. The Ideal Gas Law connects pressure, volume, and temperature, illustrating how they collectively influence gas behavior. In constant volume scenarios, as temperature increases, so does pressure, assuming no gas escapes. For comprehensive understanding, remember:
  • **Absolute Temperatures**: Calculations involving gas need temperatures in Kelvin for accurate results.
  • **Proportional Relationships**: Gas behavior can often be predicted using proportional relationships like \[\frac{P1}{T1} = \frac{P2}{T2} \]
By recognizing these principles, students can better grasp how temperature and pressure interrelate in gases.

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