Problem 62 - Chemistry and Chemical Reactivity

Question

A bicycle tire has an internal volume of \(1.52 \mathrm{L}\) and contains 0.406 mol of air. The tire will burst if its internal pressure reaches 7.25 atm. To what temperature, in degrees Celsius, does the air in the tire need to be heated to cause a blowout?

Step-by-Step Solution

Answer

The tire will burst at approximately 55.6 °C.

Step 1: Identify Known Quantities and Formula

We are given the internal volume of the tire as \(1.52 \text{ L}\), the amount of air as \(0.406 \text{ mol}\), and the pressure at which the tire bursts as \(7.25 \text{ atm}\). We need to find the temperature that would cause this pressure, using the Ideal Gas Law formula: \[PV = nRT\] where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant \((0.0821 \text{ L atm/mol K})\), and \(T\) is temperature in Kelvin.

Step 2: Rearrange the Ideal Gas Law

Rearrange the ideal gas law equation to solve for temperature \(T\): \[T = \frac{PV}{nR}\]

Step 3: Substitute the Known Values

Substitute the known values into the equation: \[T = \frac{(7.25 \text{ atm})(1.52 \text{ L})}{(0.406 \text{ mol})(0.0821 \text{ L atm/mol K})}\].

Step 4: Calculate the Temperature in Kelvin

Perform the calculations: \[T = \frac{(7.25)(1.52)}{(0.406)(0.0821)} \approx 328.745 \text{ K}\].

Step 5: Convert Temperature from Kelvin to Celsius

Convert the temperature from Kelvin to Celsius using the formula \(T_\text{C} = T_\text{K} - 273.15\): \[T_\text{C} = 328.745 - 273.15 = 55.595\text{ °C}\]. This is the temperature at which the air in the tire will cause it to burst.

Key Concepts

Pressure and Temperature

Gas pressure and temperature are intimately related in the context of the Ideal Gas Law. When gas molecules are heated, they gain kinetic energy. This increased energy causes the molecules to move more rapidly and collide with the walls of their container more frequently, which leads to an increase in pressure if the volume remains constant. This relationship is directly proportional, meaning when temperature increases, pressure also increases, as long as volume and the amount of gas remain unchanged. To understand how changes in temperature affect gas pressure, consider the bicycle tire example. If the temperature inside the tire rises, the internal pressure will increase. This increase occurs because the air molecules inside the tire are moving more vigorously, thus pushing harder against the tire walls. For safety measures, it is crucial to monitor both temperature and pressure to prevent a tire burst, especially when environmental conditions change, like during hot weather.

Gas Law Calculations

The Ideal Gas Law is a fundamental equation in chemistry for understanding how gases behave under different conditions. The equation is \[PV = nRT\]where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume it occupies.
• \(n\) is the number of moles of gas particles.
• \(R\) is the ideal gas constant \((0.0821 \text{ L atm/mol K})\).
• \(T\) is the temperature in Kelvin.

In the problem involving the bicycle tire, understanding this relationship helps us calculate the temperature at which the pressure will reach a critical point. By rearranging the formula to solve for temperature \[T = \frac{PV}{nR}\]we can substitute the known values for pressure, volume, and moles to find the temperature. This vital equation allows us to predict and control the conditions of a gas, ensuring safety and functionality in practical applications.

Thermodynamics in Chemistry

Thermodynamics is the study of energy and its transformations, and it plays a key role in understanding gas behavior through temperature and pressure analysis. According to the first law of thermodynamics, energy can neither be created nor destroyed, only transferred or transformed from one form to another. In gases, heating them increases their internal energy, mainly in the form of kinetic energy. The Ideal Gas Law provides us with a macroscopic view, linking thermodynamics principles with observable changes like pressure and volume alterations. For example, when calculating the temperature that leads a tire to burst, we're applying thermodynamics to determine how heat energy affects gas molecules and how this change translates into macroscopic parameters like pressure. By deeply grasping these principles, one learns to predict how gases will react under various conditions, which can help in designing safer and more efficient chemical processes and products.