Question

A bicycle tire is inflated to a pressure of 3.74 atm at \(15^{\circ} \mathrm{C}\). The tire is heated to \(35^{\circ} \mathrm{C}\). Calculate the pressure in the tire. Assume the tire volume doesn't change.

Short answer

The pressure in the tire at 35°C is approximately 4.00 atm.

Step-by-step solution

Identify Known Values

We are given the initial pressure of the bicycle tire as \( P_1 = 3.74 \) atm and the initial temperature as \( T_1 = 15^{\circ} \mathrm{C} \). The final temperature is given as \( T_2 = 35^{\circ} \mathrm{C} \). We are asked to find the final pressure \( P_2 \).

Convert Temperatures to Kelvin

To use the ideal gas law, we need to convert the temperatures from Celsius to Kelvin. The formula to convert Celsius to Kelvin is: \( T(K) = T(^{\circ}C) + 273.15 \).So, for \( T_1 \):\[ T_1 = 15 + 273.15 = 288.15 \text{ K} \]And for \( T_2 \):\[ T_2 = 35 + 273.15 = 308.15 \text{ K} \]

Apply the Pressure-Temperature Relationship (Gay-Lussac's Law)

According to Gay-Lussac's Law, when the volume of a gas is constant, the pressure is directly proportional to its temperature:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Rearranging for \( P_2 \):\[ P_2 = P_1 \times \frac{T_2}{T_1} \]

Calculate Final Pressure

Using the formula from Step 3 and substituting the known values:\[ P_2 = 3.74 \times \frac{308.15}{288.15} \]Calculating this gives:\[ P_2 \approx 4.00 \text{ atm} \]

Conclusion

The pressure in the tire, when heated to \(35^{\circ} \mathrm{C}\), is approximately \(4.00 \) atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume of the gas
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This equation assumes that the gas particles are in random motion and do not interact with each other except through elastic collisions. While real gases deviate from this behavior at high pressures and low temperatures, the Ideal Gas Law provides a reasonably accurate description under many conditions.
If you're solving problems with the Ideal Gas Law, remember:

• Convert all temperatures to Kelvin using the formula \( T(K) = T(^{\circ}C) + 273.15 \).
• Ensure you have consistent units for pressure, volume, and temperature.
• Use the appropriate value of \( R \) based on the units being used for your other variables.

Gay-Lussac's Law

Gay-Lussac's Law is a specific application of the Ideal Gas Law, dealing with the relationship between temperature and pressure when the volume of a gas is held constant. It states that the pressure of a given mass of gas is directly proportional to its absolute temperature. This can be expressed as: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Where:

• \( P_1 \) and \( P_2 \) are the initial and final pressures, respectively.
• \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin.

To use this law effectively, remember:

• Always convert temperature to Kelvin.
• Keep the volume of the gas constant, as it's only applicable under constant volume conditions.

In the exercise, Gay-Lussac's Law helps us find the final pressure of the gas in the bicycle tire after a temperature change, given a constant volume.

Temperature Conversion

Temperature conversion is crucial in gas law problems to ensure accuracy and consistency in calculations. In most gas law problems, including calculating changes in pressure or volume using Gay-Lussac's Law or the Ideal Gas Law, temperature must be expressed in Kelvin. This is because Kelvin is an absolute temperature scale that begins at absolute zero.To convert temperatures from Celsius to Kelvin, use the formula:

• \( T(K) = T(^{\circ}C) + 273.15 \)

This conversion is straightforward, but it's easy to overlook, leading to incorrect results in calculations. Make sure every temperature used in gas law equations is in Kelvin to apply the laws correctly and achieve reliable results.
In our specific exercise, converting the given temperatures from Celsius to Kelvin was a vital first step before applying Gay-Lussac's Law to find the new pressure at a higher temperature.