Question

A bicycle tire is filled with air to a pressure of 75 psi at a temperature of \(19^{\circ} \mathrm{C}\) . Riding the bike on asphalt on a hot day increases the temperature of the tire to \(58^{\circ} \mathrm{C}\) . The volume of the tire increases by 4.0\(\%\) . What is the new pressure in the bicycle tire?

Short answer

The new pressure in the bicycle tire is approximately \(79.64 \,\mathrm{psi}\).

Step-by-step solution

Convert Given Temperatures to Kelvin Scale

The Combined Gas Law uses the Kelvin temperature scale. So, we need to convert the given temperatures from Celsius to Kelvin: \(T_1 = 19^{\circ}\mathrm{C} + 273.15 = 292.15 \mathrm{K}\) \(T_2 = 58^{\circ}\mathrm{C} + 273.15 = 331.15 \mathrm{K}\)

Calculate the Volume Increase

The volume of the tire increases by 4%, so we need to find the relationship between initial and final volumes: \(V_2 = V_1 + 0.04V_1 = 1.04V_1\)

Apply the Combined Gas Law

While applying the Combined Gas Law, the terms related to volume and initial pressure will cancel out to get the final pressure. Substitute the known values and solve for \(P_2\): \(\frac{P_1V_1}{T_1}=\frac{P_2(1.04V_1)}{T_2}\) \(\frac{75(1)}{292.15}=\frac{P_2(1.04)}{331.15}\)

Solve for the Final Pressure

Next, we will isolate the final pressure, \(P_2\), by solving the equation: \(\frac{75(1)(331.15)}{292.15(1.04)} = P_2\) \(P_2 \approx 79.64 \,\mathrm{psi}\) Thus, the new pressure in the bicycle tire is approximately 79.64 psi.

Key concepts

Temperature Conversion

When working with gases, especially when applying the Combined Gas Law, it is critical to convert temperatures from Celsius to the Kelvin scale.
The Kelvin scale is an absolute temperature scale, starting at absolute zero, which provides a consistent basis for calculations in thermodynamics.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:

• The formula for conversion is: \[ T(K) = T(^{\circ}C) + 273.15 \]
• For example, converting \(19^{\circ}\mathrm{C}\) to Kelvin gives: \[ 19 + 273.15 = 292.15 \, \mathrm{K} \]
• Similarly, \(58^{\circ}\mathrm{C}\) converts to: \[ 58 + 273.15 = 331.15 \, \mathrm{K} \]

This conversion ensures accuracy and consistency when dealing with gas law equations and calculations.

Pressure Calculation

Calculating the pressure of a gas involves understanding how variables like temperature and volume affect it, especially when these factors change.
In this exercise, the initial pressure (\( P_1 \)) is given as 75 psi (pounds per square inch) and the task is to find the new pressure after changes in temperature and volume.
Using the Combined Gas Law, which is \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \], we can relate the initial and final states of pressure, volume, and temperature. Here are the steps:

• Substitute known values into the equation.
• Rearrange the equation to solve for the final pressure \( P_2 \).
• Ensure that the volume increase and temperature changes are accurately represented.

Finally, solving the equation gives us the new pressure: approximately 79.64 psi.

Volume Change

In gas law problems, changes in volume significantly impact pressure and temperature calculations.
In this scenario, the volume of the tire increases by 4%. This means that if the initial volume is denoted as \( V_1 \), the increased volume \( V_2 \) can be calculated as:

• \[ V_2 = 1.04 \times V_1 \]
• This indicates that the final volume is 104% of the initial volume.

This increase must be factored into the Combined Gas Law to accurately assess how it affects the pressure.Volume changes, especially increases, generally lead to pressure adjustments when considering a gas under constant temperature conditions, due to the inverse relationship between volume and pressure.

Kelvin Scale

The Kelvin scale is the foundation for thermodynamic calculations as it provides an absolute temperature reference, unlike Fahrenheit or Celsius.
This scale will always start at absolute zero (0 K), which is the lowest possible temperature, where molecular motion stops.
Using the Kelvin scale is imperative in gas law equations because it avoids negative temperatures that can distort calculations.

• All temperatures should ideally be converted to Kelvin when using equations like the Combined Gas Law.
• If you face a problem with temperature in Celsius, always remember to convert it to Kelvin first.

This ensures that all thermodynamic relationships uphold their proportional differences, maintaining consistency and accuracy in physical sciences.