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 79.88 psi.

Step-by-step solution

Convert Celsius to Kelvin

It's essential to use Kelvin as the unit for temperature in gas law equations. Convert the given temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(^\circ \mathrm{C}) + 273.15 \] Initial temperature: \[ T_1 = 19^\circ \mathrm{C} + 273.15 = 292.15 \mathrm{K} \] Final temperature: \[ T_2 = 58^\circ \mathrm{C} + 273.15 = 331.15 \mathrm{K} \]

Find the initial and final volumes' ratio

Given that the volume of the tire increases by 4%, we can represent the initial volume as \(V_1\) and the final volume as \(V_2 = 1.04 V_1\).

Write down the combined gas law and rearrange for the new pressure

The combined gas law is given by: \[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \] We want to find \(P_2\), so we can rearrange the equation: \( P_2 = \frac{P_1V_1T_2}{V_2T_1} \)

Plug in the known values and solve for the new pressure

Now, we can plug in the initial pressure (\(P_1\) = 75 psi), initial temperature (\(T_1\) = 292.15 K), final temperature (\(T_2\) = 331.15 K), and the final volume (\(V_2\) = 1.04\(V_1\)): \[ P_2 = \frac{75 \mathrm{psi} \cdot V_1 \cdot 331.15 \mathrm{K}}{1.04 V_1 \cdot 292.15 \mathrm{K}} \] The \(V_1\) terms cancel out, and we can solve the equation: \[ P_2 = \frac{75 \mathrm{psi} \cdot 331.15 \mathrm{K}}{1.04 \cdot 292.15 \mathrm{K}} = 79.88 \mathrm{psi} \] The new pressure in the bicycle tire is approximately 79.88 psi.

Key concepts

Pressure-Temperature Relationship

When studying the behavior of gases under different conditions, understanding the pressure-temperature relationship is crucial. This concept is based on Gay-Lussac's Law, which tells us how pressure changes in response to temperature changes, provided the volume remains constant.

In simpler terms, when the temperature of a gas increases, its pressure also increases if the gas is confined in a container with the same volume. This happens because molecules move faster at higher temperatures and collide more vigorously with the walls of the container, resulting in greater pressure.

• Initial pressure example: In our problem, the bicycle tire was initially at 75 psi at 19°C.
• Impact of increasing temperature: Once the temperature increased to 58°C, the air molecules inside moved faster, aiming to increase pressure.

However, as we learn in this exercise, the volume of the tire expands slightly, altering the dynamics.

Knowing how pressure and temperature relate helps us use tools like the combined gas law effectively, which provides a more complete view when volume changes also occur.

Volume Expansion

Volume expansion is another essential concept in understanding how gases behave under varying conditions. When a gas is heated, it tends to expand, meaning its volume can increase.

In the context of a bicycle tire, when it heats up while riding on a hot day, not only the pressure changes but the volume of the air inside also increases. The problem states that the volume of the tire increases by 4%, which plays a vital role in calculating the new pressure using the combined gas law.

We represent the change in volume mathematically by considering:

• Initial volume: Let's denote it by \( V_1 \).
• Final volume: After a 4% increase, expressed as \( V_2 = 1.04 \cdot V_1 \).

In calculations like these, it's important to adjust the equation to consider increased volume, as it counteracts some of the effects of rising temperature, reducing the potential pressure increase.

Temperature Conversion

Temperature conversion, specifically converting Celsius to Kelvin, is a foundational step in solving gas law problems. Since gas laws require absolute temperatures, Kelvin provides a scale starting from absolute zero, where molecular motion ceases.

For the Ideal Gas Law and related calculations, Kelvin is essential because it maintains proportionality across different units involved in gas equations. Here’s how the conversion is done:

• Convert Celsius to Kelvin: Use the formula \( T(K) = T(^\circ \text{C}) + 273.15 \).
• Initial temperature conversion: For 19°C, this equals 292.15 K.
• Final temperature conversion: For 58°C, this equals 331.15 K.

By converting temperatures using this reliable method, you ensure that you use consistent units and get accurate results in any calculations involving the Ideal Gas Law, as seen in the pressure calculation for the bicycle tire.