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 $4.0\mathrm{\%}$. What is the new pressure in the bicycle tire?

Short answer

The new pressure in the bicycle tire at a temperature of ${58}^{\circ }\mathrm{C}$ and a 4.0% increase in volume is approximately 66.15 psi.

Step-by-step solution

Convert temperatures to Kelvin

First, we need to convert the given Celsius temperatures to Kelvin, as the ideal gas law requires temperatures to be in Kelvin. $T_1=19+273.15=292.15K$ $T_2=58+273.15=331.15K$

Calculate the ratio of the volume

Since the volume increases by 4.0%, we can express the final volume as 104% of the initial volume: $V_2=1.04V_1$

Use the ideal gas law to find the final pressure

Rewrite the ideal gas law equation as: $P_1{V}_1/T_1=P_2{V}_2/T_2$ Substitute the given values and expressions: $(75psi)V_1/(292.15K)=P_2(1.04V_1)/(331.15K)$ Observe that the initial volume $V_1$ can be canceled out from both sides of the equation. Now, solve for the final pressure $P_2$: $P_2=75psi\times \frac{292.15K}{331.15K}\times \frac{1.04}{1}$ $P_2=75psi\times 0.882$ $P_2=66.15psi$

Conclusion

The new pressure in the bicycle tire at a temperature of ${58}^{\circ }\mathrm{C}$ and a 4.0% increase in volume is approximately 66.15 psi.

Key concepts

Pressure Calculation

When dealing with gases, especially in practical situations like adjusting the tire's pressure, understanding how to calculate pressure changes is vital. The ideal gas law is essential in this calculation. This law relates pressure (P), volume (V), and temperature (T) in a simple equation: $$PV=nRT$$ where $n$ is the number of moles and $R$ is the ideal gas constant. For the bicycle tire, we are given the initial pressure and must find the final pressure after the temperature change and volume expansion. First, note that if the volume increases, as in our example, the pressure decreases if the temperature stays constant. However, with a rising temperature, as in our scenario, the pressure tends to increase. The relationship for cases with constant moles and a changing state is: $$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$$ This equation helps us derive the final pressure $P_2$ by substituting values and rearranging the formula to solve for $P_2$. This equation intuitively balances the impact of changing volume and temperature on pressure.

Temperature Conversion

Temperature conversion is a fundamental step when using the ideal gas law because this law operates with temperatures in Kelvin. Kelvin is the standard unit for thermodynamic temperature measurement and avoids the negative values Celsius can have, making calculations more straightforward. To convert from Celsius to Kelvin, use the formula: - Add 273.15 to the Celsius temperature. - For example, a temperature of ${19}^{\circ }C$ converts to $292.15K$. Ensuring accurate conversion is crucial because using incorrect values can lead to errors in further calculations such as pressure and volume changes. This step can seem simple but is highly significant in keeping consistency and accuracy in calculations that involve the ideal gas law.

Volume Expansion

Volume expansion is another critical factor when working with gases. It describes how the volume of a gas changes in response to conditions like temperature and pressure shifts. For a bicycle tire, a rise in external temperature causes the air inside the tire to expand, thereby increasing its volume. In this example, we deal with a 4% increase in volume. This is mathematically represented as $V_2=1.04V_1$, signifying that the final volume $V_2$ is 104% of the initial volume $V_1$. A key point to remember: as the volume increases in a closed system, it can affect the pressure, typically lowering it if the temperature is unchanged. But combined with rising temperature, a dual effect is observed, impacting pressure differently. By acknowledging these changes, we can accurately predict and calculate pressure alterations resulting from volume expansion.

Gas Laws

Gas laws, particularly the ideal gas law, provide a framework to predict how gases respond to changes in pressure, volume, and temperature. These laws are foundational to understanding and solving real-world problems involving gases.The ideal gas law, with its formula $PV=nRT$, shows that in a closed system, where the amount of gas $n$ remains constant, the state of a gas can be adjusted and calculated by varying pressure (P), volume (V), and temperature (T). Boyle's Law, Charles' Law, and Avogadro's Law are all special cases of the ideal gas law:

• Boyle's Law shows that pressure is inversely proportional to volume at a constant temperature.
• Charles' Law illustrates that volume is directly proportional to temperature at a constant pressure.
• Avogadro's Law tells us that volume is directly proportional to the number of moles at constant temperature and pressure.

Being aware of these individual laws helps to understand the ideal gas law's overarching principles when applied to situations like tire pressure changes. They help explain the behavior of gases under various conditions, guiding us in practical applications like calculating gas pressure. Understanding these laws ensures accurate comprehension and calculation when dealing with the dynamics of gases.