Question

(a) What volume of air at 1.0 atm and \(22^{\circ} \mathrm{C}\) is needed to fill a \(0.98-\mathrm{L}\) bicycle tire to a pressure of \(5.0 \mathrm{~atm}\) at the same temperature? (Note that the 5.0 atm is the gauge pressure, which is the difference between the pressure in the tire and atmospheric pressure. Before filling, the pressure in the tire was \(1.0 \mathrm{~atm} .\) ) (b) What is the total pressure in the tire when the gauge pressure reads 5.0 atm? (c) The tire is pumped by filling the cylinder of a hand pump with air at 1.0 atm and then, by compressing the gas in the cylinder, adding all the air in the pump to the air in the tire. If the volume of the pump is 33 percent of the tire's volume, what is the gauge pressure in the tire after three full strokes of the pump? Assume constant temperature.

Short answer

(a) 5.88 L of air is needed; (b) Total pressure is 6 atm; (c) Gauge pressure after 3 strokes is 3.01 atm.

Step-by-step solution

Understanding the Problem and Initial Conditions

First, we need to determine the volumes and pressures involved. Initially, the bicycle tire has a volume of 0.98 L and is filled to a pressure of 1 atm. We aim to increase the total pressure in the tire to a gauge pressure of 5 atm, which is actually 6 atm total, since we add 1 atm (atmospheric pressure).

Calculating Air Volume to Fill Tire

Using Boyle's Law, where the initial state is air at 1 atm and 22°C (V₁, P₁ = 1 atm), and the final state is air in a 0.98 L tire at 6 atm:\[ P_1V_1 = P_2V_2 \]Insert values: \(1 \text{ atm} \times V_1 = 6 \text{ atm} \times 0.98 \text{ L}\)Solve for \(V_1\):\[ V_1 = \frac{6 \times 0.98}{1} = 5.88 \text{ L} \]Thus, 5.88 L of air at 1 atm and 22°C is required to fill the tire.

Calculating Total Pressure in Tire

Gauge pressure reads 5 atm, adding atmospheric pressure, total pressure becomes:\[ P_{total} = P_{gauge} + P_{atm} = 5 + 1 = 6 \text{ atm} \]

Volume of Pump and Number of Strokes

The pump's volume is 33% of the tire's volume. Calculate pump volume:\[ V_{pump} = 0.33 \times V_{tire} = 0.33 \times 0.98 = 0.3234 \text{ L} \]Each pump stroke adds 0.3234 L of air at 1 atm to the tire.

Calculating New Pressure in Tire After Three Strokes

After three strokes, we add the volume of air from the pump 3 times, each time at 1 atm, to the tire:Total volume added after 3 strokes is:\[ V_{added} = 3 \times V_{pump} = 3 \times 0.3234 = 0.9702 \text{ L} \]Total volume of air in tire:\[ V_{total} = 0.98 \text{ L} + 0.9702 \text{ L} = 1.9502 \text{ L} \]Using Boyle's Law, calculate new gauge pressure:\[ P_2 = \left( \frac{6 \text{ atm} \times 0.98 \text{ L}}{1.9502 \text{ L}} \right) - 1 \text{ atm} \]Solve:\[ P_2 = \left( \frac{5.88}{1.9502} \right) - 1 = 3.014 \text{ atm} \]

Conclusion

The total pressure in the tire after three full pump strokes, maintaining constant temperature, results in a gauge pressure of approximately 3.01 atm.

Key concepts

Gas Laws

Gas laws are fundamental in understanding how gases behave under different conditions. One of the most important is **Boyle's Law**, which describes the relationship between pressure and volume. It states that at a constant temperature for a fixed amount of gas, pressure is inversely proportional to volume. This means as pressure increases, volume decreases, and vice versa. It's mathematically represented as:\[ P_1V_1 = P_2V_2 \]Here,

• \( P_1 \) and \( V_1 \) are the initial pressure and volume, respectively.
• \( P_2 \) and \( V_2 \) are the final pressure and volume.

Understanding this law is crucial to solving problems like filling a bicycle tire, where we need to calculate how the volume of gas changes as we compress it into the tire. It's about balancing pressure and volume, always ensuring that when one changes, the other counters it to maintain equilibrium.

Pressure Calculation

Pressure calculation is a key step in working with gases. It's important to distinguish between **gauge pressure** and **total pressure**. Gauge pressure is the extra pressure in a system over atmospheric pressure, while total pressure includes both atmospheric and gauge pressures. In our exercise:

• Gauge pressure is given as 5 atm.
• Atmospheric pressure is approximately 1 atm.

Thus, the total pressure becomes:\[ P_{total} = P_{gauge} + P_{atm} \]For a tight fit to work right, calculating the total pressure helps ensure that the tire does not burst from too much pressure. Knowing the total pressure allows us to determine how much air is needed and how it adjusts the actual physical state of the tire.

Temperature

Temperature is a vital component in gas behavior but interestingly does not change in this problem, making analysis simpler. This is known as the **isothermal process**, where the gas temperature remains constant throughout the procedure. In our exercise:- The temperature is held at \(22^{\circ} C\). - This stability allows us to focus solely on the interplay between pressure and volume without worrying about temperature fluctuations changing the dynamics.When temperature remains constant, Boyle's Law can be directly applied without modifications for temperature changes, simplifying our calculations and yielding clear results.

Volume Expansion

Volume expansion is a key concept in gas laws applied to real-world scenarios like filling a tire. It's about understanding how added air volume affects an enclosed space. In the exercise: - Air from the pump is repeatedly added to increase the total air volume in the tire. - Each pump stroke introduces additional air, expanding the volume of gas inside the tire. The pump's volume is calculated as 33% of the tire's volume. Thus each pump action significantly impacts the tire's volume and pressure:

• Volume per stroke: 0.3234 L is added.
• After three strokes, approximately 0.9702 L is added.

Remember, as volume increases, unless the container's capacity is exceeded, pressure should theoretically decrease, according to Boyle's Law, unless we add more air to counterbalance this expansion. By understanding volume expansion, we control how gas behaves, ensuring that a tire inflates correctly and safely.