Question

Before embarking on a car trip from Michigan to Florida to escape the winter cold, you inflate the tires of your car to a manufacturersuggested pressure of 33.0 psi while the outside temperature is \(25.0^{\circ} \mathrm{F}\) and then make sure your valve caps are airtight. You arrive in Florida two days later, and the temperature outside is a pleasant \(72.0^{\circ} \mathrm{F}\). a) What is the new pressure in your tires, in SI units? b) If you let air out of the tires to bring the pressure back to the recommended 33.0 psi, what percentage of the original mass of air in the tires will you release?

Short answer

Answer: The new pressure in the tires after the temperature change will be approximately 253 kPa. When bringing the pressure back to the recommended value, approximately 9.75% of the original mass of air will be released.

Step-by-step solution

Convert the given values to SI units

First, we need to convert the given values to SI units. The pressure needs to be converted from psi to pascals (Pa), and the temperature needs to be converted from Fahrenheit to Kelvin. Given pressure, \(P_1\) = 33 psi To convert from psi to Pa, we use the conversion factor 1 psi = 6,894.757 Pa \(P_1 = 33 \times 6,894.757 = 227,527 \text{ Pa}\) Initial temperature, \(T_1\) = 25.0 °F To convert from °F to K, we use the formula: \((°F - 32) * (5/9) + 273.15\) \(T_1 = (25 - 32) * (5/9) + 273.15 = 266.483 \text{ K}\) Final temperature, \(T_2\) = 72.0 °F \(T_2 = (72 - 32) * (5/9) + 273.15 = 295.372 \text{ K}\)

Apply the ideal gas law to find the new pressure

Since we assume the tire's volume doesn't change, we can use this simplified version of the ideal gas law: \(P_{1}/T_{1} = P_{2}/T_{2}\) Solving for final pressure, \(P_2\), we have: \(P_2 = (P_{1}/T_{1}) * T_{2}\) \(P_2 = (227,527\;\text{Pa} / 266.483\;\text{K}) * 295.372\;\text{K}\) \(P_2 = 252,838\;\text{Pa}\) The new pressure in the tires is \(252,838\;\text{Pa}\) or approximately \(253\;\text{kPa}\).

Calculate the initial and final moles of air

To find the initial moles of air in the tire, we can use the ideal gas law: \(PV = n_1RT_1\) Solving for the initial moles, \(n_1\), we have: \(n_1 = \dfrac{P_1V}{RT_1}\) To find the final moles of air after releasing some air to bring the pressure back to the recommended value, we use the ideal gas law again with the new pressure: \(PV = n_2RT_2\) Solving for the final moles, \(n_2\), we have: \(n_2 = \dfrac{P_1V}{RT_2}\)

Calculate the percentage of the original mass of air released

To find the percentage of the original mass of air released, we can use the following formula: Percentage released = \(\frac{n_1-n_2}{n_1}\times 100\) Substituting the values from the previous step, we have: Percentage released = \(\frac{(\dfrac{P_1V}{RT_1}) - (\dfrac{P_1V}{RT_2})}{ (\dfrac{P_1V}{RT_1})}\times 100\) Since \(P_1V\) is common in each term, they will cancel out: Percentage released = \(\frac{1 - (\dfrac{T_1}{T_2})}{1}\times 100 = (1 - \dfrac{266.483\;\text{K}}{295.372\;\text{K}})\times 100\) Percentage released = \(\approx 9.75\%\) By letting out air to bring the tire pressure back to the recommended 33.0 psi, approximately 9.75% of the original mass of air in the tires will be released.

Key concepts

SI Units Conversion

Understanding the SI units conversion is critical for students and professionals alike, especially when dealing with the ideal gas law and physics problems in general. SI units, or International System of Units, provide a standard that enables clear communication of quantities. For instance, in the textbook exercise about car tire pressure, SI units enabled us to convert psi to pascals and Fahrenheit to Kelvin, ensuring compatibility with the ideal gas law.

Conversions often involve multiplying by a conversion factor, like converting pressure from psi to pascals by using the factor 6,894.757 Pa/psi. Similarly, temperature conversion requires a formula, converting Fahrenheit to Kelvin by adjusting the zero point to absolute zero and scaling from the Fahrenheit to the Celsius scale before adding 273.15 for Kelvin. Practical use of these conversions enables accurate applications across diverse fields like meteorology, automotive maintenance, and aerodynamics.

Pressure-Volume Temperature Relation

The relationship between pressure, volume, and temperature of a gas is elegantly described by the ideal gas law. When a gas is sealed within a car tire, as in our exercise, its behavior is congruent with this law, provided the gas acts ideally.

The law is most commonly expressed as PV=nRT, stating that the product of pressure (P) and volume (V) equals the product of the number of moles of gas (n), the ideal gas constant (R), and the temperature in Kelvin (T). When the volume and number of moles remain constant, as with a tire's fixed volume, the law simplifies to a direct proportionality between pressure and temperature. This P/T relation is invaluable, not just for adjusting tire pressure but also for understanding atmospheric conditions, operating engines, and managing various scientific experiments.

Gas Law Applications

The applications of the ideal gas law permeate many aspects of science and engineering. As seen in the tire pressure problem, one can determine the change in pressure due to temperature alterations—a principle that's important in predicting how materials behave under different conditions. These calculations are not only relevant to automotive scenarios but also extend to designing pressure vessels, evaluating the atmosphere in weather systems, and controlling conditions in laboratory experiments.

In medicine, the law assists in the management of respiratory systems by accounting for lung pressure-volume changes. Engineers apply it when developing propulsion systems, where fuel and oxidizer mixtures are closely monitored for pressure consistency. Moreover, the ideal gas law plays a role in environmental science by helping to predict the spread of pollutants in the atmosphere. Regardless of the field, understanding gas behavior through this law is indispensable for controlling and predicting the interaction between temperature, pressure, and volume.