Question

The pressure of an automobile tire is measured to be \(190 \mathrm{kPa}(\text { gage })\) before a trip and \(215 \mathrm{kPa}(\text { gage })\) after the trip at a location where the atmospheric pressure is 95 kPa. If the temperature of air in the tire before the trip is \(25^{\circ} \mathrm{C},\) the air temperature after the trip is \((a) 51.1^{\circ} \mathrm{C}\) (b) \(64.2^{\circ} \mathrm{C}\) \((c) 27.2^{\circ} \mathrm{C}\) \((d) 28.3^{\circ} \mathrm{C}\) \((e) 25.0^{\circ} \mathrm{C}\)

Short answer

a) 51.1°C b) 58.3°C c) 65.7°C d) 71.9°C Answer: a) 51.1°C

Step-by-step solution

Identify given information and convert to absolute pressure

We are given the initial gage pressure \(P_1 = 190\,\text{kPa (gage)}\), the final gage pressure \(P_2 = 215\,\text{kPa (gage)}\), the atmospheric pressure \(P_{atm} = 95\,\text{kPa}\), and the initial air temperature \(T_1 = 25\, ^{\circ}\text{C}\). We need to find the final air temperature \(T_2\) in the tire. First, we should convert the initial and final gage pressures to absolute pressures by adding the atmospheric pressure: \(P_{abs1} = P_1 + P_{atm} = 190\,\text{kPa} + 95\,\text{kPa} = 285\,\text{kPa}\) \(P_{abs2} = P_2 + P_{atm} = 215\,\text{kPa} + 95\,\text{kPa} = 310\,\text{kPa}\)

Convert temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin, using the formula \(T_{K} = T_{C} + 273.15\). \(T_{1K} = 25^{\circ} \mathrm{C} + 273.15 = 298.15\,\text{K}\)

Apply the ideal gas law

We'll assume the air in the tire behaves like an ideal gas and the amount of air inside the tire doesn't change throughout the trip. Therefore, we can apply the ideal gas law: \(P_1 V_1 / T_1 = P_2 V_2 / T_2\) Since the amount of air and tire volume remain constant during the trip, \(V_1=V_2\). Hence, we can write the equation as: \(P_1 / T_1 = P_2 / T_2\)

Solve for the final temperature

Now, we can solve for the final air temperature \(T_2\): \(T_2 = \frac{P_2}{P_1} \cdot T_1\) \(T_2 = \frac{310\,\text{kPa}}{285\,\text{kPa}} \cdot 298.15\,\text{K} \approx 322.26\,\text{K}\)

Convert the temperature back to Celsius

Finally, we'll convert the temperature back to Celsius using the formula \(T_{C} = T_{K} - 273.15\): \(T_2 \approx 322.26\,\text{K} - 273.15 = 49.11^{\circ} \mathrm{C}\) The final air temperature is closest to \((a)\: 51.1^{\circ} \mathrm{C}\), so the correct answer is (a).

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. An essential part of thermodynamics is the study of thermal systems and how their states can change in response to variations in their environment. Thermodynamics is built upon three fundamental laws.

The First Law, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form into another. In the case of the automobile tire problem, when the tire's temperature increases after a trip, it suggests that the work done by the car onto the tire (through friction and air compression) has been transformed into thermal energy, leading to an increase in the tire's temperature.

The Second Law introduces the concept of entropy and states that total entropy, or disorder, always increases in an isolated system. This helps us understand why heat flows from hot to cold and why certain processes are irreversible.

The Third Law of Thermodynamics states that as the temperature approaches absolute zero, the entropy of a perfect crystal approaches zero as well. This law is more theoretical in nature and does not directly apply to everyday situations like the tire problem.

Absolute Pressure

Absolute pressure is a vital concept in the study of gases and fluids. It is the actual pressure at a point, measured relative to the zero pressure of a complete vacuum. On the other hand, gauge pressure is the pressure relative to the atmospheric pressure. In other words, gauge pressure is what is read on most pressure measuring devices and does not account for atmospheric pressure.

To get absolute pressure (denoted as \( P_{abs} \)), one must add the gauge pressure (\( P_{gage} \)) to the atmospheric pressure (\( P_{atm} \)). The formula is: \( P_{abs} = P_{gage} + P_{atm} \). This conversion is crucial in applying the ideal gas law correctly, as it requires the use of absolute pressure. In our exercise, understanding the difference between gauge and absolute pressure was necessary to assess accurately the change in temperature of the air within the tire due to changing conditions.

Temperature Conversion

Temperature conversion is a crucial process in many scientific calculations, including those involving the ideal gas law. To work with the ideal gas law, temperatures must be in Kelvin, which is an absolute temperature scale. This is because Kelvin is directly proportional to the average kinetic energy of the particles in a substance.

The conversion from Celsius (°C) to Kelvin (K) is straightforward: \( T_K = T_C + 273.15 \). Similarly, to convert back from Kelvin to Celsius, use: \( T_C = T_K - 273.15 \). Remember, there is no such thing as a negative temperature in Kelvin, as it starts from absolute zero, which is the theoretical point where particles have minimal thermal motion.

In our textbook example, the temperature of the air in the tire needed to be converted from Celsius to Kelvin before applying the ideal gas law. After calculating the final temperature in Kelvin, converting it back to Celsius was necessary to answer the question posed in the problem. This process exemplifies how temperature conversion is essential in thermodynamics.