Question

The gage pressure of an automobile tire is measured to be \(200 \mathrm{kPa}\) before a trip and \(220 \mathrm{kPa}\) after the trip at a location where the atmospheric pressure is 90 kPa. Assuming the volume of the tire remains constant at \(0.035 \mathrm{m}^{3},\) determine the percent increase in the absolute temperature of the air in the tire.

Short answer

Answer: The percent increase in the absolute temperature of the air in the tire is approximately 6.90%.

Step-by-step solution

Convert gage pressure to absolute pressure

We must first convert the given gage pressures into absolute pressures. To do this, add the given atmospheric pressure to both initial gage pressure and final gage pressure: \(P_1 = 200\,\text{kPa} + 90\,\text{kPa} = 290\,\text{kPa}\) \(P_2 = 220\,\text{kPa} + 90\,\text{kPa} = 310\,\text{kPa}\)

Apply the proportion of pressures and temperatures

Since the ratio P/T is constant in this case, we can say: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) \(T_2 = T_1 \frac{P_2}{P_1}\) Plug in the values of \(P_1\) and \(P_2\): \(T_2 = T_1 \frac{310\,\text{kPa}}{290\,\text{kPa}}\)

Calculate the percentage increase in temperature

To find the percent increase in temperature, we can use the following formula: \(\text{Percent increase} = \frac{T_2 - T_1}{T_1} \times 100\%\) First, let's rewrite the expression for \(T_2\) in terms of \(T_1\): \(T_2 = T_1 \frac{310}{290}\) Now substitute this expression into the percent increase formula: \(\text{Percent increase} = \frac{\left(T_1 \frac{310}{290}\right) - T_1}{T_1} \times 100\%\) This simplifies to: \(\text{Percent increase} = \frac{310}{290} \times 100\% - 100\%\)

Calculate the final percent increase

Calculate the final percentage increase in the temperature of the air in the tire: \(\text{Percent increase} = \frac{310}{290} \times 100\% - 100\%\) \(\text{Percent increase} \approx 6.90\%\) The percent increase in the absolute temperature of the air in the tire is approximately 6.90%.

Key concepts

Thermodynamics

Thermodynamics is the study of energy transformations, particularly the conversion of thermal energy into other forms of energy and vice versa. One of the most important principles in thermodynamics is the ideal gas law, which helps us understand how gases will behave under different conditions of pressure, volume, and temperature.

The ideal gas law is often expressed as PV = nRT, where P stands for pressure, V for volume, n for the amount of substance (in moles), R for the ideal gas constant, and T for temperature in Kelvin. The relationship described by this equation is crucial in many practical situations, such as when we're looking at the changes in an automobile tire's pressure and temperature before and after a trip, like in our exercise.

Absolute Pressure Calculation

When we talk about pressure in thermodynamics, it's important to differentiate between gage pressure and absolute pressure. Gage pressure is the pressure measured relative to the ambient atmospheric pressure, while absolute pressure is the sum of the gage pressure and the atmospheric pressure.

In our example with a tire's gage pressure, we need to add the atmospheric pressure to obtain the absolute pressure. This calculation is fundamental when using the ideal gas law since it operates with absolute pressures. It is how we step from the relative measurement (what the tire's pressure gauge reads) to an absolute value that can be used in further calculations.

Ideal Gas Law Application

The ideal gas law is a fundamental equation in thermodynamics, allowing us to model how a gas behaves under various conditions. Its application is crucial for solving our exercise, as it relates pressure (P) and temperature (T) of a gas in a constant volume setting.

When applying the ideal gas law, it's basic to ensure all units are consistent and that temperature is in Kelvin. In situations where the volume and amount of gas remain constant, such as in an inflated tire during a trip, the law simplifies to the direct proportionality between pressure and temperature (P/T = constant). This property allows us to compare the initial and final states of the tire's gas and determine how temperature changes as pressure changes.

Percentage Increase in Temperature

The percentage increase in temperature is a way to express the relative change in temperature as a percentage. To calculate it, one needs to find the final temperature, determine the difference from the initial temperature, and put it in relation to the initial temperature.

The percentage increase helps provide a clearer picture of the magnitude of temperature change, which is particularly useful in many practical situations, including engineering and environmental studies. In our tire example, the final percentage increase indicates how much warmer the air inside the tire became after the trip. Interpreting percentage changes is instrumental for professionals who need to ensure safety and performance, such as verifying that the tire’s temperature stays within safe limits.