Problem 7 - University Physics with Modern Physics

Question

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 \(\mathrm{cm}^{3}\) of air at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right)\) and a temperature of \(27.0^{\circ} \mathrm{C}\) At the end of the stroke, the air has been compressed to a volume of 46.2 \(\mathrm{cm}^{3}\) and the gauge pressure has increased to \(2.72 \times 10^{6}\) Pa. Compute the final temperature.

Step-by-Step Solution

Answer

The final temperature is approximately 1718 K.

Step 1: Identify the Initial Conditions

The initial volume, pressure, and temperature of the gas are provided. The initial volume \( V_1 = 499 \, \mathrm{cm}^3 \), the initial pressure \( P_1 = 1.01 \times 10^5 \, \mathrm{Pa} \), and the initial temperature \( T_1 = 27.0^\circ \mathrm{C} \). We need to convert the temperature to Kelvin: \( T_1 = 27.0 + 273.15 = 300.15 \, \mathrm{K} \).

Step 2: Identify the Final Conditions

The problem provides the final volume \( V_2 = 46.2 \, \mathrm{cm}^3 \) and the final gauge pressure \( P_2 = 2.72 \times 10^6 \, \mathrm{Pa} \). The total final pressure is the atmospheric pressure plus the gauge pressure: \( P_2 = 2.72 \times 10^6 + 1.01 \times 10^5 = 2.821 \times 10^6 \, \mathrm{Pa} \).

Step 3: Use the Combined Gas Law

The combined gas law relates pressure, volume, and temperature: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We are solving for \( T_2 \), the final temperature.

Step 4: Solve for Final Temperature \( T_2 \)

Start by rearranging the equation to solve for \( T_2 \): \( T_2 = \frac{P_2 \cdot V_2 \cdot T_1}{P_1 \cdot V_1} \). Substitute the values: \[ T_2 = \frac{(2.821 \times 10^6 \, \mathrm{Pa}) \times (46.2 \, \mathrm{cm}^3) \times (300.15 \, \mathrm{K})}{(1.01 \times 10^5 \, \mathrm{Pa}) \times (499 \, \mathrm{cm}^3)} \]. Calculate \( T_2 \) to find the final temperature.

Step 5: Calculate \( T_2 \) Numerically

Perform the calculation: \[ T_2 = \frac{(2.821 \times 10^6) \times 46.2 \times 300.15}{1.01 \times 10^5 \times 499} \approx 1717.77 \mathrm{K} \].

Key Concepts

Ideal Gas Law

The Ideal Gas Law is an essential principle in thermodynamics that describes the behavior of an ideal gas. This law combines several simpler gas laws, including Boyle’s Law, Charles’s Law, and Avogadro's Law, into a single equation. The equation is presented as:

• \( PV = nRT \)

Where:

• \( P \) stands for pressure.
• \( V \) is the volume.
• \( n \) represents the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature, measured in Kelvin.

This law helps predict how a gas will respond to changes in pressure, volume, and temperature. In practice, while real gases show slight deviations from this law, especially at high pressures and low temperatures, the Ideal Gas Law serves as a reliable approximation in many situations. It is crucial for understanding fundamental gas behaviors and is essential for various applications, such as calculating changes during gas compression.

Gas Compression

Gas compression is a process where the volume of gas is reduced, which in turn increases its pressure, according to Boyle's Law. During compression, when the volume decreases, the molecules have less space to move around, leading to increased collisions with the walls of the container. This results in a higher pressure.
In the initial step of solving the problem, we note the initial and final conditions of the gas, such as volume and pressure. Here, the engine's cylinder compresses air from 499 cm³ to 46.2 cm³, while the pressure rises from 1.01 x 10⁵ Pa to 2.821 x 10⁶ Pa.
It's important to correct gauge pressure to absolute pressure by adding atmospheric pressure, as pressure differences are crucial in calculating the effects of gas compression using thermodynamic principles. This knowledge allows us to understand how engines, like those in cars, manage air-gas mixtures to perform work efficiently.

Temperature Calculation

Calculating the temperature change during gas compression involves using the Combined Gas Law. This version of the Ideal Gas Law doesn't require the quantity of moles to remain constant and can measure changes while keeping one of the properties fixed.
The Combined Gas Law is depicted as:

• \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

In this exercise, the relationship between initial and final states of pressure \( (P) \), volume \( (V) \), and temperature \( (T) \) is used to determine the unknown. The formula is rearranged to isolate \( T_2 \), the final temperature:

• \( T_2 = \frac{P_2 \cdot V_2 \cdot T_1}{P_1 \cdot V_1} \)

This calculation provides the temperature at the end of the compression process. In our problem, the final temperature, initially sourced from ambient conditions (27.0 °C converted to 300.15 K) to a calculated result of approximately 1717.77 K, reflects how significantly gas compression can heat air within engines.