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 \(2.72 \times 10^{5}\) Pa. Compute the final temperature.

Step-by-Step Solution

Answer

The final temperature is approximately 906.1 K.

Step 1: Convert Temperature to Kelvin

The initial temperature is given as 27.0°C. We first convert this temperature to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \]Thus, the initial temperature is \( T_i = 27.0 + 273.15 = 300.15 \) K.

Step 2: Identify Known Values

List down all known values:- Initial volume \( V_i = 499 \) cm³- Final volume \( V_f = 46.2 \) cm³- Initial pressure \( P_i = 1.01 \times 10^5 \) Pa- Gauge pressure \( P_g = 2.72 \times 10^5 \) Pa- Atmospheric pressure contributes to the total, making final pressure \( P_f = P_g + 1.01 \times 10^5 = 3.73 \times 10^5 \) Pa.

Step 3: Use Ideal Gas Law (Relation between states)

According to the ideal gas law for constant number of moles:\[ \frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f} \]Substitute all known values to find \( T_f \):\[ T_f = \frac{P_f V_f T_i}{P_i V_i} \]

Step 4: Calculate Final Temperature

Plug the values into the formula:\[ T_f = \frac{3.73 \times 10^5 \, \mathrm{Pa} \times 46.2 \, \mathrm{cm}^3 \times 300.15 \, \mathrm{K}}{1.01 \times 10^5 \, \mathrm{Pa} \times 499 \, \mathrm{cm}^3} \]Calculate to find \( T_f \approx 906.1 \, \mathrm{K} \).

Key Concepts

Compression Stroke

When an engine operates, it goes through several cycles, and one of these crucial phases is the compression stroke. This step involves the compression of the air-fuel mixture in an engine's cylinder. In the case of the Jaguar XK8 convertible, during the compression stroke, the volume of air in the cylinder reduces significantly, going from 499 cm³ to just 46.2 cm³.

• The primary purpose of this stroke is to increase the pressure and temperature of the air-fuel mixture, making it more efficient for subsequent combustion.
• As the piston moves upward, it compresses the air, decreasing its volume and increasing its pressure.
• The compression of this air ensures that when the fuel is ignited, it combusts more explosively, maximizing the power output.

This initial compression is critical for an engine's efficiency, effectiveness, and overall performance, as it dictates how much power can be extracted from the combustion process.

Final Temperature Calculation

To find the final temperature in the cylinder after compression, we apply the ideal gas law, which provides a relation between pressure, volume, and temperature. This relationship is crucial as it helps us understand how these variables interact within a closed system like a car engine's cylinder.

Converting Temperature

Initially, the temperature needs to be in Kelvin, which is suitable for thermodynamic calculations. The given initial temperature in Celsius is converted to Kelvin by adding 273.15, resulting in 300.15 K.

Using the Ideal Gas Law

To compute the final temperature, we use the formula that relates initial and final conditions: \[\frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f}\]
By arranging the formula to solve for the final temperature \( T_f \), and substituting the known values into this equation, we get:\[T_f = \frac{3.73 \times 10^5 \, \mathrm{Pa} \times 46.2 \, \mathrm{cm}^3 \times 300.15 \, \mathrm{K}}{1.01 \times 10^5 \, \mathrm{Pa} \times 499 \, \mathrm{cm}^3}\]
Completing the calculation provides the final temperature of approximately 906.1 K.

Pressure and Volume Relationship

In an engine like the Jaguar XK8's, the relationship between pressure and volume is defined by physical principles stated in the ideal gas law. This relationship helps understand how variations in volume and pressure directly influence temperature during the compression stroke.

• According to the law, when the volume of a gas is decreased, as happens during the compression stroke, its pressure increases if the temperature and the amount of gas remain constant.
• This is expressed by Boyle’s Law, a part of the ideal gas equation, stating \( P_1 V_1 = P_2 V_2 \) for a constant temperature.
• However, in a practical scenario inside an engine, both pressure and temperature increase, following the combined form of the ideal gas law: \( PV = nRT \).

Understanding this relationship allows automotive engineers and physicists to design engines that effectively harness the energy produced during the explosion of fuel, combining mechanical elegance with thermodynamic efficiency.