Problem 13 - Chemistry and Chemical Reactivity

Question

One of the cylinders of an automobile engine has a volume of \(400 . \mathrm{cm}^{3} .\) The engine takes in air at a pressure of 1.00 atm and a temperature of \(15^{\circ} \mathrm{C}\) and compresses the air to a volume of \(50.0 \mathrm{cm}^{3}\) at \(77^{\circ} \mathrm{C} .\) What is the final pressure of the gas in the cylinder? (The ratio of before and after volumesin this case, 400: 50 or \(8: 1-\) is called the compression ratio.

Step-by-Step Solution

Answer

The final pressure is approximately 9.72 atm.

Step 1: Understand the Problem and Given Data

You start with the initial state of an ideal gas in the cylinder with a volume of \( V_1 = 400 \, \text{cm}^3 \), pressure \( P_1 = 1.00 \, \text{atm} \), and temperature \( T_1 = 15^{\circ} \text{C} \). This gas is compressed to a volume of \( V_2 = 50 \, \text{cm}^3 \) and temperature \( T_2 = 77^{\circ} \text{C} \). We need to calculate the final pressure \( P_2 \).

Step 2: Convert Temperatures to Kelvin

The gas law calculations require temperature in Kelvin. Convert the initial and final temperatures: \[ T_1 = 15 + 273.15 = 288.15 \, \text{K} \] \[ T_2 = 77 + 273.15 = 350.15 \, \text{K} \]

Step 3: Use the Ideal Gas Law Relation

The ideal gas law states that \( PV = nRT \). For a given amount of gas, \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]. Rearrange to find \( P_2 \): \[ P_2 = P_1 \times \frac{V_1}{V_2} \times \frac{T_2}{T_1} \].

Step 4: Substitute Known Values

Substitute the known values into the equation: \[ P_2 = 1.00 \times \frac{400}{50} \times \frac{350.15}{288.15} \].

Step 5: Calculate the Final Pressure

Calculate \( P_2 \): - Calculate the volume ratio: \( \frac{400}{50} = 8 \).- Calculate the temperature ratio: \( \frac{350.15}{288.15} \approx 1.215 \).- Therefore, \( P_2 = 1.00 \times 8 \times 1.215 \approx 9.72 \, \text{atm} \).

Key Concepts

Compression Ratio

In the context of an engine, the compression ratio is a critical concept that helps in understanding how much the air-fuel mixture is compressed in a cylinder. It is the ratio of the volume of the cylinder when the piston is at its lowest point (bottom dead center) to the volume when the piston is at its highest point (top dead center). This is an important factor in engine efficiency.
For the given problem, the initial and final volumes are essential to compute the compression ratio. The volumes are 400 cm³ to 50 cm³ respectively. Thus, the compression ratio can be calculated as:

• Initial volume = 400 cm³ (when the piston is at bottom dead center)
• Compressed volume = 50 cm³ (when the piston is at top dead center)

Hence, the compression ratio is 8:1. A higher compression ratio often indicates more power output and efficiency, but it may require higher octane fuel to prevent knocking.

Gas Pressure Calculation

Understanding how to calculate the pressure of a gas after compression involves utilizing the Ideal Gas Law. This law combines pressure, volume, and temperature to describe the state of an ideal gas. In this scenario, we calculate the final pressure after compression using this key equation:

• Ideal Gas Law: \( PV = nRT \)

For a closed system with a constant amount of gas, we use the relation: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \).
By rearranging this equation, we solve for the final pressure \( P_2 \) as follows:

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

This equation shows that the final pressure\( P_2 \) depends on the initial pressure \( P_1 \), the initial and final volumes \( V_1 \) and \( V_2 \), and the initial and final temperatures \( T_1 \) and \( T_2 \). Performing the calculations, given the ratios provided in the problem, yields a final pressure of approximately 9.72 atm.

Temperature Conversion to Kelvin

Converting temperatures to Kelvin is crucial when dealing with the Ideal Gas Law because it is based on absolute temperature. The Kelvin scale starts at absolute zero, the point at which particle motion stops, making it ideal for gas calculations.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
In this problem:

• Initial temperature: 15°C becomes \( 15 + 273.15 = 288.15 \, K \)
• Final temperature: 77°C becomes \( 77 + 273.15 = 350.15 \, K \)

This conversion is necessary as it allows for a straightforward application in mathematical formulas, ensuring accurate and consistent results when calculating changes involving temperature.