The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3.7-m-diameter cylinder. The air inside, at cruising altitude, is 20°C at a pressure of 82 kPa. What volume of outside air, at −40°C and a pressure of 23 kPa, must be drawn in, heated, and compressed to fill the plane?

검증된 단계별 안내

Step 1: Calculate the volume of the interior of the Boeing 737-800 using the formula for the volume of a cylinder: \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length). The radius can be found by dividing the diameter by 2.

Step 2: Use the Ideal Gas Law \( PV = nRT \) to determine the number of moles of air \( n \) inside the plane at cruising altitude. Rearrange the equation to \( n = \frac{PV}{RT} \), where \( P \) is the pressure, \( V \) is the volume, \( R \) is the universal gas constant (8.314 J/(mol·K)), and \( T \) is the temperature in Kelvin.

Step 3: Convert the temperature inside the plane from Celsius to Kelvin using \( T(K) = T(°C) + 273.15 \). Similarly, convert the outside air temperature from Celsius to Kelvin.

Step 4: Use the Ideal Gas Law again to calculate the volume of outside air needed to provide the same number of moles \( n \) as calculated in Step 2. Rearrange the equation to \( V = \frac{nRT}{P} \), where \( P \) is the pressure of the outside air, \( T \) is the temperature of the outside air, and \( n \) is the number of moles determined earlier.

Step 5: Substitute the values for \( n \), \( R \), \( T \), and \( P \) into the equation from Step 4 to find the volume of outside air required. Ensure all units are consistent (e.g., pressure in Pascals, volume in cubic meters, temperature in Kelvin).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how gases behave under different conditions, allowing us to calculate the volume of air needed based on changes in temperature and pressure.

Volume of a Cylinder

The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height (or length). In this context, it helps determine the volume of the Boeing 737-800's interior, which is crucial for understanding how much air needs to be processed.

Thermodynamics

Thermodynamics is the study of energy transfer and the laws governing heat and work. In this scenario, it is important to consider how the outside air will be heated and compressed, as these processes will affect the air's volume and pressure, impacting the calculations needed to fill the plane.