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Ch. 18 - Kinetic Theory of Gases
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 18 - Kinetic Theory of GasesProblem 69
Chapter 18, Problem 69

A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

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Convert the speed of the space vehicle from km/h to m/s. Use the conversion factor: 1 km/h = 1000/3600 m/s. This will give the speed in SI units, which is necessary for further calculations.
Relate the kinetic energy of the nitrogen molecules to the temperature using the equation for the average kinetic energy of a molecule: Ek = (3/2)kT, where k is the Boltzmann constant and T is the temperature.
Equate the kinetic energy of the nitrogen molecules to the macroscopic kinetic energy of the vehicle. The macroscopic kinetic energy is given by Ek = (1/2)m², where m is the mass of a nitrogen molecule and v is the speed of the vehicle.
Solve for the temperature T by substituting the expression for kinetic energy into the equation Ek = (3/2)kT. Rearrange to isolate T: T = (m²)/(3k).
Substitute the known values into the equation: the mass of a nitrogen molecule (approximately 4.65 × 10-26 kg), the Boltzmann constant (1.38 × 10-23 J/K), and the speed of the vehicle in m/s. Perform the calculation to find the temperature.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy and Temperature

Kinetic energy is the energy possessed by an object due to its motion, which is directly related to its speed. The temperature of a gas is a measure of the average kinetic energy of its molecules. As the speed of the molecules increases, so does their kinetic energy, leading to a higher temperature. This relationship is crucial for understanding how the high speed of the space vehicle translates into thermal energy upon atmospheric entry.
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Heat Transfer and Friction

When a space vehicle reenters the Earth's atmosphere, it encounters air resistance, which generates friction. This friction converts the kinetic energy of the vehicle into thermal energy, causing the surface temperature to rise significantly. Understanding the principles of heat transfer, particularly conduction and convection, is essential to grasp how the vehicle's materials must withstand extreme temperatures during reentry.
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Material Science and Thermal Protection

Material science focuses on the properties and applications of materials, particularly under extreme conditions. In the context of space vehicles, thermal protection systems are designed to shield the vehicle from the intense heat generated during reentry. Special materials, such as ablative heat shields, are engineered to absorb and dissipate heat, preventing structural damage and ensuring the safety of the vehicle and its occupants.
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Related Practice
Textbook Question

At room temperature, it takes approximately 2.45 x 10³ J to evaporate 1.00 g of water. Estimate the average speed of evaporating molecules. What multiple of vrms (at 20°C) for water molecules is this? (Assume Eq. 18–4 holds.)

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Textbook Question

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. Can you explain why our atmosphere contains oxygen but not helium?

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Textbook Question

A sauna has 7.8 m³ of air volume, and the temperature is 85°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18–2.)

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Textbook Question

A sample of cesium vapor is in an oven at 400°C. The volume of the oven is 75 cm³, the vapor pressure of Cs at 400°C is 17 mm-Hg, and the diameter of cesium atoms in the vapor is 0.33 nm. Determine the number of collisions a single Cs atom undergoes with other cesium atoms per second.

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Textbook Question

Calculate the total water vapor pressure in the air on the following day: a hot summer day, with the temperature 30°C and the relative humidity at 75%.

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Textbook Question

A scuba tank has a volume of 3100 cm³. For very deep dives, the tank is filled with 50% (by volume) pure oxygen and 50% pure helium. What is the ratio of the average kinetic energies of the two types of molecule?