Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 18: Thermal Properties of Matter

Problem 20

Chapter 18, Problem 20

At an altitude of $11 comma 000$ m (a typical cruising altitude for a jet airliner), the air temperature is $negative 56.5$ °C and the air density is $0.364$ kg/m³ . What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example $18.4$ in Section $18.1$ doesn't apply.)

Verified step by step guidance

Start by converting the given temperature from Celsius to Kelvin. Use the formula: $ T(K) = T(°C) + 273.15 $. Substitute $ T(°C) = -56.5 $ to find the temperature in Kelvin.

Use the ideal gas law to find the pressure. The ideal gas law is given by $ PV = nRT $, where $ P $ is the pressure, $ V $ is the volume, $ n $ is the number of moles, $ R $ is the ideal gas constant, and $ T $ is the temperature in Kelvin.

Rearrange the ideal gas law to solve for pressure: $ P = \frac{nRT}{V} $. Since $ n/V $ is the density ($ \rho $), the equation becomes $ P = \rho RT $.

Substitute the given air density $ \rho = 0.364 \text{ kg/m}^3 $, the ideal gas constant $ R = 287 \text{ J/(kg·K)} $ for air, and the temperature in Kelvin from step 1 into the equation $ P = \rho RT $.

Calculate the pressure using the substituted values. This will give you the atmospheric pressure at the altitude of 11,000 m.

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Key Concepts

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

Ideal Gas Law

The Ideal Gas Law relates pressure, volume, temperature, and the number of moles of a gas through the equation PV = nRT. It is essential for calculating atmospheric pressure at high altitudes, where temperature and density differ from surface conditions. Understanding this law helps in determining how these variables interact in the atmosphere.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air above a given point. It decreases with altitude due to the thinning of air and lower density. Calculating pressure at high altitudes requires understanding how pressure changes with temperature and density, which are different from sea level conditions.

Density and Temperature Relationship

Density and temperature are inversely related in gases; as temperature decreases, density typically increases if pressure is constant. At high altitudes, both temperature and density are crucial for determining pressure using the Ideal Gas Law. Recognizing this relationship helps in understanding atmospheric conditions at different altitudes.

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