/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 The adiabatic lapse rate is the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 70

The adiabatic lapse rate is the rate at which air cools as it rises and expands adiabatically in the atmosphere (see Application: Smog Alert, on page 302 ). Express \(d T\) in terms of \(d p\) for an adiabatic process, and use the hydrostatic equation (Equation 15.2 ) to express \(d p\) in terms of \(d y .\) Then, calculate the lapse rate \(d T / d y .\) Take air's average molecular weight to be 29 u and \(\gamma=1.4,\) and remember that the altitude \(y\) is the negative of the depth \(h\) in Equation 15.2.

Short Answer

Expert verified
The lapse rate, which describes how the temperature varies with height, is approximately \(dT/dy = -0.0098/(T)\) K/m for an adiabatic process. This expression shows that the lapse rate is inversely proportional to the temperature, such that the temperature decreases faster at lower temperatures.

Step by step solution

01

Calculate dT in terms of dp

An adiabatic process is a thermodynamic process where no heat is added or removed. For ideal gases, we can apply Poisson's equation which states that \(T \cdot p^{(\gamma-1)/\gamma} = constant\). When we differentiate this expression, we get: \(dT/dp = -(\gamma-1)p^{(1-\gamma)/\gamma} / T\)
02

Substitute dp in terms of dy

We use the hydrostatic equation to substitute \(dp\) in terms of \(dy\). The hydrostatic equation is \(dp = - \rho g dh\), whereas here \(dh = -dy\). Thus, the hydrostatic equation in this context is \(dp = \rho g dy\). Therefore, \(dT/dp\) can be written as \(dT/dy = -(\gamma-1)\rho g / (\gamma T)\)
03

Calculate the lapse rate

To find the lapse rate \(dT/dy\), we need to substitute \(\rho\) with the ideal gas law, and apply the given constants. The density \(\rho = p/(RT)\), where \(R = 8.314 J/(mol \cdot K)\) is the ideal gas constant. So, we can plug this in to get: \(dT/dy = -(\gamma-1) p g / (\gamma R T^2) = -(\gamma-1) g / (\gamma R T)\). Given \(\gamma=1.4\) and \(R = 8.314 J/(mol \cdot K)\), we then get: \(dT/dy = -(1.4-1)*9.8 / (1.4*8.314*T)\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamic Processes
Thermodynamic processes encompass the various ways that energy can be transferred in or out of a system. One key type of process relevant to meteorology and atmospheric science is the adiabatic process. During an adiabatic process, a parcel of air changes temperature without gaining or losing heat to its surroundings. Instead, the temperature changes solely because of work done by or on the parcel during expansion or compression. This principle is crucial for understanding why the temperature of the air decreases as it rises and expands in the atmosphere.

Adiabatic cooling, as discussed in the context of the adiabatic lapse rate, happens because the air does work on its surroundings as it expands, leading to a decrease in temperature. Conversely, adiabatic warming occurs when air compresses as it descends, increasing its temperature. In weather patterns, these principles help explain phenomena like the formation of clouds and the behavior of air masses as they move across different elevations.
Hydrostatic Equation
The hydrostatic equation represents a fundamental relationship in fluid dynamics, particularly in the study of atmospheres. It conveys the balance of forces in a column of fluid - or in our case, a column of air in Earth's atmosphere. Mathematically, the equation can be expressed as \(d p = - \rho g d h\)), where \(d p\)) is the infinitesimal change in pressure, \(\rho\)) is the fluid density, \(g\)) is the acceleration due to gravity, and \(d h\)) refers to the infinitesimal change in height.

Applied to the atmosphere, the negative sign indicates that as one moves up through the atmosphere (increasing \(d h\))), the pressure decreases (resulting in a negative \(d p\))). This relationship is pivotal in the calculation of the adiabatic lapse rate since it connects changes in atmospheric pressure with changes in altitude. Understanding how pressure varies with altitude allows meteorologists to better predict weather patterns and the behavior of atmospheric constituents.
Ideal Gas Law
The ideal gas law is a keystone equation in thermodynamics and physical chemistry which relates pressure, volume, temperature, and the amount of substance for an ideal gas. It is concisely presented as \(PV = nRT\)), where \(P\)) is pressure, \(V\)) is volume, \(n\)) is the number of moles of gas, \(R\)) is the ideal gas constant, and \(T\)) is absolute temperature in Kelvin. In the context of atmospheric sciences and the adiabatic lapse rate calculation, the ideal gas law allows us to link atmospheric pressure and temperature with the density of air (\rho), which is essential to find the lapse rate.

When solving for the adiabatic lapse rate, we use a modified form of the ideal gas law to express density as \(\rho = \frac{p}{RT}\)). By substituting this into the adiabatic lapse rate formula, we get a way to calculate temperature changes with altitude without directly measuring the density of air, which varies depending on temperature and pressure. The ideal gas law thus provides a critical bridge connecting molecular properties of gases to large-scale atmospheric behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Experimental studies show that the \(p V\) curve for a frog's lung can be approximated by \(p=10 v^{3}-67 v^{2}+220 v,\) with \(v\) in \(\mathrm{mL}\) and \(p\) in \(\mathrm{Pa}\). Find the work done when such a lung inflates from zero to \(4.5 \mathrm{mL}\) volume.

An 8.5 -kg rock at \(0^{\circ} \mathrm{C}\) is dropped into a well-insulated vat containing a mixture of ice and water at \(0^{\circ} \mathrm{C}\). When equilibrium is reached, there are \(6.3 \mathrm{g}\) less ice. From what height was the rock dropped?

Your class on alternative habitats is designing an underwater habitat. A small diving bell will be lowered to the habitat. A hatch at the bottom of the bell is open, so water can enter to compress the air and thus keep the air pressure inside equal to the pressure of the surrounding water. The bell is lowered slowly enough that the inside air remains at the same temperature as the water. But the water temperature increases with depth in such a way that the air pressure and volume are related by \(p=p_{0} \sqrt{V_{0} / V},\) where \(V_{0}=17 \mathrm{m}^{3}\) and \(p_{0}=1\) atm are the surface values. Suppose the diving bell's air volume cannot be less than \(8.7 \mathrm{m}^{3}\) and the pressure must not exceed 1.5 atm when submerged. Are these criteria met?

Volvo's B5340 engine, used in the V70 series cars, has compression ratio \(10.2,\) and the fuel-air mixture undergoes adiabatic compression with \(\gamma=1.4 .\) If air at \(320 \mathrm{K}\) and atmospheric pressure fills an engine cylinder at its maximum volume, what will be (a) the temperature and (b) the pressure at the point of maximum compression?

A gas expands isothermally from state \(A\) to state \(B,\) in the process absorbing 35 J of heat. It's then compressed isobarically to state \(C,\) where its volume equals that of state \(A .\) During thecompression, \(22 \mathrm{J}\) of work are done on the gas. The gas is then heated at constant volume until it returns to state \(A\). (a) Draw a \(p V\) diagram for this process. (b) How much work is done on or by the gas during the complete cycle? (c) How much heat is transferred to or from the gas as it goes from \(B\) to \(C\) to \(A\) ?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Linear Momentum

Read Explanation

Electricity and Magnetism

Read Explanation

Classical Mechanics

Read Explanation

Work Energy and Power

Read Explanation

Electric Charge Field and Potential

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.