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.

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.

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.

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.