In many meteorological and chemical applications the internal energy is not the most ideal state function for describing energetic changes during transitions. The classical form of the First Law is especially useful for transitions in which the volume is held fixed (dV = 0) since the volume-work term vanishes, but in atmospheric applications most changes in the state of a parcel occur either adiabatically or isobarically. Hence, it becomes convenient to introduce a new function of state called the enthalpy, H, defined by
Take the differential to obtain dH = dU + p dV + V dp.
After substituting the earlier form of the First Law in terms of the internal energy we obtain dH = -Q + V dp [enthalpy form of the First Law].
Very often atmospheric processes take place at a fixed pressure (altitude). These include heating of a parcel by solar radiation at a particular altitude, condensation heating, and contact heating at the surface. In this case the enthalpy is a very convenient function to describe the parcel's thermodynamic state. Note that the change in enthalpy under a constant pressure process is just
Integrating the last equation:
where F(V) is an arbitrary function of volume appearing here as an integration constant. We can find this function by noting that U = McvT and using the definition of H, (3.45), to obtain
[enthalpy for an ideal gas].
In other words, the arbitrary function F(V) is identically zero for the ideal gas. The adiabatic process is expressed as and for the ideal gas, and
Tp which will quickly lead us to Poisson's equation (3.36).
Calculus refresher: partial derivatives Thermodynamic functions nearly always involve more than one variable as we have seen already, e.g., V(T,p). The "partial" of V(T,p) with respect to p holding T constant is defined by
In most fields the subscript T following the large parentheses is omitted, but in thermodynamics it is conventional (and useful) to retain this reminder of which variable is being held constant. Sometimes especially in mathematics and physics, the partial derivative is denoted by a subscript. For example, let f be a function of x and y, then df /dx = fx, etc. You simply take the ordinary derivative but hold all variables constant except the one being varied. For example, take the ideal gas: V = MRT/p, then (dV/dp)T = -MRT/p2.
This is a good time to search out your old calculus book and review the chapter on partial differentiation. An important result to remember is that if we go to second partial derivatives such as fxx or fxy, the order does not matter:
The differential off (x, y) is df = fx dx + fy dy. (3.56)
If we divide through by dx and set dy to zero we obtain
Suppose the function f (x, y) is held constant. Then fx dx + fy dy = 0 (3.58)
The notation in the last equation will be encountered often.
Example3.21 A1 kg parcel is heated at the surface (p = 1000hPa)ataratedQ/dt =
20Wkg-1 (W = watts). What is the rate of change of enthalpy?
Answer: Note that dp/dt = 0. Then, dH/dt = dQ/dt. □
Example 3.22 A parcel moves along an isobaric surface (constant pressure) and is heated at a rate dQM/dt = 10Wkg-1. What is the rate of change of T along the path of motion?
Answer: (dH/dt) = McpdT/dt = dQM/dt; dT/dt = (dQM/dt)/cp = 10Wkg-1/1004 J-1 kg-1 = 0.01 Ks-1. □
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