**GLOBAL WARMING & AEROSOLS**

Clifford E Carnicom

Feb 23 2004

**FURTHER DISCUSSION**

**The fundamental equations that address the heating of the atmosphere with the introduction
of foreign materials are the following:**

**c _{v} = sum [m_{fi} * c_{vi}]**

**which is the specific heat of a mixture (gravimetric analysis) ^{2}**

**where**

**m _{fi} is the mass fraction of the ith component, and c_{vi} is the specific
heat of the ith component in units of joules / (kg * ^{o}K)**

**and c _{v} is the specific heat of the mixture in units of joules / (kg * ^{o}K)
and ^{o}K is degrees Kelvin.**

**and the heat transfer as given by the first law of thermodynamics ^{3}**

**Q = m * c _{v} * del T**

**where Q represents the change in energy in joules, m is the total mass of the mixture,
and del T is the change in degrees of the mass in degrees Kelvin.**

**Let us assume the atmosphere as a shell around the earth of variable height, the volume
of which is given by:**

**v _{air} = ( 4 / 3) * pi * [ ( R + upper )^{3} - ( R + lower)^{3}
]**

**where v _{air} is the volume of the atmospheric shell in cubic meters, R is the
mean radius of the earth in meters, upper is the upper limit of the atmospheric shell under consideration in meters
(above sea level), and lower is the lower limit of the atmospheric shell in meters (above sea level).**

**Based upon an exponential regression of atmospheric density data in kilograms ^{4},
a suitable model for the mass of a column of air 1 meter square in dimension can be developed in the following
form:**

**m _{air} =
1.474 * exp ^{-1.424E-4 * h} dh**

**integrated with respect to the upper and lower limits of the atmospheric shell, and m _{air}
is the mass of the atmospheric shell in kilograms, and h is in meters.**

**The mass of the aerosol in kilograms within an atmospheric column of air 1 meter square in dimension is expressed
as:**

**m _{a} = d_{a} * (upper - lower)**

**where the density of a particular aerosol in units of kilograms is designated as d _{a}.**

**As the density of the aerosol and the atmosphere will be considered to be uniform throughout the shell considered,
the mass fractions of the atmosphere and the aerosol contribution, respectively, are:**

**mf _{air} = m_{air} / (m_{air} + m_{a})**

**and**

**mf _{a} = m_{a} / (m_{air} + m_{a})**

**Therefore:**

**c _{v} = ( mf_{a} * c_{va} ) + (mf_{air} * c_{vair})**

**where cva and cvair are the constant volume specific heats of the aerosol and air, respectively.**

**since Q = m * c _{v} * del T**

**and since we are interested in the change in Q that results from a change in the specific heat of the mixture,
we have:**

**dQ = m _{atotal}* del T * dc_{v}**

**where dQ represents the change in energy in joules that results from a change of temperature in the atmospheric
shell in degrees Kelvin and a change in the specific heat of the atmosphere from the introduction of an aerosol
component within this mixture. The total mass of the atmospheric shell is given by m _{atotal}.**

**where m _{atotal} = m_{air} * v_{air}**

**and dc _{v} = c_{v} - c_{vair}**

**It will be found that all introduced materials with a specific heat of less than 1003 ****joules
/ (kg * ^{o}K) (the specific heat of air) will lead to a decrease in the amount of energy required to raise
the temperature of the mass of the atmospheric shell by 1 degree Kelvin. Since the energy from the sun can
be considered as a relative constant for the problem of concern, this solar energy will result in an increase in
the temperature of the atmospheric shell. The specific heat of barium, for example is approximately 190**

**1. Clifford E Carnicom, Drought Inducement, (http://www.carnicom.com/drought1.htm),
04/07/02
2. Merle C. Potter, Thermodynamics for Engineers, (McGraw Hill, 1993), 251.
3. Potter, 251.
4. David R. Lide, CRC Handbook of Chemistry and Physics, (CRC Press, 2001), 14-19 to 14-22.
5. Carnicom, 04/07/02. **