\subsection{Problem background}
\label{Sec:Background}
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Recently, there have been lots of discussions 
regarding the economic and political implications
of climate variability, in particular global warming
as a measurable effect of an anthropogenic, 
i.e.\ human-made, climate change\TschSpace%
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\cite{%
Stockholm2006,%
IPCC2007summary,%
Svensmark1997,%
Heiss1999,%
Mann2003,%
Soon2003,%
Weart2004,%
Hardy2005,%
Avery2006,%
Khilyuk2006,%
Wegman2006,%
Jaworowski2007,%
Vatican2007%
}.
}{}%
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{%
\cite{%
Stockholm2006,%
IPCC2007summary,%
Svensmark1997,%
Heiss1999,%
Mann2003,%
Soon2003,%
Weart2004,%
Hardy2005,%
Avery2006,%
Khilyuk2006,%
Wegman2006,%
Jaworowski2007,%
Vatican2007%
}.
}{}%
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{%
\cite{%
Stockholm2006,%
IPCC2007summary,%
Svensmark1997,%
Heiss1999,%
Mann2003,%
Soon2003,%
Weart2004,%
Hardy2005,%
Avery2006,%
Khilyuk2006,%
Wegman2006,%
Jaworowski2007,%
Vatican2007%
}.
}{}%
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Many authors assume that carbon dioxide emissions 
from fossil-fuel consumption represent a serious 
danger to the health of our planet, 
since they are supposed to influence the climates, 
in particular the average temperatures of the surface 
and lower atmosphere of the Earth.
However, carbon dioxide is a rare trace gas, 
a very small part of the atmosphere 
found in concentrations 
as low as 
$0,03\,{\rm Vol}\,\%$ 
(cf.\ Table \ref{table:PB001} 
and  \ref{table:PB002},
see also
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           {Ref.~\refcite{EngineeringToolbox2007}} 
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           {Ref.~\cite{EngineeringToolbox2007}}
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\footnote{In a recent paper on 
\lq\lq 180 Years accurate CO2 Gas 
analysis of Air by Chemical Methods\rq\rq\ 
the German biologist Ernst-Georg Beck argues
that the IPCC reliance of ice core ${\rm CO}_2$ 
figures is wrong\TschSpace%
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\cite{Beck2007,Beck2007Erratum}.
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Though interesting on its own
that even the ${\rm CO}_2$ data themselves
are subject to a discussion it does not influence 
the rationale of this paper which is 
to show that ${\rm CO}_2$
is \textit{completely} irrelevant.} 
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\input{GerlichTscheuschner2007a_110_introduction_problem_background_table001_}
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\input{GerlichTscheuschner2007a_110_introduction_problem_background_table002_}
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A physicist starts his analysis of the problem by pointing 
his attention to two fundamental thermodynamic properties, 
namely
%  
\begin{itemize}
\item the \textit{thermal conductivity} $\lambda$, 
      a property that determines how much heat per time 
      unit and temperature difference flows in a medium;
\item the \textit{isochoric thermal diffusivity} 
      $a_{\rm v}$,
      a property that determines how rapidly
      a temperature change will spread,
      expressed in terms of  an area per time unit.
\end{itemize}      
%              
Both quantities are related by 
%
\begin{equation}
a_{\rm v}
=
\frac{\lambda}{\varrho \, c_{\rm v}}
\end{equation}
%
the proportionality constant 
of the heat equation
%
\begin{equation}
\frac{\partial T}{\partial t}
=
a_{\rm v}
\cdot
\Delta T
\end{equation}
%
whereby
$T$ is the temperature,
$\varrho$ the mass density, 
and
$c_{\rm v}$ the isochoric specific heat.

To calculate the relevant data from the gaseous components of the air
one has to use their mass concentrations as weights to calculate the
properties of the mixture \lq\lq air\rq\rq\ 
according to Gibbs thermodynamics\TschSpace%
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\cite{Callen1985,Huang1987}.%
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\footnote{The thermal conductivity of a mixture of two gases
           does not, in general, vary linearly with the composition 
           of the mixture.
           However for comparable molecular weight and small 
           concentrations the non-linearity is negligible\TschSpace%
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\cite{Evans1964}.%
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}       
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Data on volume concentrations
(Table \ref{table:PB002})
can be converted into mass concentrations 
with the aid of known mass densities 
(Table \ref{table:PB003}).
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\input{GerlichTscheuschner2007a_110_introduction_problem_background_table003_}
%

A comparison of volume percents and mass percents for ${\rm CO}_2$
shows that the current mass concentration, which is the physically 
relevant concentration, is approximately $0.06\,\%$ and not the
often quoted $0.03\,\%$
(Table \ref{table:PB004}).
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\input{GerlichTscheuschner2007a_110_introduction_problem_background_table004_}
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From known thermal conductivities
(Table \ref{table:PB005}), 
isochoric heat capacities,
and mass densities 
the isochoric thermal diffusivities 
of the
components of the air are determined  
(Table \ref{table:PB006}).
This allows to estimate the change of the effective
thermal conductivity of the air in dependence of a 
doubling of the ${\rm CO}_2$ concentration, expected
to happen within the next 300 years 
(Table \ref{table:PB007}).%
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\input{GerlichTscheuschner2007a_110_introduction_problem_background_table005_}
\input{GerlichTscheuschner2007a_110_introduction_problem_background_table006_}
\input{GerlichTscheuschner2007a_110_introduction_problem_background_table007_}
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It is obvious that a doubling of the concentration 
of the trace gas ${\rm CO}_2$,
whose thermal conductivity is approximately 
one half than that of nitrogen and oxygen,
does change 
the thermal conductivity at the most by $0,03\,\%$ 
and 
the isochoric thermal diffusivity at the most by $0.07\,\%$. 
These numbers lie within the range of the measuring inaccuracy 
and other uncertainties such as rounding errors and therefore have 
no significance at all.
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