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\centerline{\textbf{The Isothermal Bulk Modulus}}
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\noindent We have the following equations of state for a
\emph{monatomic ideal gas}.  The first is the famous ideal gas law.
The second is the internal energy of a \emph{monatomic} ideal gas.
The third is the Sackur-Tetrode equation for entropy, which is true
for any ideal gas.
$$
  pV = N k_B T 
$$
$$
  U = \frac32 N k_B T 
$$
$$
  S = N k_B \left\{
  \ln\left[
    \frac{V}{N} \left(\frac{m U}{3\pi N\hbar^2}\right)^\frac32
    \right] + \frac52
  \right\}
$$
Using the above equations, find the isothermal bulk modulus, which you
can think of them as an \emph{intensive} spring constant for a
three-dimensional material.  The Isothermal Bulk Modulus is
$$
  b_T = -V\left(\frac{\partial p}{\partial V}\right)_{T}
$$
Now find the \emph{adiabatic} bulk modulus, which
involves holding the \emph{entropy} constant:
$$
  b_S = -V\left(\frac{\partial p}{\partial V}\right)_{S}
$$

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\noindent

  
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\leftline{\textit{by David Roundy}}
\leftline{\copyright DATE David Roundy}
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