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An engineer is given a fixed volume $V_m$ of metal with which to construct a spherical pressure vessel. Interestingly, assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, $n$ (measured in moles), does not depend on the radius $r$ of the vessel : instead it depends only on $V_m$ (measured in $m^3$) the temperature $T$ (mensured in $K$). the ideal gas constant $R$ (measured in $J/(K\ mol$ )), and the tensile strength of the metal $\sigma $ (measured in $N/m^2$ ) . Which of the following gives $n$ in terms of these parameters?
  • A$n = \frac{2}{3}\,\frac{{{V_m}\sigma }}{{RT}}$
  • B$n = \frac{2}{3}\,\frac{{\sqrt[3]{{{V_m}\sigma }}}}{{RT}}$
  • C$n = \frac{2}{3}\,\frac{{\sqrt[3]{{{V_m}{\sigma ^2}}}}}{{RT}}$
  • D$n = \frac{2}{3}\,\frac{{\sqrt[3]{{{V_m}^2\sigma }}}}{{RT}}$
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