A sample consisting of \(22.7 \mathrm{~g}\) of a nongaseous, unstable compound
\(\mathrm{X}\) is placed inside a metal cylinder with a radius of \(8.00
\mathrm{~cm}\), and a piston is carefully placed on the surface of the compound
so that, for all practical purposes, the distance between the bottom of the
cylinder and the piston is zero. (A hole in the piston allows trapped air to
escape as the piston is placed on the compound; then this hole is plugged so
that nothing inside the cylinder can escape.) The piston-and-cylinder
apparatus is carefully placed in \(10.00 \mathrm{~kg}\) water at \(25.00^{\circ}
\mathrm{C}\). The barometric pressure is 778 torr.
When the compound spontaneously decomposes, the piston moves up, the
temperature of the water reaches a maximum of \(29.52^{\circ} \mathrm{C}\), and
then it gradually decreases as the water loses heat to the surrounding air.
The distance between the piston and the bottom of the cylinder, at the maximum
temperature, is \(59.8 \mathrm{~cm}\). Chemical analysis shows that the cylinder
contains \(0.300 \mathrm{~mol}\) carbon dioxide, \(0.250\) mol liquid water,
\(0.025\) mol oxygen gas, and an undetermined amount of a gaseous element
\(\mathrm{A}\).
It is known that the enthalpy change for the decomposition of \(X\), according
to the reaction described above, is \(-1893\) \(\mathrm{kJ} / \mathrm{mol}
\mathrm{X}\). The standard enthalpies of formation for gaseous carbon dioxide
and liquid water are \(-393.5 \mathrm{~kJ} / \mathrm{mol}\) and \(-286
\mathrm{~kJ} / \mathrm{mol}\), respectively. The heat capacity for water is
\(4.184 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\). The conversion
factor between \(\mathrm{L} \cdot \mathrm{atm}\) and \(\mathrm{J}\) can be
determined from the two values for the gas constant \(R\), namely, \(0.08206
\mathrm{~L}\). \(\mathrm{atm} / \mathrm{K} \cdot \mathrm{mol}\) and \(8.3145
\mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\). The vapor pressure of water
at \(29.5^{\circ} \mathrm{C}\) is 31 torr. Assume that the heat capacity of the
pistonand-cylinder apparatus is negligible and that the piston has negligible
mass. Given the preceding information, determine
a. The formula for \(\mathrm{X}\).
b. The pressure-volume work (in \(\mathrm{kJ}\) ) for the decomposition of the
\(22.7-\mathrm{g}\) sample of \(\mathrm{X}\).
c. The molar change in internal energy for the decomposition of \(X\) and the
approximate standard enthalpy of formation for \(X\).
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