A sample consisting of 22.7 g of nongaseous, unstable compound X is placed inside a metal cylinder with a radius of 8.00 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 kg of water at 25.00°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°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 cm. Chemical analysis shows that the cylinder contains 0.300 mol carbon dioxide, 0.250 mol liquid water, 0.025 mol oxygen gas, and an undetermined amount of gaseous element A.

It is known that the enthalpy change for the decomposition of X, according to the reaction described above, is -1893 kJ/mol X. The standard enthalpies of formation for gaseous carbon dioxide and liquid water are -393.5 kJ/mol and -286 kJ/mol, respectively. The heat capacity for water is 4.184 J/°C·g. The conversion factor between L·atm and J can be determined from the two values for the gas constant R, namely 0.08206 L·atm/mol·K and 8.3145 J/mol·K. The vapor pressure of water at 29.5°C is 30 torr. Assume that the heat capacity of the piston-and-cylinder apparatus is negligible and that the piston has negligible mass.

Given the preceding information, determine the following.

(a) The formula for X.

(b) The pressure-volume work (in kJ) for the decomposition of the 22.7 g sample of X.

(c) The molar change in internal energy for the decomposition of X and the approximate standard enthalpy of formation for X.

When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of 29.52°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 cm. Chemical analysis shows that the cylinder contains 0.300 mol carbon dioxide, 0.250 mol liquid water, 0.025 mol oxygen gas, and an undetermined amount of gaseous element A.

It is known that the enthalpy change for the decomposition of X, according to the reaction described above, is -1893 kJ/mol X. The standard enthalpies of formation for gaseous carbon dioxide and liquid water are -393.5 kJ/mol and -286 kJ/mol, respectively. The heat capacity for water is 4.184 J/°C·g. The conversion factor between L·atm and J can be determined from the two values for the gas constant R, namely 0.08206 L·atm/mol·K and 8.3145 J/mol·K. The vapor pressure of water at 29.5°C is 30 torr. Assume that the heat capacity of the piston-and-cylinder apparatus is negligible and that the piston has negligible mass.

Given the preceding information, determine the following.

(a) The formula for X.

(b) The pressure-volume work (in kJ) for the decomposition of the 22.7 g sample of X.

(c) The molar change in internal energy for the decomposition of X and the approximate standard enthalpy of formation for X.