A sample consisting of 22.7 g of a nongaseous, unstable compound X is placed inside a metal cylinder with a radius of 8.0 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 L of water at 25.00 ºC. The barometric pressure is 778 torr.
When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of 29.5 ºC, 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 0.30 mol carbon dioxide, 0.25 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/ º 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 29.5ºC is 31 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
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.
Any ideas for any sections of the question if you don't answer the whole beast?
X = C3H5O9N3
To determine the pressure-volume work for the decomposition of the 22.7-g sample of X, we need to calculate the change in volume using the ideal gas law and then use it to calculate the work done.
First, we need to calculate the initial moles of X. Given that the molar mass of X is 229.1 g/mol, we can determine the number of moles using the given mass of 22.7 g.
moles of X = mass of X / molar mass of X
moles of X = 22.7 g / 229.1 g/mol
Next, we need to calculate the initial volume occupied by X. We can assume that the compound X behaves like an ideal gas in this scenario.
Using the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
Rearranging the equation to solve for V:
V = (nRT) / P
We are given:
n = moles of X (calculated above)
T = initial temperature of the water (25.00 ºC = 25 + 273.15 K)
P = barometric pressure (778 torr)
Now, substituting the values into the equation:
V = (moles of X x R x T) / P
To calculate the change in volume, subtract the final volume from the initial volume. The final volume is the sum of the volume occupied by water and the volume of carbon dioxide gas produced.
The volume of carbon dioxide can be calculated using the ideal gas law with the given moles of CO2 and the final temperature of the water (29.5 ºC = 29.5 + 273.15 K).
Finally, the pressure-volume work (W) is given by the equation:
W = -PΔV
where P is the pressure (in atm) and ΔV is the change in volume (in L). The negative sign accounts for the fact that the work is being done on the system.
To calculate the molar change in internal energy, we can use the first law of thermodynamics:
ΔU = q + W
where ΔU is the change in internal energy, q is the heat transferred to or from the system, and W is the work done on or by the system.
In this case, since the heat capacity of the piston-and-cylinder apparatus is negligible and the piston has negligible mass, we can assume that q is zero. Therefore, the change in internal energy (ΔU) is equal to -W.
Finally, to approximate the standard enthalpy of formation for X (ΔH°f), we can use the enthalpy change for the decomposition reaction and the enthalpies of formation for the products (CO2 and H2O).
ΔH°f = Σ(nΔH°f - ΔH°f of reactants)
Summing up, the steps to solve for the pressure-volume work (b), molar change in internal energy and the approximate standard enthalpy of formation for X (c) are as follows:
1. Calculate the initial moles of X.
2. Calculate the initial volume occupied by X using the ideal gas law.
3. Calculate the final volume occupied by X after decomposition.
4. Calculate the pressure-volume work (W) using the equation W = -PΔV.
5. Calculate the molar change in internal energy (ΔU) using the equation ΔU = -W.
6. Calculate the approximate standard enthalpy of formation for X (ΔH°f) using the enthalpy change for the reaction and the enthalpies of formation for the products.
Note: Remember to convert all temperatures to Kelvin, and the final pressure and volume to atm and L, respectively, if necessary.