Acetoacetic acid, \(\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{COOH},\) a reagent used in organic synthesis, decomposes in acidic solution, producing acetone and \(\mathrm{CO}_{2}(\mathrm{g}).\) $$\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{COOH}(\mathrm{aq}) \longrightarrow \mathrm{CH}_{3} \mathrm{COCH}_{3}(\mathrm{aq})+\mathrm{CO}_{2}(\mathrm{g})$$ This first-order decomposition has a half-life of 144 min. (a) How long will it take for a sample of acetoacetic acid to be \(65 \%\) decomposed? (b) How many liters of \(\mathrm{CO}_{2}(\mathrm{g}),\) measured at \(24.5^{\circ} \mathrm{C}\) and 748 Torr, are produced as a \(10.0 \mathrm{g}\) sample of \(\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{COOH}\) decomposes for 575 min? [Ignore the aqueous solubility of \(\mathrm{CO}_{2}(\mathrm{g}) \cdot \mathrm{l}.\)

Firstly, it's necessary to use the formula for first order reactions. This formula is: \[t = \frac{0.693}{k}\], Here, \(t\) signifies the half-life of the reaction and \(k\) is the rate constant. Since the half-life is given, \(k\) can be calculated as: \[k = \frac{0.693}{144 min^{-1}}\]. Using this rate constant, the time required for a 65% decomposition needed can be calculated using the formula: \[t = \frac{\ln{(1/%remaining)}}{k}\]. Here, 1/%remaining is the fraction of reactant remaining which is \(100%-65%=35%\) or \(0.35\).

The stoichiometry of the reaction shows that each mole of CH3COCH2COOH produces one mole of CO2. The molar mass of CH3COCH2COOH is approximately 102.09 g/mol so 10.0 g * \(1 mol / 102.09 g\) gives us the moles of CH3COCH2COOH. Since the decomposition is first order, we use the formula \(N_t = N_0e^{-kt}\) to calculate the moles remaining after 575 minutes, where \(N_t\) is the moles of CH3COCH2COOH at time \(t\), \(N_0\) is the initial moles and \(k\) is the rate constant obtained in step 1. The moles of CO2 produced is then the difference between the initial and remaining moles of CH3COCH2COOH. Having the moles of CO2, apply the ideal gas law \(PV=nRT\) to calculate the volume of CO2. Here, \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the gas constant in L*atm/mol*K and \(T\) is temperature in Kelvin. Pressure is given in Torr so it's necessary to convert to atm, and temperature is given in Celsius so it's necessary to convert to Kelvin.

Plug in the obtained values into the equations to solve for the time for 65% decomposition and the volume of \(CO_2\).