Nitroglycerin, an explosive, decomposes according to the equation \(4 \mathrm{C}_{3} \mathrm{H}_{5}\left(\mathrm{NO}_{3}\right)_{3}(s) \longrightarrow\) $$12 \mathrm{CO}_{2}(g)+10 \mathrm{H}_{2} \mathrm{O}(g)+6 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)$$ Calculate the total volume of gases produced when collected at \(1.2 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) from \(2.6 \times 10^{2} \mathrm{~g}\) of nitroglycerin. What are the partial pressures of the gases under these conditions?

Firstly, calculate the moles of nitroglycerin using the molar mass. The molar mass of nitroglycerin, \(C_3H_5(NO_3)_3\), is 227 g/mol. The number of moles \(n\) can be calculated by dividing the given mass by the molar mass. Thus, \(n = \frac{2.6 \times 10^2 g}{227 g/mol} = 1.15 mol\).

According to the balanced equation, 4 moles of nitroglycerin produce 29 moles of gas (12 moles of carbon dioxide, 10 moles of water vapor, 6 moles of nitrogen, and 1 mole of oxygen). So, 1 mole of nitroglycerin will produce \(\frac{29}{4} = 7.25\) moles of gas. Thus, 1.15 moles of nitroglycerin will produce \(1.15 \times 7.25 = 8.34\) moles of gas.

Apply the ideal gas law to find the volume. As per the ideal gas law, \(PV=nRT\), where P is the pressure (1.2 atm), n is the number of moles (8.34 moles), R is the gas constant (0.0821 atm L/mol K), and T is the temperature in Kelvin (25℃ = 298 K). So, the total volume \(V\) of the gases can be calculated by rearranging the equation to find \(V = \frac{nRT}{P} = \frac{8.34 \times 0.0821 \times 298}{1.2} = 205 L\).

From Dalton's law of partial pressures, the partial pressure of a gas is the total pressure times the mole fraction of the gas. The mole fraction is calculated as the number of moles of that gas divided by total moles of all gases. According to the balanced equation, 4 moles of nitroglycerin produce 12 moles of \(CO_2\) , 10 moles of \(H_2O\), 6 moles of \(N_2\), and 1 mole of \(O_2\). Thus, the mole fractions are \( \frac{12}{29} , \frac{10}{29} , \frac{6}{29} , \, and \, \frac{1}{29} \) for \(CO_2 , H_2O , N_2 , \, and \, O_2\) respectively. The partial pressures are these mole fractions times the total pressure (1.2 atm), which give partial pressures of 0.50 atm, 0.41 atm, 0.25 atm, and 0.04 atm for \(CO_2 , H_2O , N_2 , \, and \, O_2\) respectively.