A 2.00-L tank, evacuated and empty, has a mass of \(725.6 \mathrm{~g}\). It is filled with butane gas \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\) at \(22^{\circ} \mathrm{C}\) to a pressure of \(1.78 \mathrm{~atm} .\) What is the mass of the tank after it is filled?

Use the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant (in this case, \(0.0821 \frac{L \cdot atm}{K \cdot mol} \)), and T is temperature in Kelvin. We are given P, V, and T. First, convert the given temperature from Celsius to Kelvin: T = 22 + 273.15 = 295.15 K. Now, plug the given values to find the number of moles of butane gas in the tank: \((1.78 \mathrm{~atm}) \times (2.00 \mathrm{~L}) = n \times (0.0821 \frac{L\cdot atm}{K \cdot mol}) \times (295.15 \mathrm{~K})\) Solving for n, we get: \(n = \frac{(1.78 \mathrm{~atm}) \times (2.00 \mathrm{~L})}{(0.0821 \frac{L\cdot atm}{K \cdot mol}) \times (295.15 \mathrm{~K})}= 0.146 \mathrm{~mol}\)

Now, we need to find the mass of the butane gas in the tank using the number of moles found in the previous step. The molecular formula of butane is given as \(\mathrm{C}_{4} \mathrm{H}_{10}\), which means it has 4 carbon atoms and 10 hydrogen atoms. The molar mass of butane is calculated as follows: Molar mass of \(\mathrm{C} = 12.01 \mathrm{~g/mol}\) Molar mass of \(\mathrm{H} = 1.008 \mathrm{~g/mol}\) Molar mass of butane = \(4 \times 12.01 \mathrm{~g/mol} + 10 \times 1.008 \mathrm{~g/mol} = 58.12 \mathrm{~g/mol}\) Now, we can calculate the mass of butane gas in the tank: Mass = Moles × Molar mass Mass of butane = \(0.146 \mathrm{~mol} \times 58.12 \mathrm{~g/mol} = 8.479 \mathrm{~g}\)

Lastly, we need to add the mass of the butane gas found in the previous step to the mass of the empty tank given in the problem. The mass of the empty tank is \(725.6 \mathrm{~g}\). Mass of the filled tank = Mass of the empty tank + Mass of the butane gas Mass of the filled tank = \(725.6 \mathrm{~g} + 8.479 \mathrm{~g} = 734.079 \mathrm{~g}\) Thus, the mass of the tank after it is filled with butane gas is \(734.079 \mathrm{~g}\).