Assume that \(4.19 \times 10^{6} \mathrm{kJ}\) of energy is needed to heat a home. If this energy is derived from the combustion of methane \(\left(\mathrm{CH}_{4}\right),\) what volume of methane, measured at 1.00 atm and \(0^{\circ} \mathrm{C},\) must be burned? \(\left(\Delta H_{\text {combustion }}^{\circ} \text { for } \mathrm{CH}_{4}=-891 \mathrm{kJ} / \mathrm{mol}\right)\).

We are given that the energy needed to heat the home is \(4.19 \times 10^{6} \mathrm{kJ}\), and the combustion of methane releases \(\Delta H_{\text {combustion}}^{\circ} = -891 \mathrm{kJ/mol}\). To find the moles of methane needed, we can write: Moles of methane = \(\frac{\text{Energy needed}}{\text{Energy released per mole}}\) Plugging in the values, we get: Moles of methane = \(\frac{4.19 \times 10^{6} \mathrm{kJ}}{-891 \mathrm{kJ/mol}}\)

Now, we can use the ideal gas law to find the volume of methane at the given temperature and pressure. The ideal gas law is given by: \(PV = nRT\) Where, P = Pressure V = Volume n = moles of gas R = Gas constant T = Temperature We have been given the pressure, P = 1.00 atm, and the temperature, T = \(0^{\circ}\mathrm{C}\) (which is 273.15 K in Kelvins). We can also note that the gas constant, R, for this problem is \(0.0821 \frac{L \cdot atm}{K \cdot mol}\). Rearranging the ideal gas law formula to find the volume, we get: \(V = \frac{nRT}{P}\) Now, we can plug in the values for n (moles of methane calculated in step 1), R, T, and P: V = \(\frac{n(0.0821)(273.15)}{1.00}\) Substitute the value of Moles of methane from step 1 and solve for V: V = \(\frac{\frac{4.19 \times 10^{6} \mathrm{kJ}}{-891 \mathrm{kJ/mol}}(0.0821)(273.15)}{1.00}\) Calculate the volume, V, to find the required volume of methane that must be burned to provide the given energy.