Answer
$118.2\ g$
Work Step by Step
Volume of the house:
$275\ m^2\times 2.50\ m=687.5\ m^3$
Number of moles of air:
$687.5\ m^3\times 1000\ L/m^3 \times 1.22\ g/L \div 28.9\ g/mol=29022.5\ mol$
Heat required:
$29022.5\ mol\times 29.1\ J/mol.K\times(22.0-15.0)\ K=5911.9\ kJ$
Methane combustion:
$CH_4(g)+2\ O_2(g)\rightarrow CO_2(g)+2\ H_2O(g)$
Heat of combustion, from enthalpies of formation:
$-393.509+2\times-241.83--74.87=-802.3\ kJ/mol$
Mass of methane required:
$5911.9\ kJ\div802.3\ kJ/mol\times16.04\ g/mol=118.2\ g$
