For his 18th birthday in February Peter plants to turn a hut in the garden of his parents into a swimming pool with an artifical beach. In order to estimate the consts for heating the water and the house , peter obtains the data for the natural gas combustion and its price.
Calculate the energy (in MJ) which is needed to maintain the temperature inside the house at `30.0^(@)C` during the party (12 hours) .
`1.00 m^(3)` of natural gas as delivered by PUC costs 0.40 € and `1.00 k` Wh of electricity costs 0.137 € . The rent for the equipment for gas heating will cost him about 150.00 € while the corresponding electrical heaters will only cost 100.00 €.

Answer

Energy for heating the water:
Volume of water : `V_("water") =22.5 m^(3)`
`n_("water") =V_("water") P_("water")(M_("water"))^(-1) = 22.5 xx 106 g m^(-3) xx (18.02 g "mol"^(-1))^(-1) =1.249 xx 10^(6) "mol"`
`E_("water") ="n"_("water") xx Cp xx DeltaT = 10249 xx 10^(6) "mol" xx 75.30 J K^(-1)"mol"^(-1) xx14 K =1316 MJ`
Energy for heating the air:
Volume of the house is :
`V_(air) =(15 m xx 8m xx 3m) + 0.5 xx(15mxx 8m xx 2m)=480 m^(3)`
`n_("air")=pV (RT) ^(-1) =1.013 xx 10^(5) Pa xx 480 m^(3) xx (8.314 J (" k mol")^(-1) xx 283.15 K)^(-1) =2.065 xx 10^(4) "mol"`
`C_(p)('air") =0.21 xx 29.4 J ("K mol")^(-1) + 0.79 xx 29.1 J (" k mol") ^(-1) =29.16 J ("K mol")^(-1)`
`E_("air") =n_("air") xx C_(p) ("air") xx DeltaT = 2.065 xx 10^(4) "mol"xx 29.17 J ("K mol")^(-1) xx 20K =12.5 MJ`