How do you calculate the mass of ethanol that must be burnt to increase the temperature of #210 g# of water by #65^@#, if exactly half of the heat released by this combustion is lost to the surroundings?

Step 1. Calculate the theoretical amount of heat required

The formula for the quantity of heat #q# transferred is

#color(blue)(bar(ul(|color(white)(a/a)q = mC_text(s)ΔTcolor(white)(a/a)|)))" "#

where

#m color(white)(ll) =# the mass of the object
#C_text(s) color(white)(l) =# its specific heat capacity
#ΔT =# its change in temperature

In this problem,

#m color(white)(ll)= "210 g"#
#C_text(s) color(white)(ll)= "4.184 J·°C"^"-1""g"^"-1"#
#ΔT = "65 °C"#

#q = 210 color(red)(cancel(color(black)("g"))) × 4.184 "J"·color(red)(cancel(color(black)("°C"^"-1""g"^"-1"))) ×65 color(red)(cancel(color(black)("°C"))) = "57 100"color(white)(l)"J" = "57.1 kJ"#

Step 2. Calculate the actual amount of heat required

#"Heat required" = 57.1 color(red)(cancel(color(black)("kJ theoretical"))) × "2 kJ required"/(1 color(red)(cancel(color(black)("kJ theoretical")))) = "114 kJ"#

Step 3. Calculate the moles of ethanol required

The equation for the combustion of ethanol is

#"C"_2"H"_5"OH" + "⁷/₂O"_2 → "2CO"_2 + "3H"_2"O" + "1367 kJ"#

#"Theoretical moles" = 114 color(red)(cancel(color(black)("kJ"))) × "1 mol ethanol"/(1367 color(red)(cancel(color(black)("kJ")))) = "0.0836 mol ethanol"#

Step 4. Calculate the mass of ethanol

#"Mass of ethanol" = 0.0836 color(red)(cancel(color(black)("mol ethanol"))) × "46.07 g ethanol"/(1 color(red)(cancel(color(black)("mol ethanol")))) = "3.8 g ethanol"#