Question

Your house has lost electricity and natural gas. Luckily, you still have running water but the idea of an ice cold shower doesn’t appeal to you. So you want to have a hot water bath but the only source of heat is a wood fire. You put 195L of cold water (15C) into the tub but you are going to have to carry something hot from the fireplace to the tub to heat it up. You have two choices. You can heat up granite stones that weigh 2000g each up to 485C and put those in the bath or you can heat up 2000g kettle of water to 95C. You want the bath to be 32C. Assume that no heat is lost to the surroundings. The density of water is 1g/mL, c of water is 4.18 J/gC, c of granite is 0.790 J/gC.

a) What is the minimum number of stones would you have to add to the bath in order to reach the desired temperature?

b) What is the minimum number of kettles of hot water you would have to add to the bath in order to reach the desired temperature?

c) Which heating option that involves carrying the least weight?

Answer 1

we use heat equation to find how much heat gained or lost

q = m.Cp. ΔT

{ where q = heat, m = mass of compound , Cp = heat capacity and ΔT = temperature change (T_final - T_initial) }   

heat required to rise water temperature from 15 ⁰C to 32 ⁰C

first calculate mass of water

m = volume x density

m = 195000 mL x 1 g/ml = 195000 g

now find heat required by water (ΔT = 32 ⁰C - 15 ⁰C = 17 ⁰C )

q = 195000 g x 4.18 J /g C x 17 ⁰C

q = 13856700 J

a) numbers of stone required to rise water temperature 32 C from 15 C

we assume initial temperaure of stone is also 15 ^oC so,

q = m.C.ΔT

13856700 J = m x 0.79 J/gC x (485 ^oC - 15 ^oC )

m = 37319.41 g

total mass of stone required to rise water temperaure 32 C from 15 C = ~37320 g

numbers of stone = total mass of stone / mass of each stone

numbers of stone = 37320 g / 2000 g = 18.65

numbers of stone = 19

b) same procedure for kettles

q = m Cp ΔT

13856700 J = m x 4.18 J/gC x (95^oC - 15^oC)

m = 41437.5 g

numbers of kettle = 41437.5 g / 2000 g = 20.71

numbers of kettle = 21

C) granite stones heating option that involves carrying the least weight