First Law of Thermodynamics and automobile

[Gil-H]

Homework Statement

An automobile battery that is originally fully charged gradually discharges while sitting on a shelf at a constant temperature of 40C, producing no electric work but resulting in a heat transfer of 1000kJ to its enviroment.
The battery is then recharged to its initial state by means of a process involving work input of 440Wh.
Find the heat interaction of the battery during this charging process. Justify your answer thermodynamically.

Homework Equations

The first law of thermodynamics
ΔE = Q -W

The Attempt at a Solution

in the first step, heat is moving out of the system and no work is done, so:
Q₁₂ = -1000kJ
W₁₂ = 0

in the charging process, the is done on the system, so:
W₂₁ = -440Wh = -440[Wh]*3.6[kJ/Wh] = -1584kJ

The battery is charged to its initial state, so:
ΔE = 0
ΔE₁₂ + ΔE₂₁ = 0
(Q₁₂ - W₁₂) + (Q₂₁ - W₂₁) = 0
-1000kJ + Q₂₁ +1584kJ = 0
and
Q₂₁ = -584kJ

the answer in the book is +440kJ. Why? What have I done wrong?

 

[Andrew Mason]

I am not sure what is meant by "heat interaction" of the battery.

If the battery in its initial state can supply 1000KJ of energy to its environment by discharging, and if 1584KJ (440Wh) of work is needed to recharge it to its original state (original charge and temperature), then the battery must dissipate 584KJ of heat to the environment in the charging process. $\Delta Q = \Delta U + W = 1000KJ - 1584KJ = -584KJ$ (W = +work done BY the battery)

I would say the answer given is wrong and you are correct.

AM