# A very hard question about specific heat.

#### dingdong_mustaf

A pond of water at 0°C is covered with a layer of ice 4.50 cm thick. If the air temperature stays constant at -11.0°C, how much time does it take for the thickness of the ice to increase to 9.00 cm?

Hint: To solve this problem, use the heat conduction equation,

dQ/dt = kA delta T/x

and note that the incremental energy dQ extracted from the water through the thickness x is the amount required to freeze a thickness dx of ice. That is, dQ = LpA dx, where p is the density of the ice, A is the area, and L is the latent heat of fusion. (The specific gravity and thermal conductivity for ice are, respectively, 0.917 is 2.0 W/m/°C.)

I dont have much of an idea on how to attempt this question, all ive got so far is.

dQ = LpA dx
so LpA dx/dt = kA delta T/x
x/dt = L delta T/ L p dx

I guess thats useful as it gets rid of surface area in the equation ( which isnt given), but im not sure where to go from there. Also, delta T would be zero, and so the entire equation would equal zero, which doesnt make much sense to me.

By the way im 16 and so presume that im very ignorant.

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#### f(x)

why should dT be 0? Outside temp is -11 while water is at 0.

dQ=KA.(T1-T2)/x .dt where T1-T2=11
dQ=mL=dx.A.P.L
so we have from above eqns.
dx.A.P.L=KA.11/x .dt
xdx.P.L=K.11 dt
integrate LHS from 0.045 to 0.09 and RHS from 0 to t, where t is the required time.
P.L.(x^2)/2=22t
substituing the values (L=3.36 x 10^5)& solving i get t=42.5 sec

IMO this is too small a value, anyway, do tell the answer :)

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