Volume expansion and sea level rise

[hnnhcmmngs]

Homework Statement

[/B]
Oceans cover 2/3 of the earth’s surface, with an average depth of 3.7 km. The average surface temperture is 17◦ C. Taking this temperature as representative of the entire ocean, and knowing that the coefficient of volume expansion for water at this temperature is β = 1.7 × 10^−4 (◦C)^−1, how much would the sea level rise if the average ocean temperature rose by 1◦ C?

Given that the oceans are in fact cooler at lower depths, and knowing what you know about the expansion or contraction of water at lower temperatures, would you expect the actual sea level rise to be larger or smaller than what you just calculated?

Homework Equations

[/B]
α=β/3
∆L=α*L*∆T

The Attempt at a Solution

[/B]
I used the first equation to find that α=5.667*10^-5 (◦C)−1 and then I used the second equation to find that ∆L=(5.667*10^-5 (◦C)^−1)*(3.7*10^3 m)*(1◦ C)=0.21 m

As for the second part, I would assume that the actual sea level rise would be smaller than what I calculated since the coefficient of linear expansion of water is smaller at lower depths. Is this correct?

 

[haruspex]

find that α=5.667*10^-5 (◦C)−1

You are assuming the oceans would expand equally in all directions. The land might object.

the actual sea level rise would be smaller than what I calculated since the coefficient of linear expansion of water is smaller at lower depths. Is this correct?

Seems right.

 

[hnnhcmmngs]

You are assuming the oceans would expand equally in all directions. The land might object.

So would I use ∆V=β*V*∆T instead? Do I find the volume expansion then divide by the surface area of the oceans to find the sea level rise?

 

[haruspex]

So would I use ∆V=β*V*∆T instead? Do I find the volume expansion then divide by the surface area of the oceans to find the sea level rise?

That sounds more reasonable to me, but you do not have to go through those steps. Can you see a more direct way?

 

[hnnhcmmngs]

That sounds more reasonable to me, but you do not have to go through those steps. Can you see a more direct way?

∆V=β*V*∆T
∆V=SA*∆d (SA=surface area, d=depth)
V=SA*d
SA*∆d=β*SA*d*∆T
Then the surface area cancels out.
∆d=β*d*∆T=(1.7*10^-4 (◦C)^−1)*(3.7*10^3 m)*(1◦ C)=0.629 m
Is this correct?

 

[haruspex]

∆V=β*V*∆T
∆V=SA*∆d (SA=surface area, d=depth)
V=SA*d
SA*∆d=β*SA*d*∆T
Then the surface area cancels out.
∆d=β*d*∆T=(1.7*10^-4 (◦C)^−1)*(3.7*10^3 m)*(1◦ C)=0.629 m
Is this correct?

Looks right.

 

[hnnhcmmngs]

Looks right.

Cool. Thank you so much!