“How cold would a sphere of silver with a radius of 5 meters be in order to freeze the ocean upon contact?”

Short answer is “no”.

There is a lower limit to how cold something can be, absolute zero.

It looks like the specific heat of silver is 0.233 J/gm K ( at 20 C). Let’s say it stays that value throughout this process, that should be fine.

The density of silver is about 10490 kg/m^3

A sphere of that size would be 523.598775598 m^3

Multiply the density by the volume to get a mass of 5492551.15602 kg

Multiply that by the specific heat (careful with the units) and you get

Energy per degree for the silver:

1279764419.35 J/K

The ocean across the world varies a bit in temperature.

Salt water freezes at about -2° C

Absolute Zero is −273.15° C

A change in temperature of about 271° for the silver sphere to absorb the energy needed to reach equilibrium with freezing salt water.

Energy to bring the silver to -2°

1279764419.35 J/K * 271

346816157644 J

(that could be absorbed by the silver)

How much water could the sphere freeze?

Let’s say some ocean water is 10° C. It would decrease by 12° to freeze.

Specific heat of water, 4179 J/kg

We will take the energy the silver could absorb and figure out much water at this temperature would transfer that energy to change its temperature to -2 C

346816157644 J = 4179 J/kg X

X = 82990226.7633 kg

About 8,000,000 kg, 8000 m^3 (20 m x 20 m x 20m)

The Great Lakes have about 22,700 cubic kilometers of water.

One cubic kilometer of water is 1000000000000 kg

You could lower the temperature of 8000 m^3 water with the silver sphere to -2° C. But at that point its still not frozen, you would use the latent heat of fusion to figure that out.

But no where near being able to freeze the ocean. Maybe a small pond.