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What is the density of water at a depth of where pressure is 80.0 atm, given that its density at the surface is 1.03×103kgm3?

Answer
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Hint: We know that change in volume is given by ΔV=V1V2 and and relation between volume, density and mass can be given as, V=mρ. So, using these formulas and substituting the values in the formula of change in volume we will find the density of water at the given height.

Formula used:
ΔV=V1V2, V=mρ

Complete answer:
Now, in question we are asked to find the density of water at a depth where pressure is 80.0 atm and the density at the surface is given as 1.03×103kgm3, so now we know that change in volume can be given by, ΔV=V1V2 and the relation between volume and density can be given as, V=mρ.
Where, V is volume, m is mass and ρ is density of water.
Now, let the given depth be h,
density of water at surface be, ρ1=1.03×103
ρ2 is the density of water at given depth.
Pressure at depth is 80.0 atm
Now, we know that 1 atm=1.015×105 Pa
So, 80.0 atm=80×1.01×105Pa
Now, considering the change in volume and substituting values of volume in terms of density we will get,

ΔV=V1V2
V=mρ
ΔV=m(1ρ11ρ2) ………………….(i)

Now, as go in depth in water the volumetric strain of water increases which can be expressed mathematically as,
Volumetric strain=change in volume original volume=ΔVV1
Now, substituting the values of expression (i) in this equation we will get,

Volumetric strain=ΔVV1=m(1ρ11ρ2)×ρ1m
ΔVV1=1ρ1ρ2 …………………….(ii)

Now, the bulk modulus can be given as, B=pV1ΔV, where p is pressure at depth h.
ΔVV1=pB ……………………(iii)

Now, the inverse of bulk modulus is called compressibility and the value of compressibility of water is fixed which can be given as, compressibility=1B=45.8×1011Pa1.
So, substituting this value in equation (iii) we will get,
ΔVV1=pB=80×1.01×103×45.8×1011
ΔVV1=3.71×103 …………..(iv)

Now, by equation expressions (ii) and (iv) we will get,
1ρ1ρ2=3.71×103
ρ1ρ2=(13.71×103)
ρ2=1.03×103(13.71×103)
ρ2=1.034×103kgm3
Hence, the density of water at a given height is 1.034×103kgm3.

Note:
Bulk modulus is used to show how much a substance can resist the compression acting on it due to change in pressure and volume on the object. It is one of the most important to be considered in Fluid mechanics and it is defined as a ratio of pressure increase to the resulting relative decrease of the volume. So, for such types of problems students must consider the Bulk modulus.