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Question

# Various species of hagfish, or slime eels, live on the ocean floor, where they burrow inside other fish, eating them from the inside out and secreting copious amounts of slime. Their skins are widely used to make eelskin wallets and accessories. Suppose a hagfish is caught in a trap at a depth of 200 m below the ocean surface, where the water temperature is $10^\circ C,$ then brought to the surface where the temperature is $15^\circ C.$ If the isothermal compressibility and volume expansivity are assumed constant and equal to the values for water, $\left(\beta=10^{-4} \mathrm{K}^{-1} \text { and } \kappa=4.8 \times 10^{-5} \mathrm{bar}^{-1}\right)$ what is the fractional change in the volume of the hagfish when it is brought to the surface?

Solution

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Form total differential of Volume we know the expression fractional volume in term of $\beta$ and $\kappa$

$\dfrac{dV}{V}=\beta dT-\kappa dP \hskip 0.5em \text{(1)}$

Note: If you want to know how get this expression please refer the solution of problem $1$ of exercise 3.10

Now since we know the initial and final temperature but we know only surface pressure thus we calculate pressure at 200 m below the ocean

$\dfrac{dP}{dh}=\rho g$

This equation tell how pressure vary with respect to depth $h$

$P_{200} =P_{0}+\rho g h$

The surface pressure $P_{0}=1.01325 \mathrm{bar}$ and density of water $rho=1000 \hskip 0.5em \mathrm{Kg\cdot m^{3}}$

$P_{2}=1.01325\times 10^{5}+1000\times 200\times 9.8= 19.60\times 10^{5}\hskip 0.5em \mathrm{Pa}=19.6\hskip 0.5em \mathrm{bar}$

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