To what pressure must water be compressed at 50°C to change its density by 1%? Assume that ? is independent of P.
Express the volume expansivity and the isothermal compressibility as functions of density ? and its partial derivatives. For water at 50°C and 1 bar, ? = 
Starting with equations
and 
To put this in to terms of density, the amount of mass must be fixed in the system meaning ?V is constant. This allows for the expression
and allows for the above equations to be changed to 
and 
Now that these are in terms of density and assuming a constant temperature of 50? C, the ? equation reduces to
And integrating yields
and value of 
From this the pressure that is need to change the density of water (1.00 g/mol) by 1% can be found by saying
?2 = 1.01 ? ?1, and solving for ?P
bar. So the pressure would need to be 1.00 bar + 225.22 bar = 226.22 bar to change the density by 1%.
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