To what temperature must the conTainer be heated in order to raise the pressure of the liquid to 50 bar?
A compound, in the liquid phase, has a constant isothermal compressibility kT= 5x10-4 bar-1 and a constant coefficient of thermal expansion ?V = 0.001 K-1. It is placed in a rigid (constant volume) container initially at T=300 K and P=1 bar.
Define the system as the liquid in the container. Write a total derivative that describes changes in temperature
Where mass and total volume are constant due to no material crossing system boundaries and the container being rigid. Hence, the specific volume ?V must also be constant so d?V=0.
Apply definition of isothermal compressibility factor and coefficient of thermal expansion:
Relate known information to (?T/?P)_?V from the total derivative through the triple product rule:
Which for T, P, and V becomes:
Apply basic rule of differential calculus where (?P/(??V))_T=1/((??V)/?P)_T so that it mirrors the definition of ?_V
Substitute into total derivative and solve the integral:
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