Express this partial derivative in terms of β and κ T. Then the ratio of d P to d Tis equal to ( ∂ P / ∂ T ) V, since there is no net change in volume. (c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Write the change in volume for this process, d V 2, in terms of d Pand the isothermal compressibility κ T, defined as (b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume, d V 1, in terms of d Tand the thermal expansion coefficient βintroduced in Problem 1.7. (a) First imagine slightly increasing the temperature of a material at constant pressure. To see why, estimate the pressure needed to keep Vfixed as Tincreases, as follows. Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume.
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