Final answer:
The isothermal compressibility of a fluid being independent of temperature implies that the fluid's thermal expansivity is also independent of pressure.
Step-by-step explanation:
To prove that the fluid's thermal expansivity is also independent of pressure, we need to understand the relationship between isothermal compressibility and thermal expansivity.
The isothermal compressibility, represented by the symbol κ, is defined as the ratio of relative decrease in volume to the increase in pressure while keeping the temperature constant. On the other hand, thermal expansivity, represented by the symbol α, is defined as the ratio of relative increase in volume to the increase in temperature while keeping the pressure constant.
If the isothermal compressibility of the fluid is independent of temperature, it means that the relative decrease in volume is proportional to the increase in pressure, regardless of temperature.
This implies that the fluid's volume change is solely dependent on pressure changes, and not on temperature changes. Therefore, the fluid's thermal expansivity, which is related to volume change with respect to temperature change, remains unaffected by pressure changes.
In summary, if the isothermal compressibility is independent of temperature, it implies that the fluid's thermal expansivity is also independent of pressure.