From the table of the saturated vapor pressure of the water, at 20 degrees C, the saturated vapor pressure is 2.3388 kPa.

My question is: If I put a water in a closed container, and I pressurize the container up 2.3388 kPa (above the atmosphere pressure) by nitrogen or any other inert gas, will the water evaporate, or it will never evaporate because we are already above its saturated pressure?

In other words, is the saturated vapor pressure of a substance affected only by the partial pressure of the substance itself, or by the amount of pressure above its surface regardless of the substances cause that pressure?

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This is purely an equilibrium phenomenon. The vapor pressure of a fluid is generally just a function of the temperature. This pressure is just the partial pressure of the fluid, not anything else. To rationalize why this is, consider what the reaction looks like:

$$\ce{A(l) <=> A(g)}$$

$$\mathrm{K}_{eq} = \frac{P_{\ce{A(g)}}}{\ce{[A(l)]}}$$

The partial pressure is only dependent on factors that affect the activity of the pure substance. Since the liquid is essentially incompressible, the total external pressure does not have an effect.

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