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Homework posted in wrong forum, so no template

The pressure of a liquid in equilibrium, is equal to the pressure of it's vapour, or to the sum of vapour pressure plus atmospheric pressure?

My doubt starts from this problem:

At 293 K and 1 atm, the vapour pressure of water is 565.8 Pa. Calculate the vapour pressure of water, when the total pressure is 2 atm, knowing that the molar volume of water is 1.8*10^-5 m^3/mol.

The formula that I'm supposed to use is the one relative to the chemical potential at equilibrium between to phases, at constant temperature:

V(liq)dP= RT d(lnp) in which P stands for the pressure on the liquid, and p the vapour pressure.

Integrating the two quantities the resulting formula is ln(p'/p) = V(liq)*(P-p)/(RT) (p' stands for the new vapour pressure)

Now the solution to the problem on the book is:

ln(p'/565.8) = [1.8*10^-5*(202650 - 565.8)]/(R*T)

My question is, why there is the subtraction between the final total pressure on the liquid and the initial vapour pressure, instead doing 202650 - (101325 + 565.8), in which (101325 + 565.8) is the initial total pressure? What am I getting wrong?

Thank you in advance ;)

My doubt starts from this problem:

At 293 K and 1 atm, the vapour pressure of water is 565.8 Pa. Calculate the vapour pressure of water, when the total pressure is 2 atm, knowing that the molar volume of water is 1.8*10^-5 m^3/mol.

The formula that I'm supposed to use is the one relative to the chemical potential at equilibrium between to phases, at constant temperature:

V(liq)dP= RT d(lnp) in which P stands for the pressure on the liquid, and p the vapour pressure.

Integrating the two quantities the resulting formula is ln(p'/p) = V(liq)*(P-p)/(RT) (p' stands for the new vapour pressure)

Now the solution to the problem on the book is:

ln(p'/565.8) = [1.8*10^-5*(202650 - 565.8)]/(R*T)

My question is, why there is the subtraction between the final total pressure on the liquid and the initial vapour pressure, instead doing 202650 - (101325 + 565.8), in which (101325 + 565.8) is the initial total pressure? What am I getting wrong?

Thank you in advance ;)