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What is the difference between Vapor pressure and Partial pressure?

Answer
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Hint: Partial pressure and vapor pressure are commonly used scientific terms relating to the amount of pressure exerted by the vapor in its thermodynamic equilibrium on its liquid or solid state at a given temperature in a closed system when both the vapor and the liquid are in contact. Partial Pressures which states that the total pressure exerted by an ideal mixture of gases is equal to the sum of the partial pressures of individual gases.

Complete answer:
The vapour pressure of the gas: The pressure exerted by this gas phase in equilibrium with its solid or liquid counterpart is known as vapor pressure. For example, the amount of water vapor will increase and the pressure will increase if a bottle of water is heated up.
The partial pressure of the gas: In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas if it alone occupied the entire volume of the original mixture at the same temperature. The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules.
The main difference between partial pressure and vapor pressure is that partial pressure is the pressure exerted by an individual gas in a mixture as if it were alone in that system, while the vapor pressure refers to the pressure exerted by the vapor in its thermodynamic equilibrium with its condensed state of liquid. The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture. The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules.

Note:
We can calculate partial pressure from vapor pressure. First we have to Convert pressure to the same units so $780$ torr$ = 1.03atm$. Subtract water vapor pressure from total pressure to get partial pressure of gas \[A:{\text{ }}{P_A} = 1.03{\text{ }}atm - {\text{ }}1{\text{ }}atm = 0.03{\text{ }}atm.\]