Raoult's Law:

The partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. In consequence, the relative lowering of vapor pressure of a dilute solution of nonvolatile solute is equal to the mole fraction of solute in the solution.

Mathematically, Raoult's law for a single component in an ideal solution is stated as

{\displaystyle p_{i}=p_{i}^{\star }x_{i}}

where p_{i} is the partial pressure of the component i in the gaseous mixture (above the solution), p_{i}^{{\star }} is the vapor pressure of the pure component i, and x_{i} is the mole fraction of the component i in the mixture (in the solution).

Once the components in the solution have reached equilibrium, the total vapor pressure of the solution can be determined by combining Raoult's law with Dalton's law of partial pressures to give

{\displaystyle p=p_{\rm {A}}^{\star }x_{\rm {A}}+p_{\rm {B}}^{\star }x_{\rm {B}}+\cdots }

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Henry's law:

Henry's law is a gas law that states that the amount of dissolved gas is proportional to its partial pressure in the gas phase. The proportionality factor is called the Henry's law constant. It was formulated by the English chemist William Henry, 

Henry's law states that the solubility of a gas in a liquid is proportional to the pressure of the gas over the solution.

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Relation between the laws:

The equation is c = kP , where c is the molar concentration in mol/L of the dissolved gas, P is the pressure (in atm) of the gas over the solution at equilibrium and k is a constant that depends only on temperature for a given gas.

Which means that when you want to mix a gas and a liquid, the amount of gas that will actually dissolve in the liquid is proportional to two things: the pressure of that gas at equilibrium over the solution and a constant k that changes depending on the gas and the temperature. By using the equation up there, you will find the molar concentration (mol/L) of the gas that will dissolve in the solution. 

On the other hand,

Raoult's law is used in a case where the solute (the smallest component of the solution) is non-volatile. The law considers that the vapor pressure of the whole solution will always be less than that of the pure solvent. Therefore, the vapor pressure of the solution will depends on the concentration of the solute. 

Raoult's law equation is P1 = X1 P1* (they are multiplied), where P1 is the vapor pressure of the solvent over the solution, X1 is the mole faction of the solvent in the solution and P1* is the vapor pressure of the pure solvent (if it was alone in the solution). 

Overall, the difference is that Henry's law takes care of what happen IN the solution when you have gas over it, while Raoult's law looks at what is happening OVER the solution when you mix a non-volatile solute to a solvent that has a known vapor pressure when it's pure (e.g. water). 

Henry's law will give you the molar concentration of a dissolved gas in the solution, Raoult's law will give you a vapor pressure over a solution after you mixed a solvent with a non-volatile solute. 

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