State Raoult’s law. The vapour pressure of pure benzene at a certain temperature is 0.850bar. a non-volatile, non-electrolyte solid weighing 0.5g when added to 39.0g of benzene (molar mass 78g/mol). Vapour pressure of the solution, then is 0.845 bar. What is the molar mass of the substance?

Raoult's law states that the vapor pressure of a component in an ideal solution is directly proportional to its mole fraction in the solution. Mathematically, it can be expressed as:

P_A = X_A * P°_A

where:
P_A is the vapor pressure of component A in the solution,
X_A is the mole fraction of component A in the solution, and
P°_A is the vapor pressure of pure component A.

In this case, benzene is the component we are interested in, and we have the following information:

Vapor pressure of pure benzene (P°_benzene) = 0.850 bar
Vapor pressure of the benzene solution (P_benzene) = 0.845 bar
Mass of the non-volatile, non-electrolyte solid added = 0.5 g
Mass of benzene (solvent) = 39.0 g
Molar mass of benzene = 78 g/mol

First, we need to calculate the mole fraction of benzene in the solution. The mole fraction (X_benzene) is defined as the moles of benzene divided by the total moles of both components (benzene and the non-volatile solid). We can calculate the moles of benzene as follows:

moles of benzene = mass of benzene / molar mass of benzene
= 39.0 g / 78 g/mol
= 0.5 mol

Now, let's calculate the moles of the non-volatile solid:

moles of non-volatile solid = mass of non-volatile solid / molar mass of non-volatile solid

Since we don't know the molar mass of the non-volatile solid, let's call it M:

moles of non-volatile solid = 0.5 g / M g/mol

Now, we can express the mole fraction of benzene (X_benzene) as:

X_benzene = moles of benzene / (moles of benzene + moles of non-volatile solid)
= 0.5 mol / (0.5 mol + 0.5 g / M g/mol)

According to Raoult's law, the vapor pressure of the solution is equal to the mole fraction of benzene multiplied by the vapor pressure of pure benzene:

P_benzene = X_benzene * P°_benzene

Substituting the given values:

0.845 bar = X_benzene * 0.850 bar

Now, let's solve this equation for X_benzene:

X_benzene = 0.845 bar / 0.850 bar
= 0.9941

Substituting this value back into the equation for X_benzene, we get:

0.9941 = 0.5 mol / (0.5 mol + 0.5 g / M g/mol)

Now, we can solve this equation for the molar mass of the non-volatile solid (M):

0.9941 * (0.5 mol + 0.5 g / M g/mol) = 0.5 mol

0.9941 * (0.5 mol) = 0.5 mol + 0.5 g / M g/mol

0.49705 mol = 0.5 mol + 0.5 g / M g/mol

0.49705 mol - 0.5 mol = 0.5 g / M g/mol

-0.00295 mol = 0.5 g / M g/mol

M g/mol = -0.5 g / -0.00295 mol