Derive the relation between molarity and molality. Calculate the molality of 1M HCl solution having density 1.5365 gmol^-1.

To derive the relationship between molarity and molality, we first need to understand what each term means. Molarity (M) is defined as the number of moles of solute per liter of solution, while molality (m) is defined as the number of moles of solute per kilogram of solvent. The key difference lies in the denominator: molarity uses the volume of the entire solution, whereas molality focuses solely on the mass of the solvent.

Deriving the Relationship

Let's break down the relationship step by step:

• Molarity (M): M = n/V, where n is the number of moles of solute and V is the volume of the solution in liters.
• Molality (m): m = n/m_solvent, where m_solvent is the mass of the solvent in kilograms.

To connect these two definitions, we need to express the mass of the solvent in terms of the volume of the solution and its density. The density (d) of the solution is defined as:

d = mass of solution/volume of solution

From this, we can rearrange to find the mass of the solution:

mass of solution = d × V

Now, if we denote the mass of the solute as m_solute and the mass of the solvent as m_solvent, we can express the mass of the solution as:

mass of solution = m_solute + m_solvent

Rearranging gives us:

m_solvent = (mass of solution) - m_solute

Substituting this into the equation for molality, we can express it in terms of molarity:

m = n / [(d × V) - (n × M_solute)]

Where M_solute is the molar mass of the solute. This shows how molarity and molality are interrelated, depending on the density of the solution and the molar mass of the solute.

Calculating Molality of 1M HCl Solution

Now, let's calculate the molality of a 1M HCl solution with a density of 1.5365 g/mL. First, we need to determine the number of moles of HCl in 1 liter of solution:

Since the molarity is 1M, this means there is 1 mole of HCl in 1 liter of solution.

Next, we calculate the mass of the solution using its density:

mass of solution = density × volume = 1.5365 g/mL × 1000 mL = 1536.5 g

Now, we need to find the mass of the solute (HCl). The molar mass of HCl is approximately 36.46 g/mol. Therefore, the mass of HCl in 1 mole is:

mass of HCl = 1 mole × 36.46 g/mol = 36.46 g

Now we can find the mass of the solvent (water) in the solution:

mass of solvent = mass of solution - mass of solute = 1536.5 g - 36.46 g = 1500.04 g

To convert this mass into kilograms, we divide by 1000:

mass of solvent = 1500.04 g / 1000 = 1.50004 kg

Finally, we can calculate the molality:

m = n / m_solvent = 1 mole / 1.50004 kg ≈ 0.6667 mol/kg

Final Result

The molality of the 1M HCl solution is approximately 0.667 mol/kg. This calculation illustrates how molarity and molality can be derived and calculated, emphasizing the importance of understanding both concepts in solution chemistry.