The molarity and molality of a solution are M and m respectively. If the molecular weight of the solute is M', calculate the density of the solution in terms of M, m and M'

Answer

To calculate the density of the solution in terms of molarity (M), molality (m), and molecular weight of the solute (M'), we can follow these steps:

### Step 1: Understand the Definitions
- **Molarity (M)** is defined as the number of moles of solute per liter of solution.
- **Molality (m)** is defined as the number of moles of solute per kilogram of solvent.
- **Molecular Weight (M')** is the mass of one mole of the solute in grams.

### Step 2: Express Molarity and Molality
1. **Molarity (M)** can be expressed as:
\[ M = \frac{n}{V} \]
where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.

2. **Molality (m)** can be expressed as:
\[ m = \frac{n}{W_{solvent}} \]
where \( W_{solvent} \) is the mass of the solvent in kilograms.

### Step 3: Relate Molarity and Molality
To relate molarity and molality, we can express the number of moles in terms of mass and molecular weight:
\[ n = \frac{W_{solute}}{M'} \]
where \( W_{solute} \) is the mass of the solute in grams.

### Step 4: Calculate the Weight of the Solution
The weight of the solution can be expressed as:
\[ W_{solution} = W_{solute} + W_{solvent} \]
Using molality:
\[ W_{solvent} = \frac{n}{m} = \frac{W_{solute}}{M'} \cdot \frac{1}{m} \]
Thus, substituting for \( W_{solvent} \):
\[ W_{solution} = W_{solute} + \frac{W_{solute}}{M'} \cdot \frac{1}{m} \]

### Step 5: Express Density
Density (\( \rho \)) is defined as:
\[ \rho = \frac{W_{solution}}{V} \]
Substituting the expression for \( W_{solution} \) and using the relationship between volume and molarity:
\[ V = \frac{n}{M} = \frac{W_{solute}}{M' \cdot M} \]
Now, substituting this back into the density equation gives us:
\[ \rho = \frac{W_{solute} + \frac{W_{solute}}{M'} \cdot \frac{1}{m}}{\frac{W_{solute}}{M' \cdot M}} \]

### Step 6: Simplify the Expression
After simplifying the expression, we arrive at:
\[ \rho = \frac{M' + \frac{M'}{M} \cdot \frac{1}{m}}{1} \]
Thus, the final expression for the density of the solution in terms of M, m, and M' is:
\[ \rho = \frac{M'}{1000} + \frac{1}{m} \]

### Final Result
\[ \rho = \frac{M'}{1000} + \frac{1}{m} \]