A `5.25%` solution of a substance is isotonic with a `1.5%` solution of urea (molar mass `= 60g mol^(-1)`) in the same solvent. If the densities of both the solutions are assumed to be equal to `1.0 g cm^(-3)`, molar mass of the substance will be:

Options

A. `210 g mol^(-1)`
B. `90. 0 g mol^(-1)`
C. `115.0 g mol^(-1)`
D. `105.0 g mol^(-1)`

Answer

To find the molar mass of the substance that is isotonic with a 1.5% solution of urea, we can follow these steps:

### Step 1: Understand Isotonic Solutions
Isotonic solutions have the same osmotic pressure. Therefore, we can use the relationship between the concentrations of the two solutions to set up an equation.

### Step 2: Write the Osmotic Pressure Equation
The osmotic pressure (π) is given by the formula:
\[ \pi = CRT \]
Where:
- \(C\) = concentration (in molarity)
- \(R\) = gas constant
- \(T\) = temperature (in Kelvin)

For two isotonic solutions, we can equate their osmotic pressures:
\[ C_1 = C_2 \]

### Step 3: Define the Concentrations
Let:
- \(C_1\) be the concentration of the 5.25% solution of the substance.
- \(C_2\) be the concentration of the 1.5% solution of urea.

### Step 4: Calculate the Number of Moles
The number of moles (\(n\)) can be calculated using the formula:
\[ n = \frac{W}{M} \]
Where:
- \(W\) = mass of the solute
- \(M\) = molar mass of the solute

For the 5.25% solution:
- \(W_1 = 5.25 \, \text{g}\) (in 100 g of solution)
- \(M_1 = ?\) (molar mass of the substance)

For the 1.5% urea solution:
- \(W_2 = 1.5 \, \text{g}\) (in 100 g of solution)
- \(M_2 = 60 \, \text{g/mol}\) (given)

### Step 5: Set Up the Equation
Since the volumes of the solutions are the same, we can equate the number of moles:
\[ \frac{W_1}{M_1} = \frac{W_2}{M_2} \]
Substituting the values we have:
\[ \frac{5.25}{M_1} = \frac{1.5}{60} \]

### Step 6: Solve for Molar Mass
Cross-multiplying gives:
\[ 5.25 \times 60 = 1.5 \times M_1 \]
Calculating the left side:
\[ 315 = 1.5 \times M_1 \]
Now, divide both sides by 1.5 to find \(M_1\):
\[ M_1 = \frac{315}{1.5} = 210 \, \text{g/mol} \]

### Final Answer
The molar mass of the substance is **210 g/mol**.

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