In isotonic solution of protein A and protein B, 0.73g of protein A is dissolved in 250ml of solution while 1.65g of protein B is dissolved in 1L solutionthen what is ratio of molecular mass of protein A and protein B?

Answer

To solve the problem of finding the ratio of the molecular mass of protein A and protein B in isotonic solutions, we can follow these steps:

### Step 1: Understand the Concept of Isotonic Solutions
Isotonic solutions have the same osmotic pressure. Therefore, the osmotic pressure (π) of protein A is equal to the osmotic pressure of protein B.

### Step 2: Write the Formula for Osmotic Pressure
The osmotic pressure is given by the formula:
\[ \pi = C \cdot RT \]
Where:
- \( C \) is the concentration of the solution (in moles per liter),
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

Since both solutions are isotonic, we can set the osmotic pressures equal to each other:
\[ \pi_A = \pi_B \]

### Step 3: Calculate the Concentration of Protein A
The concentration of protein A can be calculated using the formula:
\[ C_A = \frac{\text{moles of A}}{\text{volume of solution A}} \]
The moles of protein A can be expressed as:
\[ \text{moles of A} = \frac{W_A}{M_A} \]
Where:
- \( W_A = 0.73 \, \text{g} \) (mass of protein A),
- \( M_A \) is the molar mass of protein A.

The volume of solution A is 250 mL, which is equivalent to 0.250 L. Thus:
\[ C_A = \frac{0.73 / M_A}{0.250} = \frac{0.73}{0.250 \cdot M_A} \]

### Step 4: Calculate the Concentration of Protein B
Similarly, for protein B:
\[ C_B = \frac{\text{moles of B}}{\text{volume of solution B}} \]
Where:
\[ \text{moles of B} = \frac{W_B}{M_B} \]
With:
- \( W_B = 1.65 \, \text{g} \) (mass of protein B),
- \( M_B \) is the molar mass of protein B.

The volume of solution B is 1 L. Thus:
\[ C_B = \frac{1.65 / M_B}{1} = \frac{1.65}{M_B} \]

### Step 5: Set the Concentrations Equal
Since the osmotic pressures are equal:
\[ C_A \cdot RT = C_B \cdot RT \]
The \( RT \) cancels out, leading to:
\[ \frac{0.73}{0.250 \cdot M_A} = \frac{1.65}{M_B} \]

### Step 6: Cross-Multiply to Solve for the Ratio
Cross-multiplying gives us:
\[ 0.73 \cdot M_B = 1.65 \cdot (0.250 \cdot M_A) \]
Rearranging this gives:
\[ \frac{M_A}{M_B} = \frac{1.65 \cdot 0.250}{0.73} \]

### Step 7: Calculate the Ratio
Now, we can calculate the ratio:
\[ \frac{M_A}{M_B} = \frac{1.65 \cdot 0.250}{0.73} = \frac{0.4125}{0.73} \approx 1.769 \]

### Final Answer
The ratio of the molecular mass of protein A to protein B is approximately:
\[ \frac{M_A}{M_B} \approx 1.769 \]

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