The osmotic pressure of 0.200 g of haemoglobin in 20.0 ml of solution is 2.88 torr at `25^(@)C`. Calculate the molecular weight of haemoglobin.

Explanation

To calculate the molecular weight of hemoglobin using the given osmotic pressure, we can use the formula for osmotic pressure:

\[ \Pi = \frac{n}{V}RT \]

Where:
- \(\Pi\) = osmotic pressure (in atm)
- \(n\) = number of moles of solute
- \(V\) = volume of solution (in liters)
- \(R\) = universal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin

### Step 1: Convert osmotic pressure from torr to atm
Given that 1 atm = 760 torr, we can convert the osmotic pressure:

\[ \Pi = \frac{2.88 \text{ torr}}{760 \text{ torr/atm}} = 0.00378947 \text{ atm} \]

### Step 2: Convert the volume of the solution from mL to L
The volume of the solution is given as 20.0 mL. We convert this to liters:

\[ V = \frac{20.0 \text{ mL}}{1000 \text{ mL/L}} = 0.0200 \text{ L} \]

### Step 3: Convert the temperature from Celsius to Kelvin
The temperature is given as 25°C. We convert this to Kelvin:

\[ T = 25 + 273.15 = 298.15 \text{ K} \]

### Step 4: Rearrange the osmotic pressure formula to find the number of moles (n)
We can rearrange the formula to solve for \(n\):

\[ n = \frac{\Pi V}{RT} \]

Substituting the values we have:

\[ n = \frac{(0.00378947 \text{ atm})(0.0200 \text{ L})}{(0.0821 \text{ L·atm/(K·mol)})(298.15 \text{ K})} \]

Calculating this gives:

\[ n = \frac{0.0000757894}{24.475} \approx 0.000003096 \text{ mol} \]

### Step 5: Calculate the molecular weight (M)
Molecular weight (M) is calculated using the formula:

\[ M = \frac{\text{mass}}{n} \]

Where the mass of hemoglobin is given as 0.200 g. Substituting the values we have:

\[ M = \frac{0.200 \text{ g}}{0.000003096 \text{ mol}} \approx 64557.6 \text{ g/mol} \]

### Final Answer
The molecular weight of hemoglobin is approximately **64557.6 g/mol**.

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