The osmotic pressure of blood is 8.2 atm at `37^(@)C` . How much glucose should be used per litre of for an intravenous injection that is isotonic with blood ?

Explanation

To determine how much glucose should be used per litre for an intravenous injection that is isotonic with blood, we will follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature given is \(37^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T = 37 + 273 = 310 \, K \]

### Step 2: Use the Osmotic Pressure Formula
The osmotic pressure (\(\pi\)) is given by the formula:
\[ \pi = iCRT \]
Where:
- \(i\) = van 't Hoff factor (for glucose, which does not dissociate, \(i = 1\))
- \(C\) = concentration of the solution in moles per litre
- \(R\) = universal gas constant = \(0.082 \, \text{L atm K}^{-1} \text{mol}^{-1}\)
- \(T\) = temperature in Kelvin

### Step 3: Set Up the Equation
Given that the osmotic pressure of blood is \(8.2 \, \text{atm}\):
\[ 8.2 = 1 \cdot C \cdot 0.082 \cdot 310 \]

### Step 4: Solve for Concentration \(C\)
Rearranging the equation to find \(C\):
\[ C = \frac{8.2}{0.082 \cdot 310} \]
Calculating the denominator:
\[ 0.082 \cdot 310 = 25.42 \]
Now substituting back:
\[ C = \frac{8.2}{25.42} \approx 0.322 \, \text{mol/L} \]

### Step 5: Calculate the Weight of Glucose Required
The molecular weight of glucose (\(C_6H_{12}O_6\)) is calculated as follows:
\[ \text{Molecular weight} = (6 \times 12) + (12 \times 1) + (6 \times 16) = 180 \, \text{g/mol} \]
Now, to find the weight of glucose needed per litre:
\[ \text{Weight} = C \times \text{Molecular weight} \times \text{Volume} \]
Substituting the values:
\[ \text{Weight} = 0.322 \, \text{mol/L} \times 180 \, \text{g/mol} \times 1 \, \text{L} \approx 58.06 \, \text{g} \]

### Final Answer
The amount of glucose required per litre for an intravenous injection that is isotonic with blood is approximately **58.06 grams**.

---