To calculate the osmotic pressure of a 1-liter aqueous solution containing \(2 \times 10^{-2}\) kg of glucose at \(25^\circ C\), we can follow these steps:
### Step 1: Identify the formula for osmotic pressure
The osmotic pressure (\(\pi\)) can be calculated using the formula:
\[
\pi = CRT
\]
where:
- \(C\) = concentration of the solution in moles per liter (M)
- \(R\) = universal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin (K)
### Step 2: Calculate the molar mass of glucose
The molecular formula of glucose is \(C_6H_{12}O_6\). To find the molar mass:
- Carbon (C): \(12 \, \text{g/mol} \times 6 = 72 \, \text{g/mol}\)
- Hydrogen (H): \(1 \, \text{g/mol} \times 12 = 12 \, \text{g/mol}\)
- Oxygen (O): \(16 \, \text{g/mol} \times 6 = 96 \, \text{g/mol}\)
Adding these together:
\[
\text{Molar mass of glucose} = 72 + 12 + 96 = 180 \, \text{g/mol}
\]
### Step 3: Convert the mass of glucose to grams
The mass of glucose given is \(2 \times 10^{-2}\) kg. Converting this to grams:
\[
2 \times 10^{-2} \, \text{kg} = 2 \times 10^{-2} \times 1000 \, \text{g} = 20 \, \text{g}
\]
### Step 4: Calculate the number of moles of glucose
Using the formula for moles:
\[
\text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}
\]
Substituting the values:
\[
\text{Number of moles} = \frac{20 \, \text{g}}{180 \, \text{g/mol}} \approx 0.1111 \, \text{mol}
\]
### Step 5: Calculate the concentration (C) in moles per liter
Since the solution volume is 1 liter, the concentration \(C\) is:
\[
C = \frac{\text{Number of moles}}{\text{Volume (L)}} = \frac{0.1111 \, \text{mol}}{1 \, \text{L}} = 0.1111 \, \text{M}
\]
### Step 6: Convert the temperature to Kelvin
The temperature given is \(25^\circ C\). Converting this to Kelvin:
\[
T = 25 + 273 = 298 \, \text{K}
\]
### Step 7: Substitute the values into the osmotic pressure formula
Now we can calculate the osmotic pressure:
\[
\pi = C \cdot R \cdot T
\]
Substituting the values:
\[
\pi = 0.1111 \, \text{M} \times 0.0821 \, \text{L·atm/(K·mol)} \times 298 \, \text{K}
\]
### Step 8: Calculate the osmotic pressure
Calculating the above expression:
\[
\pi \approx 0.1111 \times 0.0821 \times 298 \approx 2.72 \, \text{atm}
\]
### Final Answer
The osmotic pressure of the solution is approximately \(2.72 \, \text{atm}\).
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