Alright, let's break this problem down step-by-step to understand and solve it.
### Given Data:
1. Osmotic Pressure ([tex]\(\Pi\)[/tex]) = 1.779 atm
2. Mass of Solute (m) = 1.37 grams
3. Volume of Solution (V) = 100 cm[tex]\(^3\)[/tex] = 0.1 liters (since 1 liter = 1000 cm[tex]\(^3\)[/tex])
4. Temperature (T) = 17°C = 290.15 K (we convert from Celsius to Kelvin by adding 273.15)
5. Universal Gas Constant (R) = 0.0821 L atm K[tex]\(^{-1}\)[/tex] mol[tex]\(^{-1}\)[/tex]
### Steps to Find the Molar Mass:
1. Convert Osmotic Pressure to Required Units:
[tex]\[
\Pi = 1.779 \text{ atm}
\][/tex]
2. Convert Volume to Liters (already given as 0.1 liters):
[tex]\[
V = 0.1 \, \text{L}
\][/tex]
3. Convert Temperature to Kelvin (already given conversion):
[tex]\[
T = 290.15 \, \text{K}
\][/tex]
4. Calculate the Number of Moles of Solute ([tex]\(n\)[/tex]) using the osmotic pressure formula:
The osmotic pressure formula is given by:
[tex]\[
\Pi = \frac{nRT}{V}
\][/tex]
Rearranging to solve for [tex]\(n\)[/tex] (number of moles):
[tex]\[
n = \frac{\Pi \cdot V}{R \cdot T}
\][/tex]
Substituting the given values:
[tex]\[
n = \frac{1.779 \, \text{atm} \times 0.1 \, \text{L}}{0.0821 \, \text{L atm K}^{-1} \times 290.15 \, \text{K}}
\][/tex]
From the given calculation, we know the number of moles:
[tex]\[
n = 0.0074681015720584705 \, \text{moles}
\][/tex]
5. Calculate the Molar Mass (M):
The molar mass (M) can be found using the formula:
[tex]\[
M = \frac{m}{n}
\][/tex]
Where [tex]\(m\)[/tex] is the mass of the solute in grams and [tex]\(n\)[/tex] is the number of moles calculated above.
Substituting the values:
[tex]\[
M = \frac{1.37 \, \text{g}}{0.0074681015720584705 \, \text{moles}}
\][/tex]
From the given result, we know the molar mass:
[tex]\[
M = 183.44688898257448 \, \text{g/mol}
\][/tex]
### Conclusion:
The relative molecular mass of the solute, given the osmotic pressure conditions, is approximately [tex]\(183.45 \, \text{g/mol}\)[/tex].
This step-by-step solution provides a detailed and comprehensive method for calculating the molar mass of the solute using the provided data and formulas.
