Question

A solution containing 200 g of a protein per litre is isotonic with 1.71g of sucrose ($C_{12}H_{22}O_{11}$) per litre. The molecular mass of protein is $a \times 10^x$u. Find the value of a + x.

Answer 1

8

Answer 2

1. Isotonic Solutions: Isotonic solutions have equal molar concentrations.
2. Molar Concentration Formula: Molar concentration ($C$) is given by $C = \frac{w}{M \cdot V}$, where $w$ is the mass, $M$ is the molar mass, and $V$ is the volume.
3. Equality for Isotonic Solutions: For isotonic solutions with the same volume $V$, their molar concentrations are equal: $\frac{w_1}{M_1 \cdot V} = \frac{w_2}{M_2 \cdot V}$. This simplifies to $\frac{w_1}{M_1} = \frac{w_2}{M_2}$.
4. Molar Mass of Sucrose: The molar mass of sucrose ($C_{12}H_{22}O_{11}$) is calculated as $(12 \times 12.011) + (22 \times 1.008) + (11 \times 15.999) \approx 342$ g/mol.
5. Given Values:
   • Mass of protein ($w_1$) = 200 g/L
   • Mass of sucrose ($w_2$) = 1.71 g/L
   • Molar mass of sucrose ($M_2$) = 342 g/mol
6. Substituting Values: Substitute these values into the simplified equation: $\frac{200 \text{ g}}{M_{protein}} = \frac{1.71 \text{ g}}{342 \text{ g/mol}}$.
7. Solving for Protein's Molecular Mass: $M_{protein} = \frac{200 \text{ g} \times 342 \text{ g/mol}}{1.71 \text{ g}}$ $M_{protein} = \frac{200 \times 342}{1.71} \text{ g/mol}$ To simplify, multiply the numerator and denominator by 100: $M_{protein} = \frac{200 \times 342 \times 100}{171} \text{ g/mol}$ Recognize that $342 = 2 \times 171$: $M_{protein} = \frac{200 \times (2 \times 171) \times 100}{171} \text{ g/mol}$ Cancel out 171: $M_{protein} = 200 \times 2 \times 100 \text{ g/mol} = 40000 \text{ g/mol}$.
8. Scientific Notation: The molecular mass of the protein is given as $a \times 10^x$ u. So, $40000 \text{ u} = a \times 10^x$ u.
9. Expressing in Scientific Notation: Express 40000 in scientific notation, where $1 \le a < 10$: $40000 = 4 \times 10^4$.
10. Identifying a and x: Therefore, $a = 4$ and $x = 4$.
11. Final Calculation: The value of $a + x$ is $4 + 4 = 8$.