To determine the minimum temperature required to dissolve 100 grams of sucrose ([tex]\(C_{12}H_{22}O_{11}\)[/tex]) in 100 grams of water, we need to understand how the solubility of sucrose in water varies with temperature. Solubility is the maximum amount of a substance that can be dissolved in a solvent at a given temperature.
Here is the step-by-step solution:
1. Understanding Solubility and Saturation at Different Temperatures:
- At room temperature (around 20°C), the solubility of sucrose in water is significantly lower than at higher temperatures.
- As the temperature increases, the solubility of sugars like sucrose in water also increases because the molecules move faster and can dissolve more solute.
2. Analysis of Saturation Point:
- We need to find a point where the solubility of sucrose reaches at least 100 grams per 100 grams of water.
- Typically, scientists study the solubility curve of a substance, which shows how much solute can dissolve in a solvent at different temperatures.
3. Evaluating Given Options:
- [tex]\(0^{\circ} C\)[/tex]: At this temperature, the solubility of sucrose is quite low, and it is highly unlikely that 100 grams of sucrose will dissolve in 100 grams of water.
- [tex]\(20^{\circ} C\)[/tex]: While the solubility increases at this temperature, it is still not sufficient to dissolve 100 grams of sucrose in 100 grams of water.
- [tex]\(50^{\circ} C\)[/tex]: The solubility of sucrose improves significantly but generally does not reach the required threshold.
- [tex]\(90^{\circ} C\)[/tex]: Close to boiling, the amount of sucrose that can dissolve is much higher and may approach or exceed required levels.
- [tex]\(100^{\circ} C\)[/tex]: This is the boiling point of water, where solubility for many substances, including sucrose, reaches its maximum. At this temperature, it is almost certain that 100 grams of sucrose can dissolve in 100 grams of water.
4. Conclusion:
- Based on the typical solubility behavior of sucrose in water, the minimum temperature required to dissolve 100 grams of [tex]\(C_{12}H_{22}O_{11}\)[/tex] in 100 grams of water is closest to [tex]\(100^{\circ}C\)[/tex].
Therefore, the minimum temperature needed to dissolve 100 grams of sucrose ([tex]\(C_{12}H_{22}O_{11}\)[/tex]) in 100 grams of water is [tex]\(100^{\circ} C\)[/tex].
