Calculating Boiling Point And Freezing Point Of Sucrose Solution A Step-by-Step Guide

In this comprehensive exploration, we delve into the fascinating realm of colligative properties, specifically focusing on the impact of dissolving a non-volatile solute, sucrose, in a solvent, water. Colligative properties, those intriguing characteristics of solutions that depend solely on the concentration of solute particles rather than their nature, play a pivotal role in various scientific and industrial applications. Our primary objective is to meticulously calculate the boiling point elevation and freezing point depression of a sucrose solution, providing a clear understanding of how solute concentration influences these crucial physical properties.

At the heart of our investigation lies the concept of colligative properties, which govern the behavior of solutions based on the number of solute particles present. Boiling point elevation and freezing point depression, two prominent colligative properties, dictate the temperature at which a solution boils and freezes, respectively. By unraveling the underlying principles behind these phenomena, we can gain valuable insights into the behavior of solutions and their diverse applications. Understanding these properties is not only crucial in chemistry but also has significant implications in fields like biology, medicine, and engineering. For instance, in the pharmaceutical industry, colligative properties are considered during formulation to maintain stability and efficacy. In the food industry, they are important for determining the texture and preservation of food products. Moreover, in environmental science, they help in understanding the behavior of aquatic ecosystems, particularly concerning the salinity and freezing points of water bodies. This broad applicability underscores the importance of a thorough grasp of these concepts.

Our journey begins with a meticulous examination of the problem statement. We are presented with a sucrose solution, meticulously prepared by dissolving 6.84 grams of sucrose (molecular mass 342 u) in 100 grams of water at a controlled temperature of 298 K. To embark on our calculations, we must first determine the molality of the solution, a crucial parameter that represents the number of moles of solute per kilogram of solvent. This value serves as the cornerstone for calculating both the boiling point elevation and the freezing point depression. We will then employ the appropriate colligative property equations, incorporating the molality and the respective constants (Kb for boiling point elevation and Kf for freezing point depression) to arrive at our final answers. The process involves not only mathematical precision but also a conceptual understanding of the underlying principles. By meticulously working through this problem, we aim to not only find the numerical solutions but also to reinforce our understanding of the relationship between solute concentration and the colligative properties of solutions. This exercise will serve as a foundation for tackling more complex problems in solution chemistry and related fields.

Our first objective is to calculate the boiling point of the sucrose solution. To achieve this, we will employ the colligative property equation for boiling point elevation, which states that the elevation in boiling point (ΔTb) is directly proportional to the molality (m) of the solution and the ebullioscopic constant (Kb) of the solvent. The ebullioscopic constant, specific to each solvent, quantifies the extent to which the boiling point is elevated per unit molality of solute. For water, the Kb value is given as 0.52 K kg mol⁻¹. This value is crucial as it links the concentration of the solution to the change in its boiling point. Understanding the significance of Kb allows us to appreciate how different solvents respond differently to the addition of solutes. For instance, solvents with higher Kb values will exhibit a more pronounced boiling point elevation for the same molality of solute compared to solvents with lower Kb values.

Before we can apply the boiling point elevation equation, we must first determine the molality of the sucrose solution. Molality, defined as the number of moles of solute per kilogram of solvent, provides a concentration measure that is independent of temperature changes, making it particularly suitable for colligative property calculations. To calculate the molality, we begin by converting the mass of sucrose (6.84 g) to moles using its molecular mass (342 u). This conversion yields the number of moles of sucrose present in the solution. Next, we convert the mass of water (100 g) to kilograms, as molality is expressed in moles per kilogram. Finally, we divide the number of moles of sucrose by the mass of water in kilograms to obtain the molality of the solution. The calculation of molality is a fundamental step in understanding the concentration of the solution and its impact on colligative properties. Accurate determination of molality is essential for precise calculations of boiling point elevation and freezing point depression.

With the molality calculated, we can now apply the boiling point elevation equation: ΔTb = Kb * m. By substituting the values of Kb (0.52 K kg mol⁻¹) and the calculated molality (which we will determine in the next step), we can find the elevation in boiling point (ΔTb). This value represents the difference between the boiling point of the solution and the boiling point of the pure solvent (water). To obtain the boiling point of the solution, we add the elevation in boiling point (ΔTb) to the boiling point of pure water (373.15 K). This final calculation gives us the temperature at which the sucrose solution will boil under the given conditions. This process highlights the direct relationship between solute concentration and the boiling point of the solution, illustrating how the presence of a solute elevates the boiling point above that of the pure solvent. The boiling point elevation is a testament to the colligative nature of solutions, where the number of solute particles, rather than their identity, dictates the physical properties of the solution.

Having successfully determined the boiling point of the sucrose solution, we now turn our attention to calculating its freezing point. Similar to boiling point elevation, freezing point depression is another colligative property that depends on the concentration of solute particles in the solution. The presence of a solute lowers the freezing point of the solvent, a phenomenon described by the freezing point depression equation. This equation states that the depression in freezing point (ΔTf) is directly proportional to the molality (m) of the solution and the cryoscopic constant (Kf) of the solvent. The cryoscopic constant, analogous to the ebullioscopic constant, is specific to each solvent and quantifies the extent to which the freezing point is depressed per unit molality of solute. For water, the Kf value is provided as 1.86 K kg mol⁻¹. Understanding the cryoscopic constant is crucial for predicting how much the freezing point will decrease for a given concentration of solute. The magnitude of Kf indicates the sensitivity of the solvent's freezing point to the presence of solutes; a higher Kf value implies a greater freezing point depression for the same molality of solute.

The freezing point depression equation, ΔTf = Kf * m, forms the cornerstone of our calculation. As we have already determined the molality (m) of the sucrose solution in the previous section, we can now directly substitute this value, along with the given Kf value for water (1.86 K kg mol⁻¹), into the equation. This substitution will yield the depression in freezing point (ΔTf), which represents the difference between the freezing point of the pure solvent (water) and the freezing point of the solution. It is important to note that the freezing point depression is a negative value, indicating a decrease in the freezing point compared to the pure solvent. The direct proportionality between molality and freezing point depression underscores the colligative nature of this property, highlighting that the number of solute particles, not their chemical identity, is the key factor influencing the freezing point of the solution.

To obtain the freezing point of the solution, we subtract the depression in freezing point (ΔTf) from the freezing point of pure water (273.15 K). This subtraction accounts for the lowering of the freezing point due to the presence of sucrose in the solution. The resulting value represents the temperature at which the sucrose solution will freeze under the specified conditions. This calculation provides a clear and quantitative understanding of how the addition of a solute affects the freezing point of a solvent. The freezing point depression phenomenon has practical implications in various applications, such as the use of antifreeze in car radiators to prevent freezing in cold temperatures and the addition of salt to icy roads to melt the ice. Understanding and calculating freezing point depression is therefore essential in diverse scientific and engineering contexts.

In this section, we will meticulously walk through the detailed calculations required to determine the boiling point and freezing point of the sucrose solution. Our approach will involve a step-by-step breakdown of each calculation, ensuring clarity and accuracy in our results. We will begin by calculating the molality of the solution, which serves as the foundation for both boiling point elevation and freezing point depression calculations. Subsequently, we will apply the appropriate colligative property equations, substituting the calculated molality and the given constants (Kb and Kf) to determine the changes in boiling point and freezing point. Finally, we will arrive at the boiling point and freezing point of the solution by adjusting the respective values for pure water.

2.4.1 Molality Calculation

The first crucial step in our analysis is to determine the molality of the sucrose solution. As previously defined, molality (m) is the number of moles of solute per kilogram of solvent. To calculate molality, we need to know the number of moles of sucrose and the mass of water in kilograms. We are given that 6.84 grams of sucrose (molecular mass 342 u) is dissolved in 100 grams of water. To convert the mass of sucrose to moles, we divide the mass by the molecular mass: Moles of sucrose = mass of sucrose / molecular mass of sucrose = 6.84 g / 342 g/mol = 0.02 moles. Next, we convert the mass of water from grams to kilograms: Mass of water in kg = mass of water in grams / 1000 = 100 g / 1000 = 0.1 kg. Now we can calculate the molality by dividing the moles of sucrose by the mass of water in kilograms: Molality (m) = moles of sucrose / mass of water in kg = 0.02 moles / 0.1 kg = 0.2 mol/kg. This calculation of molality is essential as it provides the concentration of the solution in a unit that is independent of temperature, making it suitable for colligative property calculations. The accurate determination of molality is paramount for the subsequent calculations of boiling point elevation and freezing point depression.

2.4.2 Boiling Point Elevation Calculation

With the molality of the solution determined, we can now calculate the boiling point elevation (ΔTb) using the equation: ΔTb = Kb * m, where Kb is the ebullioscopic constant for water (0.52 K kg mol⁻¹) and m is the molality of the solution (0.2 mol/kg). Substituting the values into the equation, we get: ΔTb = 0.52 K kg mol⁻¹ * 0.2 mol/kg = 0.104 K. This result indicates that the boiling point of the solution is elevated by 0.104 K compared to pure water. To find the boiling point of the solution, we add this elevation to the boiling point of pure water (373.15 K): Boiling point of solution = boiling point of pure water + ΔTb = 373.15 K + 0.104 K = 373.254 K. Therefore, the boiling point of the sucrose solution is 373.254 K. This calculation demonstrates how the presence of a non-volatile solute like sucrose elevates the boiling point of the solvent, a direct consequence of the colligative properties of solutions. The magnitude of the boiling point elevation is directly proportional to the molality of the solution, as dictated by the boiling point elevation equation.

2.4.3 Freezing Point Depression Calculation

Finally, we calculate the freezing point depression (ΔTf) using the equation: ΔTf = Kf * m, where Kf is the cryoscopic constant for water (1.86 K kg mol⁻¹) and m is the molality of the solution (0.2 mol/kg). Substituting the values into the equation, we get: ΔTf = 1.86 K kg mol⁻¹ * 0.2 mol/kg = 0.372 K. This result indicates that the freezing point of the solution is depressed by 0.372 K compared to pure water. To find the freezing point of the solution, we subtract this depression from the freezing point of pure water (273.15 K): Freezing point of solution = freezing point of pure water - ΔTf = 273.15 K - 0.372 K = 272.778 K. Therefore, the freezing point of the sucrose solution is 272.778 K. This final calculation underscores the colligative nature of freezing point depression, where the presence of a solute lowers the freezing point of the solvent. The extent of the freezing point depression is directly proportional to the molality of the solution, as described by the freezing point depression equation. The ability to calculate freezing point depression is crucial in various applications, such as determining the appropriate concentration of antifreeze in automotive cooling systems.

In conclusion, we have successfully calculated the boiling point and freezing point of a sucrose solution prepared by dissolving 6.84 g of sucrose in 100 g of water at 298 K. Our calculations, based on the principles of colligative properties, have demonstrated the significant impact of solute concentration on the physical properties of solutions. We meticulously determined the molality of the solution, a crucial parameter for colligative property calculations, and subsequently applied the boiling point elevation and freezing point depression equations to arrive at our final answers.

Our findings reveal that the boiling point of the sucrose solution is elevated to 373.254 K, while the freezing point is depressed to 272.778 K. These values provide a clear quantitative understanding of how the presence of sucrose, a non-volatile solute, alters the boiling and freezing behavior of water. The elevation in boiling point and depression in freezing point are direct consequences of the colligative nature of solutions, where the number of solute particles, rather than their chemical identity, dictates the magnitude of the changes in these physical properties. The results underscore the importance of considering solute concentration when predicting and manipulating the boiling and freezing points of solutions.

The calculations and analysis presented in this study highlight the practical applications of colligative properties in various scientific and engineering disciplines. Understanding and applying these principles is essential in fields such as chemistry, biology, medicine, and environmental science. For instance, in the pharmaceutical industry, colligative properties are carefully considered during drug formulation to ensure stability and efficacy. In the food industry, they play a crucial role in determining the texture and preservation of food products. Furthermore, in environmental science, colligative properties are essential for understanding the behavior of aquatic ecosystems, particularly concerning salinity and freezing points. The broader implications of colligative properties extend to numerous real-world applications, making their study and understanding of paramount importance for scientists and engineers alike. The principles discussed here provide a foundation for further exploration of solution chemistry and its diverse applications.