Question

(5 балів) У чашку налили розчин кави при температурі 100 °С і кинули туди кілька кубиків льоду, взятого при температурі 0°С. Коли лід розтанув, мпература розчину виявилася рівною 50 °C. На скільки відсотків зменшилася нцентрація кави в розчині? Теплообмін розчину кави з навколишнім середовищем враховувати. Питомі теплоємності розчину кави і води однакові та дорівнюють кДж/кг*°С, питома теплота плавлення льоду 330 кДж /кг. Зауваження: під центрацією розуміється відношення маси чистої кави до всієї маси розчину. ​

Answer 1

To calculate the percentage decrease in the concentration of coffee in the solution, we need to consider the heat exchange that occurs when the ice melts and the final temperature of the solution.

Given information:

- Initial temperature of the coffee solution: 100 °C - Temperature of the ice: 0 °C - Final temperature of the solution after the ice melts: 50 °C - Specific heat capacities of the coffee solution and water are the same: 1 kJ/kg*°C - Specific heat of fusion for ice: 330 kJ/kg

When the ice is added to the hot coffee solution, heat is transferred from the coffee solution to the ice until the ice melts. This heat transfer causes the temperature of the coffee solution to decrease.

To calculate the heat transferred, we can use the equation:

Q = m * c * ΔT

Where: - Q is the heat transferred - m is the mass of the coffee solution - c is the specific heat capacity of the coffee solution - ΔT is the change in temperature

Let's assume the mass of the coffee solution is 1 kg for simplicity.

The heat transferred from the coffee solution to the ice can be calculated as follows:

Q1 = m * c * (T1 - T2)

Where: - T1 is the initial temperature of the coffee solution (100 °C) - T2 is the final temperature of the solution after the ice melts (50 °C)

Substituting the values into the equation:

Q1 = 1 kg * 1 kJ/kg*°C * (100 °C - 50 °C) = 50 kJ

This means that 50 kJ of heat is transferred from the coffee solution to the ice.

The heat required to melt the ice can be calculated using the equation:

Q2 = m * L

Where: - L is the specific heat of fusion for ice (330 kJ/kg)

Substituting the values into the equation:

Q2 = 1 kg * 330 kJ/kg = 330 kJ

This means that 330 kJ of heat is required to melt the ice.

The total heat transferred from the coffee solution to the ice and to melt the ice is:

Q_total = Q1 + Q2 = 50 kJ + 330 kJ = 380 kJ

Now, let's calculate the change in concentration of coffee in the solution.

Assuming the initial concentration of coffee in the solution is 100%, the mass of coffee in the solution is equal to the mass of the coffee solution.

After the ice melts, the mass of the coffee in the solution remains the same, but the total mass of the solution increases due to the addition of the melted ice.

Let's assume the mass of the melted ice is 1 kg for simplicity.

The final mass of the solution is the sum of the mass of the coffee solution and the mass of the melted ice:

Final mass of the solution = mass of coffee solution + mass of melted ice = 1 kg + 1 kg = 2 kg

The concentration of coffee in the solution after the ice melts can be calculated as follows:

Concentration of coffee in the solution = (mass of coffee / final mass of the solution) * 100%

Substituting the values into the equation:

Concentration of coffee in the solution = (1 kg / 2 kg) * 100% = 50%

Therefore, the concentration of coffee in the solution decreased by 50%.

Please note that this calculation assumes ideal conditions and does not take into account any other factors that may affect the concentration of coffee in the solution.