To solve this problem, we need to analyze the heat transfer between the ice cubes and the drink until they reach thermal equilibrium. Let's break it down step by step.
Understanding the Problem
We have four ice cubes, each measuring 2 cm x 2 cm x 2 cm. The total volume of the ice is calculated first, and then we can determine the mass of the ice. The drink has a volume of 200 ml and is initially at a temperature of 10°C. Our goal is to find the final temperature of the drink when thermal equilibrium is reached, and if the ice does not completely melt, we will also calculate the amount of ice that melts.
Step 1: Calculate the Mass of the Ice
The volume of one ice cube is:
- Volume of one cube = side³ = 2 cm × 2 cm × 2 cm = 8 cm³
- Volume of four cubes = 4 × 8 cm³ = 32 cm³
Now, converting this volume to cubic meters (since density is given in kg/m³):
Using the density of ice (900 kg/m³), we can find the mass:
- Mass of ice = Density × Volume = 900 kg/m³ × 32 × 10^-6 m³ = 0.0288 kg
Step 2: Calculate the Heat Gained by the Ice
When the ice is placed in the drink, it will absorb heat to first melt and then warm up to the final temperature. The heat gained by the ice can be divided into two parts:
- Heat required to melt the ice (Q1)
- Heat required to raise the temperature of the melted ice (Q2)
The heat required to melt the ice is given by:
- Q1 = mass of ice × latent heat of fusion = 0.0288 kg × 3.4 × 10^5 J/kg = 97920 J
Next, if we assume that all the ice melts, the temperature of the melted ice (initially at 0°C) will rise to the final equilibrium temperature (Tf). The heat gained by the melted ice is:
- Q2 = mass of melted ice × specific heat capacity of water × change in temperature
- Q2 = 0.0288 kg × 4200 J/kg°C × (Tf - 0°C) = 120.96 × Tf J
Step 3: Calculate the Heat Lost by the Drink
The drink will lose heat as it cools down from its initial temperature (10°C) to the final temperature (Tf). The heat lost by the drink is given by:
- Q = mass of drink × specific heat capacity × change in temperature
- Mass of drink = density × volume = 1000 kg/m³ × 0.2 m³ = 0.2 kg
- Q = 0.2 kg × 4200 J/kg°C × (10°C - Tf) = 840 × (10 - Tf) J
Step 4: Set Up the Equation for Thermal Equilibrium
At thermal equilibrium, the heat gained by the ice equals the heat lost by the drink:
- Q1 + Q2 = Q
- 97920 J + 120.96 × Tf = 840 × (10 - Tf)
Step 5: Solve for Tf
Expanding the equation:
- 97920 + 120.96Tf = 8400 - 840Tf
Combining like terms:
- 120.96Tf + 840Tf = 8400 - 97920
- 960.96Tf = -89520
- Tf = -93.2°C (not physically possible)
This indicates that not all ice has melted. We need to find the amount of ice that melts.
Step 6: Calculate the Amount of Ice That Melts
Let’s denote the mass of ice that melts as m. The heat required to melt m kg of ice is:
The heat lost by the drink when it cools to 0°C is:
- Q = 0.2 kg × 4200 J/kg°C × 10°C = 8400 J
Setting the heat gained by the melting ice equal to the heat lost by the drink:
- m × 3.4 × 10^5 = 8400
- m = 8400 / 3.4 × 10^5 = 0.0247 kg
Final Results
In summary:
- The final temperature of the drink when thermal equilibrium is reached is 0°C.
- The amount of ice that melts is approximately 0.0247 kg or 24.7 grams.
This analysis illustrates the principles of heat transfer and thermal equilibrium, showcasing how energy is conserved in a closed system. If you have any further questions or need clarification on any part, feel free to ask!
