Temperature Mixing of Water: Equal Masses vs. Volumes

When mixing water at different temperatures, it's important to understand how the final temperature will be affected. In this article, we'll delve into the principle of conservation of energy to predict the temperature of a mixture when you combine water at 100°C with water at 0°C. We'll explore the role of time and volume in this process and clarify why the details matter.

The Principle of Conservation of Energy

To find the final temperature when mixing water at 100°C with water at 0°C, we can use the principle of conservation of energy. This principle states that the total energy before mixing is equal to the total energy after mixing. In simpler terms, the heat lost by the hotter water is equal to the heat gained by the colder water.

Equal Masses of Water

Assuming you mix equal masses of hot water at 100°C and cold water at 0°C, the equation for the final temperature can be expressed as:

T_f (m_1 T_1 m_2 T_2) / (m_1 m_2)

Where:

T_f is the final temperature. m_1 and m_2 are the masses of the hot and cold water respectively. T_1 is the temperature of the hot water (100°C). T_2 is the temperature of the cold water (0°C).

When m_1 m_2, the formula simplifies to:

T_f (100 0) / 2 50°C

Thus, if you mix equal volumes (or masses) of water at 100°C and 0°C, the final temperature of the mixture will be approximately 50°C.

Equal Volumes of Water

When the volumes or masses of the water aren't equal, the final temperature will be closer to the temperature of the larger volume of water. For instance, if you mix 1ml of 100°C water with 1 gallon of 0°C water, the temperature of the hot water won't significantly change because the larger volume of cold water will dominate. Conversely, if you mix equal volumes of hot and cold water, the temperature will change more dramatically.

The Role of Time and Volume

The analysis above assumes an instantaneous mixing process and equal volumes. However, in real-world scenarios, the mixing time and volumes can play significant roles. The mixing time of 5 seconds, while short, does not significantly affect the final temperature if the volumes are equal and well-mixed.

Examples of Mixing Different Quantities

The critical detail lies in the quantities involved. If you're mixing 1ml of 100°C water with 1ml of 0°C water, the temperature will change more dramatically compared to mixing 1ml of 100°C water with 1 gallon of 0°C water.

Example 1: Mixing 1ml of 100°C water with 1ml of 0°C water:

Mass of hot water: 1ml Temperature of hot water: 100°C Mass of cold water: 1ml Temperature of cold water: 0°C Final temperature: (1ml*100 1ml*0) / (1ml 1ml) 50°C

Example 2: Mixing 1 gallon of 100°C water with 1 gallon of 0°C water:

Mass of hot water: 1 gallon Temperature of hot water: 100°C Mass of cold water: 1 gallon Temperature of cold water: 0°C Final temperature: (1gallon*100 1gallon*0) / (1gallon 1gallon) 50°C

In both examples, the volumes are equal, and the final temperature will be 50°C. However, in practical scenarios, the mixing time, the specific volumes, and the thermal conductivity of the containers can all affect the outcome.

Conclusion

The temperature of the mixture when mixing water at different temperatures is governed by the principle of conservation of energy. While the mixing time of 5 seconds generally does not significantly affect the final temperature if the volumes are equal and well-mixed, the specific volumes and masses of the water do. The details matter in real-world applications, and understanding these principles can help in various scientific and engineering contexts.