Calculating the Final Temperature of Mixed Water: A Comprehensive Guide

When two bodies of water at different temperatures are mixed, the final temperature of the mixture can be determined using the principle of conservation of energy. This principle ensures that the heat lost by the hotter water is exactly equal to the heat gained by the cooler water. This article will provide a detailed explanation of how to calculate the final temperature using a formula and several practical examples.

Understanding the Principle of Conservation of Energy in Thermodynamics

The principle of conservation of energy asserts that energy cannot be created or destroyed but can be transferred from one form to another or transferred between different objects. In the context of mixing water, this means that the heat lost by the hotter water will be equal to the heat gained by the cooler water, assuming no heat is lost to the surroundings.

The Formula for Calculating Final Temperature

To find the final temperature (T_f) when two masses of water at different temperatures (T_1) and (T_2) are mixed, the following formula can be applied:

[m_1 cdot c cdot T_1 - T_f m_2 cdot c cdot T_f - T_2]

Where:

(m_1) mass of the first body of water (T_1) initial temperature of the first body of water (m_2) mass of the second body of water (T_2) initial temperature of the second body of water (c) specific heat capacity of water, which is constant and can be canceled out.

Simplifying the Equation

By canceling (c) from both sides, the equation simplifies to:

[m_1 cdot T_1 - T_f m_2 cdot T_f - T_2]

Rearranging the equation gives:

[m_1 cdot T_1 m_2 cdot T_2 m_1 cdot m_2 cdot T_f]

Thus, the final temperature (T_f) can be calculated as:

[T_f frac{m_1 cdot T_1 m_2 cdot T_2}{m_1 m_2}]

Practical Example: Mixing Two Different Temperatures of Water

To illustrate the calculation, consider the following scenario:

2 kg of water at 80°C (hot water) 3 kg of water at 20°C (cool water)

Plugging these values into the formula:

[T_f frac{2 text{ kg} cdot 80 text{°C} 3 text{ kg} cdot 20 text{°C}}{2 text{ kg} 3 text{ kg}} frac{160 text{°C} 60 text{°C}}{5} frac{220 text{°C}}{5} 44 text{°C}]

Therefore, the final temperature of the mixture would be 44°C.

Conducting a Simple Experiment

To conduct an experiment and verify the results, follow these steps:

Get two equal containers of water and measure the temperature of each. Pour the water from both containers into a third, larger container. Measure the temperature of the mixed water. Repeat the experiment multiple times with varying amounts of water to observe consistency. Possible results can be plotted on a graph to visualize the relationship between the initial temperatures and the final temperature.

Further Exploration

Understanding the principle of conservation of energy and the concept of specific heat capacity can help in various practical applications beyond just mixing water. Experimenting with different masses and initial temperatures can lead to further insights into the behavior of thermal systems.

Conclusion

Mixing two bodies of water at different temperatures is a practical application of thermodynamic principles. By understanding and applying the formula for calculating the final temperature, you can predict the outcome of mixing processes with precision. Whether you are conducting scientific experiments or simply mixing water in a practical setting, this knowledge is invaluable.