Calculating the Final Temperature When Mixing Two Bodies of Water at Different Temperatures

In thermodynamics, the principle of conservation of energy plays a crucial role in understanding how heat is transferred when two bodies of water at different temperatures are mixed. This principle underpins the calculations necessary to determine the final equilibrium temperature of the mixture. Let's explore this concept through a detailed example and the underlying physics.

Principle of Conservation of Energy

The central concept here is the principle of conservation of energy, which states that the total amount of energy in an isolated system remains constant over time. In the context of heat transfer, this means that the heat lost by the warmer body will be exactly equal to the heat gained by the cooler body.

Example Scenario

Consider two bodies of water: one at a temperature of 0°C and the other at 45°C. Both bodies of water have the same mass of 1 kg.

Mathematical Formulation

Using the principle of conservation of energy, we can set up the following equation to determine the final temperature of the mixture, denoted as (T_f):

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

Where:

(m_1) and (m_2) are the masses of the cooler and warmer bodies of water, both equal to 1 kg in this case. (c) is the specific heat capacity of water, which we assume to be constant for both bodies. (T_1) and (T_2) are the initial temperatures of the cooler and warmer bodies of water, respectively. (T_f) is the final temperature of the mixture.

Since the specific heat capacity (c) is the same for both bodies of water, it cancels out from the equation, simplifying it to:

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

Substituting the given values:

[1 cdot (T_f - 0) -1 cdot (45 - T_f)]

Solving for (T_f):

[T_f frac{45}{2} 22.5 , text{°C}]

Understanding the Result

The final temperature of the mixture, which is 22.5°C, represents the equilibrium temperature where both bodies of water no longer exchange heat with each other. This is because the heat lost by the warmer body (45°C) exactly equals the heat gained by the cooler body (0°C).

Heat Transfer Considerations

There are a few additional considerations in practical scenarios:

Sensible Heat

The initial formula used for heat transfer, (Q m cdot c cdot Delta T), where (Q) is the heat transferred, (m) is the mass, (c) is the specific heat capacity, and (Delta T) is the change in temperature, can be applied here.

Given that the mass of both bodies of water is 1 kg and the specific heat capacity of water is approximately 4.186 J/g·°C, we can write:

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

For practical purposes in this example, since the masses are the same and the specific heat capacities are equal, the equation simplifies as outlined previously to:

[1 cdot (T_f - 0) -1 cdot (45 - T_f)]

Hence, the final temperature (T_f) is 22.5°C.

Volume and Density Considerations

It is important to note that when the volume of water is considered, the density of water at 0°C and 45°C can slightly differ. At 0°C, water has a slightly lower density compared to 45°C. However, since this difference is minimal and the mass is the same, the final temperature remains approximately 22.5°C.

This slight variation in density can lead to a minor adjustment in the final temperature, making it slightly lower than 22.5°C. But for practical purposes, and to keep the explanation straightforward, we can safely use 22.5°C as the final temperature of the mixture.

Conclusion

The principle of conservation of energy and the concept of heat transfer are fundamental in understanding the behavior of two bodies of water at different temperatures when they are mixed. The final temperature of the mixture can be calculated by equating the heat lost by the warmer body to the heat gained by the cooler body. This application of thermodynamics showcases the importance of energy conservation in real-world scenarios.