Estimating Heat Quantity For Melting Ice A Comprehensive Guide

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Melting ice is a fascinating phase transition that requires a specific amount of heat energy. In this comprehensive guide, we will walk through the process of estimating the quantity of heat needed to melt 150 g of ice initially at -10°C into water at 15°C. This calculation involves several steps, each addressing different aspects of the phase transition and temperature change. To accurately estimate the total heat required, we need to consider the specific heat capacities of both ice and water, as well as the latent heat of fusion for ice. The specific heat capacity is the amount of heat required to raise the temperature of 1 kg of a substance by 1 Kelvin (or 1 degree Celsius). The latent heat of fusion is the amount of heat required to change 1 kg of a substance from a solid to a liquid state at its melting point. By breaking down the process into distinct stages and applying the relevant formulas, we can arrive at a precise estimate of the total heat required. Let's delve deeper into each stage and understand the physics behind it, ensuring a clear and thorough understanding of the entire process. This comprehensive guide will not only provide the numerical answer but also illuminate the underlying principles and steps involved in such calculations, making it a valuable resource for students and enthusiasts alike.

Understanding the Problem

Before diving into the calculations, let's clearly define the problem. We have 150 g of ice at -10°C, and we want to transform it into water at 15°C. This process can be broken down into three distinct stages, each requiring a specific amount of heat:

1. Heating the Ice: Raising the temperature of the ice from -10°C to its melting point, 0°C.
2. Melting the Ice: Changing the phase of the ice from solid to liquid at 0°C.
3. Heating the Water: Raising the temperature of the water from 0°C to 15°C.

Each of these stages involves different physical processes and requires a specific calculation. To accurately estimate the total heat, we must consider each stage separately and then sum the results. This detailed approach ensures that we account for all the energy transformations occurring during the process. The initial stage focuses on increasing the kinetic energy of the ice molecules, while the second stage overcomes the intermolecular forces holding the ice structure together. The final stage again involves increasing the kinetic energy, but this time in the liquid phase. Understanding these distinct stages is crucial for a comprehensive grasp of the heat transfer involved in the phase transition process.

Step 1 Heating the Ice from -10°C to 0°C

The first step involves heating the ice from its initial temperature of -10°C to its melting point of 0°C. During this phase, the ice remains in a solid state, and the heat energy supplied increases the kinetic energy of the water molecules within the ice structure. The formula to calculate the heat required for this step is:

$$Q_1 = m imes c_{ice} imes riangle T$$

Where:

• $Q_1$ is the heat required (in Joules).
• $m$ is the mass of the ice (in kg).
• $c_{ice}$ is the specific heat capacity of ice (in J/kg·K).
• $ riangle T$ is the change in temperature (in °C or K).

Given values:

• $m = 150 g = 0.15 kg$
• $c_{ice} = 2100 J/kg·K$
• $ riangle T = 0°C - (-10°C) = 10°C$

Plugging these values into the formula, we get:

$$Q_1 = 0.15 kg imes 2100 J/kg·K imes 10°C = 3150 J$$

Therefore, the heat required to raise the temperature of the ice from -10°C to 0°C is 3150 Joules. This initial heating phase is crucial as it prepares the ice to undergo the phase transition from solid to liquid. The energy supplied during this stage primarily increases the vibrational motion of the water molecules within the ice lattice, bringing them closer to the point where they can break free from their fixed positions and transition into the liquid state. This step is a clear illustration of how heat energy is used to increase the temperature of a substance without changing its phase.

Step 2 Melting the Ice at 0°C

The second step involves the phase transition from ice to water at a constant temperature of 0°C. This process requires energy to break the hydrogen bonds holding the ice structure together, allowing the water molecules to move more freely in the liquid state. The heat required for this phase change is calculated using the formula:

$$Q_2 = m imes L_f$$

Where:

• $Q_2$ is the heat required (in Joules).
• $m$ is the mass of the ice (in kg).
• $L_f$ is the latent heat of fusion for ice (in J/kg).

Given values:

• $m = 0.15 kg$
• $L_f = 3.36 imes 10^5 J/kg$

Plugging these values into the formula, we get:

$$Q_2 = 0.15 kg imes 3.36 imes 10^5 J/kg = 50400 J$$

Thus, the heat required to melt the ice at 0°C is 50400 Joules. This phase transition is a critical part of the process, as it requires significantly more energy than simply changing the temperature. The latent heat of fusion represents the energy needed to overcome the intermolecular forces that maintain the solid structure of ice. During melting, the supplied heat is used to increase the potential energy of the water molecules, rather than their kinetic energy, which is why the temperature remains constant. Understanding this distinction between sensible heat (temperature change) and latent heat (phase change) is essential for grasping the thermodynamics of phase transitions.

Step 3 Heating the Water from 0°C to 15°C

The final step involves heating the water from 0°C to the desired final temperature of 15°C. In this stage, the water is already in the liquid phase, and the heat energy supplied increases the kinetic energy of the water molecules, resulting in a temperature rise. The formula to calculate the heat required for this step is:

$$Q_3 = m imes c_{water} imes riangle T$$

Where:

• $Q_3$ is the heat required (in Joules).
• $m$ is the mass of the water (in kg).
• $c_{water}$ is the specific heat capacity of water (in J/kg·K).
• $ riangle T$ is the change in temperature (in °C or K).

Given values:

• $m = 0.15 kg$
• $c_{water} = 4200 J/kg·K$
• $ riangle T = 15°C - 0°C = 15°C$

Plugging these values into the formula, we get:

$$Q_3 = 0.15 kg imes 4200 J/kg·K imes 15°C = 9450 J$$

Therefore, the heat required to raise the temperature of the water from 0°C to 15°C is 9450 Joules. This final heating stage is similar to the initial heating of the ice, but it occurs in the liquid phase. The specific heat capacity of water is higher than that of ice, indicating that it takes more energy to raise the temperature of water by the same amount. This higher specific heat capacity is due to the more complex interactions between water molecules in the liquid state compared to the solid state. Understanding the thermal properties of different phases of matter is crucial for accurate heat transfer calculations.

Total Heat Required

To find the total heat required to melt the ice and heat the resulting water, we sum the heat from each of the three steps:

$$Q_{total} = Q_1 + Q_2 + Q_3$$

Substituting the values we calculated:

$$Q_{total} = 3150 J + 50400 J + 9450 J = 63000 J$$

Therefore, the total heat required to melt 150 g of ice at -10°C to water at 15°C is 63000 Joules. This final result represents the cumulative energy needed to complete the entire process, from raising the ice's temperature to melting it and then heating the resulting water. The significant amount of energy required highlights the importance of considering all phases and temperature changes when dealing with thermal processes. Understanding these principles is crucial for various applications, including climate modeling, industrial processes, and everyday phenomena.

Conclusion

In summary, we have estimated that 63000 Joules of heat are required to melt 150 g of ice at -10°C and heat the resulting water to 15°C. This calculation involved three key steps: heating the ice to its melting point, melting the ice into water, and then heating the water to the final temperature. Each step required a separate calculation using the principles of specific heat capacity and latent heat of fusion. By breaking down the problem into manageable stages, we were able to accurately estimate the total heat required. This detailed process not only provides the numerical answer but also enhances our understanding of the underlying physics involved in phase transitions and heat transfer. The concepts discussed here are fundamental to thermodynamics and have broad applications in various scientific and engineering fields. Mastering these principles allows for a deeper understanding of how energy interacts with matter and the transformations that can occur.

This comprehensive guide has provided a step-by-step approach to solving this problem, making it a valuable resource for anyone studying thermodynamics or needing to perform similar calculations. The clear explanations and detailed calculations aim to enhance understanding and provide a solid foundation for further exploration of thermal physics. The application of specific heat capacity and latent heat concepts are crucial in many real-world scenarios, from designing efficient heating and cooling systems to understanding the effects of climate change. This knowledge empowers us to make informed decisions and develop innovative solutions in various fields.