Heat Required To Transform Ice To Steam Calculating KW For Phase Transition

by Scholario Team 76 views

In the realm of thermodynamics, calculating the energy required to change the state of a substance is a fundamental concept. This article delves into the intricate process of determining the amount of heat, measured in kilowatts (kW), needed to transform 2500 grams of ice initially at -10°C into steam at 220°C. We will explore the various stages involved, including the heat required for temperature changes and phase transitions, while also considering the latent heats of fusion and vaporization.

Understanding the Concepts: Heat, Temperature, and Phase Transitions

Before we dive into the calculations, it's crucial to grasp the underlying concepts. Heat is the transfer of thermal energy between objects or systems due to a temperature difference. Temperature, on the other hand, is a measure of the average kinetic energy of the particles within a substance. When heat is added to a substance, its temperature generally increases. However, at certain points, the added energy doesn't raise the temperature but instead causes a phase transition, such as melting or boiling.

A phase transition involves a change in the physical state of a substance. For instance, ice (solid) melts into water (liquid), and water boils into steam (gas). These transitions require energy to overcome the intermolecular forces holding the substance in its initial state. This energy is known as latent heat. The latent heat of fusion is the energy required to melt a solid into a liquid, while the latent heat of vaporization is the energy needed to vaporize a liquid into a gas.

In our case, we're dealing with both temperature changes and phase transitions. The ice needs to be warmed from -10°C to 0°C, then melted into water at 0°C. The water then needs to be heated from 0°C to 100°C, boiled into steam at 100°C, and finally, the steam needs to be heated from 100°C to 220°C. Each of these steps requires a specific amount of heat, which we will calculate in detail.

Step-by-Step Calculation: From Ice at -10°C to Steam at 220°C

To accurately determine the total heat required, we need to break down the process into five distinct stages:

  1. Heating the ice from -10°C to 0°C: In this stage, we're increasing the temperature of the ice without changing its phase. The heat required can be calculated using the formula:

    Q1 = m * c_ice * ΔT1

    Where:

    • Q1 is the heat required (in calories)
    • m is the mass of the ice (2500 g)
    • c_ice is the specific heat capacity of ice (approximately 0.5 cal/g°C)
    • ΔT1 is the change in temperature (0°C - (-10°C) = 10°C)

    Plugging in the values, we get:

    Q1 = 2500 g * 0.5 cal/g°C * 10°C = 12500 calories

  2. Melting the ice at 0°C: Here, we're dealing with a phase transition from solid to liquid. The heat required is calculated using the latent heat of fusion:

    Q2 = m * Lf

    Where:

    • Q2 is the heat required (in calories)
    • m is the mass of the ice (2500 g)
    • Lf is the latent heat of fusion of ice (80 cal/g)

    So:

    Q2 = 2500 g * 80 cal/g = 200000 calories

  3. Heating the water from 0°C to 100°C: Now we're heating the water, again without a phase change. We use the specific heat capacity of water:

    Q3 = m * c_water * ΔT2

    Where:

    • Q3 is the heat required (in calories)
    • m is the mass of the water (2500 g)
    • c_water is the specific heat capacity of water (1 cal/g°C)
    • ΔT2 is the change in temperature (100°C - 0°C = 100°C)

    Therefore:

    Q3 = 2500 g * 1 cal/g°C * 100°C = 250000 calories

  4. Vaporizing the water at 100°C: This is another phase transition, from liquid to gas. We use the latent heat of vaporization, but this time, we need to use the value provided in Joules (J) and convert it to calories. There are approximately 4.184 Joules in 1 calorie.

    Q4 = m * Lv

    Where:

    • Q4 is the heat required (in Joules)
    • m is the mass of the water (2500 g)
    • Lv is the latent heat of vaporization of water (2254 J/g)

    Q4 = 2500 g * 2254 J/g = 5635000 Joules

    Converting to calories:

    Q4 = 5635000 J / 4.184 J/cal ≈ 1346800 calories

  5. Heating the steam from 100°C to 220°C: Finally, we heat the steam. The specific heat capacity of steam is approximately 0.48 cal/g°C.

    Q5 = m * c_steam * ΔT3

    Where:

    • Q5 is the heat required (in calories)
    • m is the mass of the steam (2500 g)
    • c_steam is the specific heat capacity of steam (0.48 cal/g°C)
    • ΔT3 is the change in temperature (220°C - 100°C = 120°C)

    So:

    Q5 = 2500 g * 0.48 cal/g°C * 120°C = 144000 calories

Total Heat and Conversion to Kilowatts

Now, we sum up the heat required for each stage:

Q_total = Q1 + Q2 + Q3 + Q4 + Q5

Q_total = 12500 cal + 200000 cal + 250000 cal + 1346800 cal + 144000 cal = 1953300 calories

To convert calories to Joules, we multiply by 4.184:

Q_total = 1953300 cal * 4.184 J/cal ≈ 8171677.2 Joules

To express this in kilojoules (kJ), we divide by 1000:

Q_total ≈ 8171.68 kJ

To express the heat requirement as power in kilowatts (kW), we need to consider the time it takes to supply this heat. However, the problem does not provide a time frame. If we assume this transformation occurs over 1 second, then the power required would be:

Power = Q_total / time

Power = 8171.68 kJ / 1 s = 8171.68 kW

Important Note: This is a very high power requirement and assumes the entire process happens in just one second. In a more realistic scenario, the transformation would take significantly longer, resulting in a lower power requirement. For example, if the process takes 10 minutes (600 seconds), the power would be:

Power = 8171.68 kJ / 600 s ≈ 13.62 kW

Key Takeaways and Practical Implications

This calculation demonstrates the significant amount of energy required to change the phase of a substance, particularly during vaporization. The latent heat of vaporization is substantially higher than the latent heat of fusion, highlighting the energy-intensive nature of boiling a liquid into a gas. Understanding these energy requirements is crucial in various applications, including industrial processes, climate modeling, and even cooking.

In practical terms, this means that heating water to boil requires considerably more energy than simply raising its temperature. This is why kettles and boilers are designed with powerful heating elements to deliver the necessary energy for vaporization. Similarly, industrial processes that involve phase transitions, such as steam power generation, require careful energy management to optimize efficiency.

Moreover, the concepts of heat transfer and phase transitions play a vital role in understanding climate phenomena. The evaporation of water from oceans and other bodies of water absorbs a tremendous amount of energy, influencing global weather patterns and temperature regulation. Accurate modeling of these processes is essential for predicting climate change and its impacts.

Conclusion: The Significance of Thermodynamic Calculations

Calculating the heat required for phase transitions and temperature changes is a cornerstone of thermodynamics. By understanding the principles of specific heat capacity, latent heat, and heat transfer, we can accurately predict and manage energy requirements in various applications. This detailed calculation, from ice at -10°C to steam at 220°C, underscores the importance of these concepts in both scientific and practical contexts. Whether it's designing efficient heating systems or understanding global climate patterns, a solid grasp of thermodynamics is indispensable.

In summary, the transformation of 2500 g of ice at -10°C to steam at 220°C requires a substantial amount of energy due to the temperature changes and phase transitions involved. The calculation highlights the significance of latent heat, especially the latent heat of vaporization, and underscores the practical implications of these thermodynamic principles in various fields.