Heat Required To Melt Ice Into Water A Physics Calculation

by Admin 59 views

In this article, we will delve into the fascinating world of thermodynamics and explore the process of calculating the amount of heat energy required to transform ice at a sub-zero temperature into water at a warmer temperature. This is a fundamental concept in physics and has numerous practical applications, from understanding weather patterns to designing efficient cooling systems. Our specific problem involves determining the heat needed to melt 150 grams of ice initially at $-10^{\circ} C$ into water at $15^{\circ} C$. To solve this, we will break down the process into distinct stages, each requiring a specific amount of heat. We'll utilize key concepts such as specific heat capacity (SHC) and latent heat of fusion, along with the provided constants, to arrive at our final answer. So, let's embark on this journey of thermal calculations and unravel the intricacies of phase transitions.

Before we dive into the calculations, let's solidify our understanding of the key concepts involved in this process. The first concept is specific heat capacity (SHC), which is the amount of heat energy required to raise the temperature of 1 kilogram of a substance by 1 Kelvin (or 1 degree Celsius). Different substances have different specific heat capacities; for instance, water has a higher specific heat capacity than ice, meaning it takes more energy to raise the temperature of water compared to ice. The formula for calculating the heat required for a temperature change is:

Q=mcΞ”TQ = mc\Delta T

Where:

  • Q is the heat energy (in Joules)
  • m is the mass (in kilograms)
  • c is the specific heat capacity (in Joules per kilogram per Kelvin)

  • \Delta T$ is the change in temperature (in Kelvin or degrees Celsius)

The second crucial concept is latent heat, specifically the latent heat of fusion in this case. Latent heat is the energy absorbed or released during a phase change (like melting or freezing) without a change in temperature. The latent heat of fusion is the energy required to change a substance from a solid to a liquid at its melting point. For ice, this is the energy needed to break the bonds holding the water molecules in a solid structure, allowing them to move more freely as a liquid. The formula for calculating the heat required for a phase change is:

Q=mLQ = mL

Where:

  • Q is the heat energy (in Joules)
  • m is the mass (in kilograms)
  • L is the latent heat of fusion (in Joules per kilogram)

With these concepts in mind, we can now break down the melting process into three distinct stages:

  1. Heating the ice from $-10^{\circ} C$ to $0^{\circ} C$: This involves raising the temperature of the ice to its melting point.
  2. Melting the ice at $0^{\circ} C$: This is the phase transition where the ice changes into water at a constant temperature.
  3. Heating the water from $0^{\circ} C$ to $15^{\circ} C$: This involves raising the temperature of the water to the final desired temperature.

We will calculate the heat required for each stage separately and then sum them up to find the total heat needed for the entire process. This step-by-step approach will provide a clear and accurate solution to our problem.

Now, let's apply these concepts to our specific problem and calculate the heat required for each stage of the melting process. Remember, we are dealing with 150 grams of ice initially at $-10^{\circ} C$ that needs to be transformed into water at $15^{\circ} C$. We have the following constants:

  • Specific heat capacity of water ($c_w$) = 4200 J kg$^{-1}$ K$^{-1}$
  • Specific heat capacity of ice ($c_i$) = 2100 J kg$^{-1}$ K$^{-1}$
  • Latent heat of fusion of ice ($L_f$) = 3.36 x 10$^5$ J kg$^{-1}$

Let's convert the mass of ice from grams to kilograms:

m=150g=0.15kgm = 150 g = 0.15 kg

Stage 1: Heating the ice from $-10^{\circ} C$ to $0^{\circ} C$

In this stage, we need to raise the temperature of the ice by 10 degrees Celsius. Using the formula $Q = mc\Delta T$, we can calculate the heat required:

Q1=mβˆ—ciβˆ—Ξ”TQ_1 = m * c_i * \Delta T

Q1=0.15kgβˆ—2100Jkgβˆ’1Kβˆ’1βˆ—(0βˆ’(βˆ’10))KQ_1 = 0.15 kg * 2100 J kg^{-1} K^{-1} * (0 - (-10)) K

Q1=0.15kgβˆ—2100Jkgβˆ’1Kβˆ’1βˆ—10KQ_1 = 0.15 kg * 2100 J kg^{-1} K^{-1} * 10 K

Q1=3150JQ_1 = 3150 J

Therefore, 3150 Joules of heat are required to raise the temperature of the ice from $-10^{\circ} C$ to its melting point of $0^{\circ} C$.

Stage 2: Melting the ice at $0^{\circ} C$

Here, we need to change the phase of the ice from solid to liquid at a constant temperature of $0^{\circ} C$. We use the formula for latent heat of fusion, $Q = mL$:

Q2=mβˆ—LfQ_2 = m * L_f

Q2=0.15kgβˆ—3.36x105Jkgβˆ’1Q_2 = 0.15 kg * 3.36 x 10^5 J kg^{-1}

Q2=50400JQ_2 = 50400 J

Thus, 50400 Joules of heat are required to melt the ice completely into water at $0^{\circ} C$.

Stage 3: Heating the water from $0^{\circ} C$ to $15^{\circ} C$

Finally, we need to raise the temperature of the water from $0^{\circ} C$ to $15^{\circ} C$. Again, we use the formula $Q = mc\Delta T$, but this time with the specific heat capacity of water:

Q3=mβˆ—cwβˆ—Ξ”TQ_3 = m * c_w * \Delta T

Q3=0.15kgβˆ—4200Jkgβˆ’1Kβˆ’1βˆ—(15βˆ’0)KQ_3 = 0.15 kg * 4200 J kg^{-1} K^{-1} * (15 - 0) K

Q3=0.15kgβˆ—4200Jkgβˆ’1Kβˆ’1βˆ—15KQ_3 = 0.15 kg * 4200 J kg^{-1} K^{-1} * 15 K

Q3=9450JQ_3 = 9450 J

Therefore, 9450 Joules of heat are required to raise the temperature of the water from $0^{\circ} C$ to $15^{\circ} C$.

To find the total heat required for the entire process, we simply add the heat from each stage:

Qtotal=Q1+Q2+Q3Q_{total} = Q_1 + Q_2 + Q_3

Qtotal=3150J+50400J+9450JQ_{total} = 3150 J + 50400 J + 9450 J

Qtotal=63000JQ_{total} = 63000 J

So, the total heat required to melt 150 grams of ice at $-10^{\circ} C$ to water at $15^{\circ} C$ is 63000 Joules. This comprehensive calculation, broken down into stages, highlights the importance of considering both specific heat capacity and latent heat when analyzing phase transitions.

In conclusion, we have successfully estimated the quantity of heat needed to melt 150 grams of ice at $-10^{\circ} C$ to water at $15^{\circ} C$. By meticulously breaking down the process into three distinct stages – heating the ice to its melting point, melting the ice into water, and then heating the water to the final temperature – we were able to accurately calculate the heat required for each stage. We utilized the concepts of specific heat capacity and latent heat of fusion, along with the provided constants, to arrive at a total heat requirement of 63000 Joules. This exercise not only demonstrates the application of fundamental thermodynamic principles but also highlights the importance of understanding phase transitions and heat transfer in various real-world scenarios. Whether it's in meteorology, engineering, or even everyday life, the principles we've explored here are essential for comprehending and predicting how materials behave under different thermal conditions. Understanding these concepts allows us to analyze and design systems that involve heat transfer and phase changes, making it a cornerstone of many scientific and engineering disciplines. The step-by-step approach we've taken in this article provides a clear and concise method for solving similar problems, making it a valuable resource for students and professionals alike. The ability to accurately estimate heat requirements is crucial for a wide range of applications, from designing efficient heating and cooling systems to understanding the thermal behavior of materials in various environments. This problem serves as an excellent example of how theoretical knowledge can be applied to practical situations, bridging the gap between academic concepts and real-world applications.