Thermodynamic Analysis Of Aluminum And Iron Blocks Equilibrium Temperature And Entropy Change
In the fascinating realm of thermodynamics, understanding the interplay of heat, energy, and entropy is crucial for comprehending the behavior of physical systems. This article delves into a classic thermodynamics problem involving two blocks of different materials, aluminum and iron, brought into thermal contact within an isolated container. Our objective is to determine the final equilibrium temperature of the system and the total entropy change that occurs during the process. This exploration will not only enhance our understanding of heat transfer and equilibrium but also illuminate the concept of entropy as a measure of disorder in a system.
Problem Statement: Aluminum and Iron Blocks in Thermal Contact
Consider a 20 kg aluminum block initially at 200°C placed in contact with a 20 kg iron block at 100°C inside an isolated container. Our goal is to meticulously determine two key parameters: 1) the final equilibrium temperature that the system will reach once thermal equilibrium is established, and 2) the total entropy change for this process, providing insights into the system's thermodynamic behavior.
Thermodynamic Principles: Guiding Our Analysis
To embark on this analysis, we'll lean on fundamental thermodynamic principles:
- First Law of Thermodynamics (Conservation of Energy): In an isolated system, the total energy remains constant. Energy can transform from one form to another, but it cannot be created or destroyed. This principle will be pivotal in determining the equilibrium temperature.
- Heat Transfer: Heat naturally flows from a region of higher temperature to a region of lower temperature until thermal equilibrium is reached, where both objects have the same temperature.
- Specific Heat Capacity: Every substance possesses a unique specific heat capacity (c), which quantifies the amount of heat required to raise the temperature of 1 kg of the substance by 1°C. This property plays a crucial role in calculating heat transfer.
- Entropy: Entropy (S) is a thermodynamic property that measures the disorder or randomness of a system. The second law of thermodynamics dictates that the total entropy of an isolated system can only increase or remain constant in a reversible process; it never decreases. This principle will guide our calculation of the total entropy change.
Parameters and Material Properties: Setting the Stage
Before diving into the calculations, let's gather the necessary parameters and material properties:
- Mass of Aluminum Block (mA): 20 kg
- Initial Temperature of Aluminum Block (TAi): 200°C
- Mass of Iron Block (mI): 20 kg
- Initial Temperature of Iron Block (TIi): 100°C
- Specific Heat Capacity of Aluminum (cA): 0.900 kJ/kg·°C
- Specific Heat Capacity of Iron (cI): 0.450 kJ/kg·°C
With these values in hand, we're well-equipped to unravel the thermal dynamics of our system.
Determining the Final Equilibrium Temperature: A Step-by-Step Approach
The cornerstone of our analysis is the principle of energy conservation. In an isolated system, the heat lost by the aluminum block must equal the heat gained by the iron block. This fundamental concept allows us to derive an equation for the final equilibrium temperature (Tf). Mathematically, we express this as:
Heat Lost by Aluminum = Heat Gained by Iron
Expanding this equation using the specific heat capacity formula (Q = mcΔT), we get:
mA * cA * (TAi - Tf) = mI * cI * (Tf - TTi)
Where:
- Q represents heat transfer
- m denotes mass
- c signifies specific heat capacity
- ΔT represents the change in temperature
Now, let's plug in the values we gathered earlier:
20 kg * 0.900 kJ/kg·°C * (200°C - Tf) = 20 kg * 0.450 kJ/kg·°C * (Tf - 100°C)
Simplifying the equation:
18 (200 - Tf) = 9 (Tf - 100)
3600 - 18Tf = 9Tf - 900
Combining like terms:
27Tf = 4500
Finally, solving for Tf:
Tf = 4500 / 27 ≈ 166.67°C
Therefore, the final equilibrium temperature of the system is approximately 166.67°C. This value represents the temperature at which both the aluminum and iron blocks will reach thermal equilibrium within the isolated container.
Calculating the Total Entropy Change: Unveiling the System's Disorder
Entropy, a measure of a system's disorder, is a cornerstone of the second law of thermodynamics. To calculate the total entropy change for this process, we need to determine the entropy change for each block individually and then sum them up. The formula for entropy change (ΔS) in a process involving heat transfer is:
ΔS = m * c * ln(Tf / Ti)
Where:
- ΔS is the change in entropy
- m is the mass
- c is the specific heat capacity
- Tf is the final temperature (in Kelvin)
- Ti is the initial temperature (in Kelvin)
Let's calculate the entropy change for each block:
Entropy Change for Aluminum
First, convert the temperatures to Kelvin:
- TAi (Kelvin) = 200°C + 273.15 = 473.15 K
- Tf (Kelvin) = 166.67°C + 273.15 = 439.82 K
Now, plug the values into the entropy change formula:
ΔSA = 20 kg * 0.900 kJ/kg·°C * ln(439.82 K / 473.15 K)
ΔSA ≈ 18 * ln(0.9295) ≈ -1.32 kJ/K
Thus, the entropy change for the aluminum block is approximately -1.32 kJ/K. The negative sign indicates that the aluminum block's entropy decreases as it cools down.
Entropy Change for Iron
Convert the initial temperature of iron to Kelvin:
- TIi (Kelvin) = 100°C + 273.15 = 373.15 K
Now, calculate the entropy change for the iron block:
ΔSI = 20 kg * 0.450 kJ/kg·°C * ln(439.82 K / 373.15 K)
ΔSI ≈ 9 * ln(1.1787) ≈ 1.43 kJ/K
The entropy change for the iron block is approximately 1.43 kJ/K. The positive sign signifies that the iron block's entropy increases as it warms up.
Total Entropy Change: A Sum of Disorder
To determine the total entropy change for the process, we simply add the entropy changes of the aluminum and iron blocks:
ΔS_total = ΔSA + ΔSI
ΔS_total ≈ -1.32 kJ/K + 1.43 kJ/K ≈ 0.11 kJ/K
Therefore, the total entropy change for this process is approximately 0.11 kJ/K. The positive value confirms that the overall entropy of the isolated system has increased. This increase aligns with the second law of thermodynamics, which dictates that the entropy of an isolated system can only increase or remain constant.
Conclusion: Entropy's Dance Towards Equilibrium
In this exploration of thermodynamics, we've successfully determined the final equilibrium temperature and total entropy change for a system of aluminum and iron blocks brought into thermal contact. The equilibrium temperature of approximately 166.67°C represents the point where heat transfer ceases, and both blocks attain a uniform temperature. The total entropy change, calculated to be approximately 0.11 kJ/K, underscores the fundamental principle that entropy in an isolated system tends to increase. This increase in entropy reflects the system's natural progression toward a state of greater disorder as energy is redistributed.
This analysis not only deepens our understanding of heat transfer and equilibrium but also provides a tangible illustration of entropy's role in thermodynamic processes. The positive entropy change serves as a compelling reminder of the second law of thermodynamics and the universe's relentless march towards increasing disorder.