The Third Law of Thermodynamics, as stated by Walther Nernst, is a fundamental principle in thermodynamics that establishes the zero for entropy as that of a perfect, pure crystalline solid at 0 Kelvin (0 K). This law is based on the principle that as the absolute temperature of a substance approaches zero, so does its entropy.
Understanding the Third Law of Thermodynamics
Entropy and Absolute Zero
The Third Law of Thermodynamics states that as a system approaches absolute zero (0 K), its entropy approaches a constant, typically zero, value. This means that the entropy of a perfect crystalline solid at 0 K is defined as zero. This is because at absolute zero, the atoms or molecules in a perfect crystal are completely ordered and have no thermal motion, resulting in a minimum entropy state.
Nernst’s Formulation
The Third Law of Thermodynamics was first formulated by the German chemist Walther Nernst in the early 20th century. Nernst’s formulation states that the entropy of a perfect crystalline substance at absolute zero is zero. This is known as the Nernst heat theorem or the Nernst postulate.
The mathematical expression of Nernst’s formulation is:
lim(T→0) [S(T) / T] = 0
where S(T) is the entropy of the system at temperature T.
Implications of the Third Law
The Third Law of Thermodynamics has several important implications:
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Absolute Entropy Measurement: The Third Law provides a fixed reference point (zero entropy at 0 K) for the measurement of absolute entropy of substances. This allows for the calculation of entropy changes in chemical reactions and physical changes.
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Spontaneous Processes: The Third Law establishes that a spontaneous process increases the entropy of the universe, and if the entropy change of the universe is less than zero, the process is nonspontaneous. If the entropy change of the universe is equal to zero, the system is at equilibrium.
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Unattainability of Absolute Zero: The Third Law implies that it is impossible to reach absolute zero (0 K) in a finite number of steps, as this would require an infinite amount of work to remove the last bit of thermal energy from the system.
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Cryogenic Applications: The Third Law has important implications for cryogenic applications, such as the design of refrigeration systems and the study of low-temperature phenomena in materials science and condensed matter physics.
Experimental Determination of Absolute Entropy
In practice, chemists determine the absolute entropy of a substance by measuring the molar heat capacity (Cp) as a function of temperature and then plotting the quantity Cp/T versus T. The area under the curve between 0 K and any temperature T is the absolute entropy of the substance at T. This method allows for the calculation of absolute entropy values under specific conditions.
The formula for calculating the absolute entropy of a substance at temperature T is:
S(T) = ∫(0 to T) (Cp/T) dT
where Cp is the molar heat capacity of the substance.
Theoretical Foundations of the Third Law
Nernst’s Heat Theorem
Nernst’s heat theorem, also known as the Third Law of Thermodynamics, states that the entropy change of a system undergoing a reversible isothermal process approaches zero as the temperature approaches absolute zero. Mathematically, this can be expressed as:
lim(T→0) [ΔS / ΔT] = 0
where ΔS is the change in entropy and ΔT is the change in temperature.
Quantum Mechanical Interpretation
The Third Law of Thermodynamics can also be understood from a quantum mechanical perspective. At absolute zero, the quantum mechanical ground state of a system is the only accessible state, and the system has no thermal energy. This results in a highly ordered and low-entropy state, consistent with the Third Law.
Statistical Mechanical Interpretation
The Third Law can also be interpreted in terms of statistical mechanics. As the temperature approaches absolute zero, the number of accessible microstates of the system approaches a single state, corresponding to the ground state. This results in a vanishing entropy, as the system becomes perfectly ordered.
Applications of the Third Law
Calculation of Absolute Entropy
The Third Law provides a fixed reference point for the measurement of absolute entropy, allowing for the calculation of entropy changes in chemical reactions and physical changes. This is important for understanding the spontaneity and feasibility of various processes.
Cryogenic Applications
The Third Law has important implications for cryogenic applications, such as the design of refrigeration systems and the study of low-temperature phenomena in materials science and condensed matter physics. Understanding the behavior of systems at temperatures approaching absolute zero is crucial for these applications.
Thermodynamic Modeling
The Third Law is a fundamental principle in thermodynamics and is essential for the accurate modeling and prediction of the behavior of systems, particularly at low temperatures. It is used in the development of equations of state, phase diagrams, and other thermodynamic models.
Astrophysics and Cosmology
The Third Law of Thermodynamics has implications for the study of astrophysical and cosmological phenomena, such as the evolution of stars and the early universe. Understanding the behavior of matter and energy at extremely low temperatures is crucial for these fields.
Numerical Examples and Problems
Example 1: Calculating Absolute Entropy
Consider a substance with the following molar heat capacity (Cp) data:
| Temperature (K) | Cp (J/mol·K) |
|---|---|
| 0 | 0 |
| 50 | 25.6 |
| 100 | 29.1 |
| 150 | 29.9 |
| 200 | 30.0 |
Calculate the absolute entropy of the substance at 200 K.
Solution:
1. Plot the Cp/T vs. T curve.
2. Calculate the area under the curve from 0 K to 200 K using numerical integration.
3. The area under the curve is the absolute entropy of the substance at 200 K.
Example 2: Entropy Change in a Spontaneous Process
Consider a spontaneous process where the entropy change of the universe is positive. Explain how this is consistent with the Third Law of Thermodynamics.
Solution:
1. The Third Law states that the entropy of a perfect crystalline solid at 0 K is zero.
2. For a spontaneous process, the entropy of the universe must increase, as per the Second Law of Thermodynamics.
3. The increase in entropy of the universe is consistent with the Third Law, as the entropy of the system at 0 K is defined as zero.
4. The spontaneous process increases the overall entropy of the universe, which is in agreement with the Third Law.
Example 3: Unattainability of Absolute Zero
Explain why it is impossible to reach absolute zero (0 K) in a finite number of steps, as per the Third Law of Thermodynamics.
Solution:
1. The Third Law states that the entropy of a perfect crystalline solid approaches zero as the temperature approaches absolute zero.
2. Reaching absolute zero would require an infinite amount of work to remove the last bit of thermal energy from the system, as the entropy approaches zero asymptotically.
3. This is because the closer the system gets to absolute zero, the more difficult it becomes to remove the remaining thermal energy, as the entropy change approaches zero.
4. Therefore, the Third Law implies that it is impossible to reach absolute zero in a finite number of steps, as this would require an infinite amount of work.
Conclusion
The Third Law of Thermodynamics is a fundamental principle that has far-reaching implications in various fields of science, from chemistry and physics to astrophysics and cosmology. By understanding the theoretical foundations, experimental determination, and practical applications of the Third Law, science students can gain a deeper appreciation for the behavior of systems at low temperatures and the importance of entropy in the universe.
References:
- Cengel, Y. A., & Boles, M. A. (2015). Thermodynamics: An Engineering Approach (8th ed.). McGraw-Hill Education.
- Atkins, P., & de Paula, J. (2014). Atkins’ Physical Chemistry (10th ed.). Oxford University Press.
- Levine, I. N. (2009). Physical Chemistry (6th ed.). McGraw-Hill Education.
- Engel, T., & Reid, P. (2013). Physical Chemistry (3rd ed.). Pearson.
- Silbey, R. J., Alberty, R. A., & Bawendi, M. G. (2005). Physical Chemistry (4th ed.). Wiley.
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