Entropy & Degradation of Energy
Concept of molecular disorder, mathematical expression, SI unit, and step-by-step proof of energy degradation.
1. What is Entropy?
- Qualitative Definition: Entropy is the measure of the molecular disorder or randomness of a system.
- Quantitative Definition: Entropy is a thermodynamic state function that measures a system's thermal energy per unit absolute temperature that is unavailable for doing useful mechanical work.
Mathematical Formula:
For a reversible thermodynamic process occurring at a constant absolute temperature T, the change in entropy (ΔS) is given by:
ΔS = ΔQ / T
Where:
- ΔQ: Heat added to or removed from the system.
- T: Absolute temperature of the system in Kelvin (K).
SI Unit & Sign Convention:
- SI Unit: Joules per Kelvin (J/K or J · K-1).
- Positive Change (+ΔS): When heat is added to a system (ΔQ > 0), entropy increases.
- Negative Change (-ΔS): When heat is removed from a system (ΔQ < 0), entropy decreases.
2. Physical Meaning: Entropy as Disorder
Consider the reversible isothermal expansion of an ideal gas. When an amount of heat Q is added to the gas, it expands slowly at a constant temperature T.
According to the First Law of Thermodynamics:
ΔU = Q - W
Since internal energy U of an ideal gas depends solely on temperature (ΔU = 0 for isothermal process):
W = Q
After expansion, the gas molecules occupy a larger volume, resulting in greater randomness of position and velocity. Thus, the gas transitions into a more disordered state. The quantity ΔQ / T serves as a direct quantitative measure of this increase in molecular disorder.
3. Increase in Entropy Means Degradation of Energy
Degradation of energy means that as the entropy of the universe increases, the total energy remains conserved (First Law), but a fraction of that energy becomes continuously unavailable for performing useful work (Second Law).
Mathematical Proof:
Case 1 (Reversible Ideal State):
Suppose a quantity of heat Q is available in a thermal reservoir at high temperature T1. Let T0 be the temperature of the lowest available sink in the surroundings. A Carnot engine operating between T1 and T0 can absorb heat Q at T1 and perform maximum work W1:
W1 = η1 × Q = [1 - (T0 / T1)] × Q ——— (Equation 1)
Case 2 (Irreversible Heat Flow State):
Now consider an irreversible natural process where heat Q flows spontaneously from the higher temperature reservoir T1 to a lower temperature reservoir T2 (where T2 < T1). If a Carnot engine now operates between T2 and the cold sink T0, the work W2 obtained is:
W2 = η2 × Q = [1 - (T0 / T2)] × Q ——— (Equation 2)
Loss of Available Work:
Since T2 < T1, it follows that W2 < W1. Subtracting Equation 2 from Equation 1 gives the loss of available work (ΔW):
ΔW = W1 - W2
ΔW = [1 - (T0 / T1)] × Q - [1 - (T0 / T2)] × Q
ΔW = T0 × [(Q / T2) - (Q / T1)]
Notice that [(Q / T2) - (Q / T1)] represents the net increase in entropy (ΔSuniv) during the irreversible transfer of heat Q from T1 to T2:
Where:
- ΔS = (Q / T2) - (Q / T1): Increase in entropy of the universe (ΔS > 0).
- ΔW = T0 ΔS: Amount of energy degraded (rendered permanently unavailable for useful work).
All natural or spontaneous processes are irreversible, and every irreversible process leads to a net increase in the entropy of the universe (ΔSuniv > 0). Consequently, while the total energy of the universe remains constant, the quantity of energy available to do useful work continuously decreases. This fundamental principle is known as the Law of Degradation of Energy or the "Heat Death" concept of the universe.
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