Thermodynamics is a branch of physics that deals with heat, work, temperature, and the laws governing the transfer and transformation of energy. Here’s a discussion of the three laws of thermodynamics along with their mathematical expressions and examples.
First Law of Thermodynamics (Law of Energy Conservation)
Statement: The first law states that energy cannot be created or destroyed in an isolated system. The change in internal energy (ΔU) of a system is equal to the amount of heat (Q) added to the system minus the work (W) done by the system.
Mathematical Expression: \[ \Delta U = Q - W \]
Where:
- \(\Delta U\) = change in internal energy of the system
- \(Q\) = heat added to the system
- \(W\) = work done by the system
Example: Consider a gas in a container where heat \(Q = 500 , \text{J}\) is added to the gas, and it performs work \(W = 300 , \text{J}\) on the surroundings. The change in internal energy would be calculated as follows: \[ \Delta U = Q - W = 500 , \text{J} - 300 , \text{J} = 200 , \text{J} \]
Second Law of Thermodynamics
Statement: The second law states that the total entropy of an isolated system can never decrease over time. It implies that energy conversions are not 100% efficient and that some energy will always be converted into a less usable form (e.g., heat). This law also introduces the concept of irreversibility in natural processes.
Mathematical Expression: In the context of spontaneous processes, the change in entropy (\(ΔS\)) of the system and surroundings is given by: \[ \Delta S_{tot} = \Delta S_{system} + \Delta S_{surroundings} \geq 0 \]
Where:
- \(\Delta S_{tot}\) is the total change in entropy,
- \(\Delta S_{system}\) is the change in entropy of the system,
- \(\Delta S_{surroundings}\) is the change in entropy of the surroundings.
Example: When ice melts at \(0^\circ C\) and absorbs heat \(Q\) from the surroundings, it undergoes a change in entropy. The entropy change for the melting process can be calculated as: \[ \Delta S = \frac{Q_{\text{rev}}}{T} \] If, for instance, \(Q = 334 , \text{kJ}\) (the heat of fusion for ice) and the temperature is \(273.15 , \text{K}\): \[ \Delta S = \frac{334 \times 10^3 , \text{J}}{273.15 , \text{K}} \approx 1223.7 , \text{J/K} \] This shows an increase in entropy, in accordance with the second law.
Third Law of Thermodynamics
Statement: The third law states that as the temperature of a system approaches absolute zero, the entropy of a perfectly crystalline substance approaches zero. In other words, it is impossible to reach absolute zero in a finite number of steps.
Mathematical Expression: \[ S(0) = 0 \quad \text{for a perfect crystal} \]
Where:
- S(0) is the entropy at absolute zero.
Example: For a perfect crystal at absolute zero, we can conclude:
- The entropy of a perfectly ordered system (such as a crystalline lattice) approaches zero since there is only one way to arrange the particles at absolute zero (since they are at their lowest energy state and perfectly ordered).
However, in practice, infinite steps are required to reach absolute zero according to the third law. For example, the entropy of real substances at very low temperatures can approach but never reach zero due to the presence of imperfections and residual entropy.
Summary
- First Law: Energy conservation; \(\Delta U = Q - W\).
- Second Law: Entropy increase; \(\Delta S_{tot} \geq 0\).
- Third Law: Entropy of a perfect crystal at absolute zero is zero; \(S(0) = 0\).
These laws form the foundation of thermodynamics and have vast implications in physics, engineering, chemistry, and other fields.