Thermodynamic Description of System
Thermodynamics is the branch of physics that deals with heat and its relation to other forms of energy and work. A thermodynamic system is a quantity of matter or a region in space chosen for study.
Various Thermodynamic Processes
A thermodynamic process is a change in the state of a system from an initial equilibrium state to a final equilibrium state. These processes are characterized by how certain properties of the system remain constant.
- Isothermal Process: A process in which the temperature (T) of the system remains constant throughout. This typically occurs when a system is in thermal contact with a large heat reservoir. For an ideal gas, PV = constant.
- Adiabatic Process: A process in which no heat (Q) enters or leaves the system. This implies that the system is thermally isolated from its surroundings. For an ideal gas, PVγ = constant, where γ (gamma) is the adiabatic index (Cp/Cv).
- Isobaric Process: A process in which the pressure (P) of the system remains constant. This often occurs in systems open to the atmosphere.
- Isochoric Process (or Isovolumetric Process): A process in which the volume (V) of the system remains constant. In this process, no work is done by or on the system (W = 0).
- Cyclic Process: A process in which the system returns to its initial state after a series of changes. For a cyclic process, the change in internal energy (ΔU) is zero.
Applications of First Law of Thermodynamics
The First Law of Thermodynamics: The change in the internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system on its surroundings. ΔU = Q - W.
This law is essentially a statement of the conservation of energy. It means that energy cannot be created or destroyed, only transferred or converted from one form to another.
- Isolated Systems: For an isolated system, Q = 0 and W = 0, so ΔU = 0. The internal energy of an isolated system remains constant.
- Isochoric Process: If V = constant, then W = PΔV = 0. Thus, ΔU = Qv (heat added at constant volume).
- Isobaric Process: If P = constant, W = PΔV. So, ΔU = Qp - PΔV. This leads to the definition of enthalpy, H = U + PV, where ΔH = Qp.
- Isothermal Process (for ideal gas): If T = constant, ΔU = 0. Therefore, Q = W. Any heat added to the system is completely converted into work done by the system.
- Adiabatic Process: If Q = 0, then ΔU = -W. Any work done by the system comes from its internal energy, causing its temperature to drop. Conversely, work done on the system increases its internal energy and temperature.
General Relation between Cp and Cv
Cp is the molar specific heat at constant pressure, and Cv is the molar specific heat at constant volume. For an ideal gas, there is a fundamental relationship between them.
Cp - Cv = R
Where R is the universal gas constant. This relation shows that Cp is always greater than Cv because, at constant pressure, some of the heat supplied is used to do work against the external pressure as the gas expands, in addition to increasing its internal energy. At constant volume, all the heat supplied goes directly into increasing the internal energy.
Work Done During Isothermal and Adiabatic Processes
Work Done During Isothermal Process
For an ideal gas undergoing a reversible isothermal expansion or compression from volume V1 to V2:
W = nRT ln(V2/V1)
Where n is the number of moles, R is the universal gas constant, T is the constant temperature, and ln denotes the natural logarithm. If V2 > V1, work is positive (done by the system). If V2 < V1, work is negative (done on the system).
Work Done During Adiabatic Process
For an ideal gas undergoing a reversible adiabatic expansion or compression from state (P1, V1) to (P2, V2):
W = (P1V1 - P2V2) / (γ - 1) = nR(T1 - T2) / (γ - 1)
Where γ (gamma) = Cp/Cv is the adiabatic index. In an adiabatic expansion, the system does positive work, and its temperature decreases (T2 < T1). In an adiabatic compression, work is done on the system, and its temperature increases (T2 > T1).
Reversible & Irreversible Processes
Thermodynamic processes can be classified based on their reversibility.
| Feature | Reversible Process | Irreversible Process |
|---|---|---|
| Definition | A process that can be reversed by an infinitesimal change in external conditions, with no net change in the system or surroundings. It proceeds infinitesimally slowly, maintaining equilibrium at every step. | A process that cannot be reversed without leaving a permanent change in the system or surroundings. Most natural processes are irreversible. |
| Path | Follows an exact equilibrium path. | Does not follow an equilibrium path; typically involves finite differences in forces/temperatures. |
| Work Done | Maximum work can be extracted from a reversible expansion; minimum work required for a reversible compression. | Less than maximum work extracted from an expansion; more than minimum work required for a compression. |
| Entropy Change | Total entropy change of the universe (system + surroundings) is zero (ΔSuniverse = 0). | Total entropy change of the universe is positive (ΔSuniverse > 0). |
| Driving Force | Infinitesimal differences. | Finite differences (e.g., rapid expansion, heat transfer across large temperature difference). |
| Example | Idealized slow compression/expansion; frictionless pendulum. | Friction, heat transfer across a finite temperature difference, free expansion of a gas. |
Second Law & Entropy
The Second Law of Thermodynamics:
- Clausius Statement: It is impossible to construct a device which operates in a cycle and produces no effect other than the transfer of heat from a colder body to a hotter body.
- Kelvin-Planck Statement: It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work.
The Second Law introduces the concept of entropy, a measure of the disorder or randomness of a system. It dictates the direction of spontaneous processes.
Entropy (S): A state function that quantifies the degree of randomness or disorder in a system. For a reversible process, the change in entropy (ΔS) is defined as the heat transferred (dQrev) divided by the absolute temperature (T) at which the transfer occurs.
The Second Law implies that the entropy of an isolated system never decreases; it either remains constant (for reversible processes) or increases (for irreversible processes). The universe as a whole is an isolated system, and its entropy is constantly increasing.
Entropy Changes in Reversible & Irreversible Processes
Entropy Change in Reversible Process
For a reversible process, the change in entropy of a system is given by:
ΔS = ∫dQrev/T
For a reversible process in an isolated system, ΔSsystem + ΔSsurroundings = 0.
Entropy Change in Irreversible Process
For an irreversible process, the change in entropy of an isolated system is always positive:
ΔSisolated system > 0
And for any real (irreversible) process, the total entropy of the system and its surroundings (the universe) always increases:
ΔSuniverse = ΔSsystem + ΔSsurroundings > 0
This means that irreversible processes inherently lead to an increase in the overall disorder or randomness of the universe.
Carnot’s Theorem (Statement only)
Carnot’s Theorem: No heat engine operating between two given thermal reservoirs can be more efficient than a reversible Carnot engine operating between the same two reservoirs. Furthermore, all reversible engines operating between the same two reservoirs have the same efficiency. The efficiency of a reversible engine depends only on the temperatures of the hot (TH) and cold (TC) reservoirs: η = 1 - (TC/TH).
Thermodynamic Potentials
Thermodynamic potentials are functions that are used to predict the direction of a thermodynamic process and to calculate maximum work obtainable from a system under specific constraints (e.g., constant temperature, constant pressure, etc.). They are useful for describing the equilibrium state of a system.
Internal Energy (U)
Internal energy is the total energy contained within a thermodynamic system, excluding the kinetic and potential energy of the system as a whole. It includes the kinetic and potential energies of the molecules, atoms, and subatomic particles within the system.
U: The internal energy of a system, a state function. For a closed system, the change in internal energy is given by the First Law: ΔU = Q - W.
It is a fundamental state function and depends only on the current state of the system, not on the path taken to reach that state.
Enthalpy (H)
Enthalpy is particularly useful for processes occurring at constant pressure, which are common in chemistry and everyday life.
Enthalpy (H): A thermodynamic potential defined as H = U + PV. Where U is internal energy, P is pressure, and V is volume.
The change in enthalpy (ΔH) for a process at constant pressure is equal to the heat exchanged (Qp): ΔH = Qp. This makes it convenient for measuring heat changes in reactions, such as heats of formation, combustion, etc.
Helmholtz Free Energy (A or F)
Helmholtz free energy is useful for predicting the spontaneity of processes occurring at constant temperature and constant volume.
Helmholtz Free Energy (A): A thermodynamic potential defined as A = U - TS. Where U is internal energy, T is absolute temperature, and S is entropy.
The change in Helmholtz free energy (ΔA) represents the maximum possible work that can be extracted from a closed system at constant temperature and volume. For a spontaneous process at constant T and V, ΔA ≤ 0.
Gibbs Free Energy (G)
Gibbs free energy is crucial for understanding chemical reactions and phase transitions under constant temperature and pressure conditions, which are typical for laboratory experiments.
Gibbs Free Energy (G): A thermodynamic potential defined as G = H - TS. Where H is enthalpy, T is absolute temperature, and S is entropy.
The change in Gibbs free energy (ΔG) represents the maximum non-PV work that can be extracted from a system at constant temperature and pressure. For a spontaneous process at constant T and P, ΔG ≤ 0. If ΔG = 0, the system is at equilibrium.
Maxwell’s Relations & Applications
Maxwell's relations are a set of equations derived from the exact differentials of the thermodynamic potentials (U, H, A, G). They relate the partial derivatives of thermodynamic properties (T, P, V, S) and are very powerful for relating properties that are difficult to measure to those that are easily measurable.
The fundamental thermodynamic equations, from which Maxwell's relations are derived, are:
- dU = TdS - PdV
- dH = TdS + VdP
- dA = -SdT - PdV
- dG = -SdT + VdP
By applying the property of exact differentials (the mixed partial derivatives are equal, e.g., for dz = Mdx + Ndy, (∂M/∂y)x = (∂N/∂x)y), we obtain the four Maxwell's relations:
Maxwell's Relations
- From dU:
(∂T/∂V)S = -(∂P/∂S)V
- From dH:
(∂T/∂P)S = (∂V/∂S)P
- From dA:
(∂P/∂T)V = (∂S/∂V)T
- From dG:
(∂V/∂T)P = -(∂S/∂P)T
Applications of Maxwell's Relations
Maxwell's relations are vital for relating measurable quantities to those that are hard to measure. Some key applications include:
- Calculating Entropy Changes: They allow us to calculate changes in entropy (which is difficult to measure directly) from changes in easily measurable quantities like P, V, and T. For example, the third relation (∂S/∂V)T = (∂P/∂T)V relates the change in entropy with volume at constant temperature to the change in pressure with temperature at constant volume.
- Relating Specific Heats: They can be used to derive relationships between Cp and Cv, such as Cp - Cv = T(∂P/∂T)V(∂V/∂T)P, which reduces to R for an ideal gas.
- Joule-Thomson Effect: These relations are fundamental in understanding the Joule-Thomson effect (the change in temperature of a gas when it expands adiabatically through a valve), which is crucial for liquefaction of gases.
- Clapeyron Equation: They can be used to derive the Clapeyron equation, which describes the slope of a phase boundary in a P-T diagram.