Thermodynamic Description of System
Various Thermodynamical Processes
A thermodynamic process is a passage of a thermodynamic system from an initial state to a final state. These processes are characterized by changes in pressure, volume, temperature, and internal energy.
- Isothermal Process: A process in which the temperature (T) of the system remains constant throughout. This usually involves heat exchange with the surroundings.
- Adiabatic Process: A process in which no heat (Q) is exchanged between the system and its surroundings (Q=0). The system is thermally insulated.
- Isobaric Process: A process in which the pressure (P) of the system remains constant.
- Isochoric (or Isometric) Process: A process in which the volume (V) of the system remains constant.
- Cyclic Process: A process in which the system returns to its initial state after a series of changes. The net change in internal energy for a cyclic process is zero (ΔU = 0).
Applications of First Law of Thermodynamics
The First Law of Thermodynamics, also known as the Law of Conservation of Energy, states that energy cannot be created or destroyed in an isolated system. It can only be transformed from one form to another. Mathematically, it is expressed as:ΔU = Q - W
Where ΔU is the change in internal energy of the system, Q is the heat added to the system, and W is the work done by the system. (Some conventions use W as work done on the system, changing the sign to ΔU = Q + W). We will use W as work done by the system.
General Relation Between Cp and Cv
Cp is the molar heat capacity at constant pressure, and Cv is the molar heat capacity at constant volume. For an ideal gas, there is a fundamental relationship between them.
For an isobaric process, Q = nCpΔT and W = PΔV.
From the First Law, ΔU = Q - W = nCpΔT - PΔV.
For an isochoric process, Q = nCvΔT and W = 0.
From the First Law, ΔU = Q - W = nCvΔT.
Since internal energy for an ideal gas depends only on temperature, ΔU is the same for the same ΔT. Therefore, nCpΔT - PΔV = nCvΔT.
For an ideal gas, PV = nRT, so PΔV = nRΔT (at constant pressure).
Substituting PΔV = nRΔT into the equation:
nCpΔT - nRΔT = nCvΔT
Dividing by nΔT:
Cp - R = Cv
or
Cp - Cv = R
Where R is the universal gas constant. This is also known as Mayer's relation.
Work Done During Isothermal Processes
In an isothermal process, the temperature (T) remains constant. For an ideal gas, PV = nRT, so PV = constant.
The work done by the system is given by W = ∫P dV. Since P = nRT/V,
W = ∫V1V2 (nRT/V) dV
W = nRT ∫V1V2 (1/V) dV
W = nRT [ln V]V1V2
W = nRT ln(V2/V1)
Since P1V1 = P2V2 (Boyle's Law), V2/V1 = P1/P2.
W = nRT ln(P1/P2)
Work Done During Adiabatic Processes
In an adiabatic process, there is no heat exchange (Q=0) between the system and surroundings. For an ideal gas, the relationship between P and V is P Vγ = constant, where γ = Cp/Cv is the adiabatic index.
From the First Law, ΔU = -W. For an ideal gas, ΔU = nCvΔT.
So, W = -nCvΔT = nCv(T1 - T2).
Using PV = nRT, we can substitute T = PV/nR:
W = nCv(P1V1 - P2V2) / nR = Cv(P1V1 - P2V2) / R.
Since Cp - Cv = R, then Cv/R = Cv/(Cp - Cv) = 1/(Cp/Cv - 1) = 1/(γ - 1).
W = (P1V1 - P2V2) / (γ - 1)
Reversible & Irreversible Processes
Processes in thermodynamics can be classified based on whether they can be reversed without leaving any change in the surroundings.
| Feature | Reversible Process | Irreversible Process |
|---|---|---|
| Definition | A process that can be reversed such that both the system and surroundings return to their original states without any net change. | A process that cannot be reversed without leaving some change in the surroundings. |
| Equilibrium | System is always infinitesimally close to equilibrium state. | System deviates significantly from equilibrium. |
| Friction/Dissipation | No friction, viscosity, or other dissipative effects. | Always involves friction, viscosity, turbulence, heat transfer across finite temperature difference, or unrestrained expansion. |
| Work Output | Maximum possible work output (for expansion) or minimum work input (for compression). | Less than maximum work output or more than minimum work input. |
| Spontaneity | Idealized, non-spontaneous in nature. | Real, spontaneous processes. |
| Driving Force | Infinitesimal difference in driving forces. | Finite difference in driving forces. |
| Entropy Change | ΔSuniverse = 0 | ΔSuniverse > 0 |
Second Law of Thermodynamics & Entropy
The Second Law of Thermodynamics establishes the concept of entropy and the direction of spontaneous processes.
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 cooler 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. This implies that no heat engine can have 100% efficiency.
Entropy (S): A thermodynamic property that is a measure of the randomness or disorder of a system. It also quantifies the unavailable energy in a system for doing useful work.
Entropy Changes in Reversible Processes
For a reversible process, the change in entropy of a system is defined as the heat transferred reversibly to the system (qrev) divided by the absolute temperature (T) at which the transfer occurs.
ΔS = ∫(dqrev/T) for a general reversible process
For an isothermal reversible process, ΔS = qrev/T
For a reversible cycle, the net change in entropy of the system is zero (ΔSsystem = 0), and also ΔSsurroundings = 0, so ΔSuniverse = 0.
Entropy Changes in Irreversible Processes
Irreversible processes are real processes, and they always lead to an increase in the entropy of the universe.
For any irreversible process, ΔSuniverse > 0
This means ΔSsystem + ΔSsurroundings > 0.
The entropy of an isolated system always increases in an irreversible process and remains constant in a reversible process. It never decreases.
Carnot’s Theorem (Statement only)
Carnot’s Theorem states that no heat engine operating between two given constant-temperature reservoirs can be more efficient than a reversible heat engine operating between the same two reservoirs. Furthermore, all reversible heat engines operating between the same two reservoirs have the same efficiency.
Thermodynamic Potentials
Thermodynamic potentials are state functions that are useful for understanding the energy changes in thermodynamic systems under specific conditions (e.g., constant temperature, constant pressure).
Internal Energy (U)
Internal energy is the total energy contained within a thermodynamic system, including kinetic and potential energies of its molecules, but excluding the kinetic and potential energy of the system as a whole. It is a state function.
Differential form: dU = TdS - PdV (for reversible processes where only P-V work is done)
Enthalpy (H)
Enthalpy is a thermodynamic potential useful for processes occurring at constant pressure. It accounts for the internal energy of the system plus the energy required to make space for it and push aside its surroundings.
Definition: H = U + PV
Differential form: dH = dU + PdV + VdP
Substituting dU = TdS - PdV, we get:
dH = TdS - PdV + PdV + VdP
dH = TdS + VdP
Helmholtz Free Energy (A or F)
Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature and volume. It is often denoted by A or F.
Definition: A = U - TS
Differential form: dA = dU - TdS - SdT
Substituting dU = TdS - PdV, we get:
dA = TdS - PdV - TdS - SdT
dA = -PdV - SdT
Gibbs Free Energy (G)
Gibbs free energy is a thermodynamic potential that measures the 'useful' or process-initiating work obtainable from an isothermal, isobaric thermodynamic system. It is commonly used in chemistry and biochemical reactions.
Definition: G = H - TS = U + PV - TS
Differential form: dG = dH - TdS - SdT
Substituting dH = TdS + VdP, we get:
dG = TdS + VdP - TdS - SdT
dG = VdP - SdT
Maxwell’s Relations & Applications
Maxwell's relations are a set of equations in thermodynamics derived from the equality of the second derivatives of the thermodynamic potentials. They relate partial derivatives of various thermodynamic properties (T, P, V, S) and are very useful for calculating properties that are difficult to measure directly from those that are easier to measure.
These relations are derived from the exact differentials of U, H, A, and G. For an exact differential dz = Mdx + Ndy, we know that (∂M/∂y)x = (∂N/∂x)y.
From dU = TdS - PdV:
(∂T/∂V)S = -(∂P/∂S)V
From dH = TdS + VdP:
(∂T/∂P)S = (∂V/∂S)P
From dA = -PdV - SdT:
(∂P/∂T)V = (∂S/∂V)T
From dG = VdP - SdT:
(∂V/∂T)P = -(∂S/∂P)T
Applications of Maxwell's Relations:
- They allow the calculation of changes in entropy, which cannot be measured directly, from measurable quantities like P, V, T.
- They are used to derive other thermodynamic relations, such as the relation between Cp and Cv or properties of phase transitions.
- They help in understanding the behavior of systems under different conditions by relating seemingly unrelated thermodynamic variables. For example, the third relation (∂P/∂T)V = (∂S/∂V)T relates the change in pressure with temperature at constant volume to the change in entropy with volume at constant temperature.