Thermodynamic potentials ek fundamental concept hai thermodynamics me, jo energy functions provide karte hain jisse hum equilibrium conditions, spontaneous processes, aur work done ko analyze kar sakte hain. Ye potentials internal energy, enthalpy, Helmholtz free energy, aur Gibbs free energy jaise quantities ko define karte hain aur thermodynamic systems ka behavior predict karne me madad karte hain.
1. Introduction
Thermodynamics me energy transfer aur equilibrium ka analysis karne ke liye, different state functions ka use kiya jata hai:
- Internal energy (U): Total microscopic energy of system
- Enthalpy (H): Heat content at constant pressure
- Helmholtz free energy (A or F): Useful for constant volume and temperature
- Gibbs free energy (G): Useful for constant pressure and temperature
Thermodynamic potentials allow humko:
- Predict spontaneous processes
- Calculate maximum work
- Determine equilibrium conditions
2. Internal Energy (U)
2.1 Definition
Internal energy is the total microscopic energy (kinetic + potential) of a system.
- Function of state variables: U=U(S,V)U = U(S, V)U=U(S,V)
- SSS = entropy, VVV = volume
- First Law of Thermodynamics:
dU=TdS−PdVdU = TdS – PdVdU=TdS−PdV
- Here, T=(∂U/∂S)VT = (\partial U/\partial S)_VT=(∂U/∂S)V and P=−(∂U/∂V)SP = -(\partial U/\partial V)_SP=−(∂U/∂V)S
Applications:
- Isochoric processes (ΔV = 0 → Q = ΔU)
- Adiabatic processes (Q = 0 → W = -ΔU)
3. Enthalpy (H)
3.1 Definition
Enthalpy is defined as the sum of internal energy and product of pressure and volume:
H=U+PVH = U + PVH=U+PV
- Differential form:
dH=dU+PdV+VdP=TdS+VdPdH = dU + PdV + VdP = TdS + VdPdH=dU+PdV+VdP=TdS+VdP
- Natural variables: H=H(S,P)H = H(S, P)H=H(S,P)
3.2 Significance
- Useful in constant pressure processes, such as chemical reactions in open vessels
- Heat added at constant pressure:
dH=QPdH = Q_PdH=QP
Applications:
- Steam turbines, boilers, and exothermic/endothermic reactions
4. Helmholtz Free Energy (A or F)
4.1 Definition
Helmholtz free energy is defined as:
A=U−TSA = U – TSA=U−TS
- Natural variables: A=A(T,V)A = A(T, V)A=A(T,V)
- Differential form:
dA=dU−TdS−SdT=−SdT−PdVdA = dU – TdS – SdT = -SdT – PdVdA=dU−TdS−SdT=−SdT−PdV
4.2 Physical Significance
- Represents maximum useful work obtainable from a system at constant temperature and volume
- Spontaneous processes:
ΔA<0\Delta A < 0ΔA<0
Applications:
- Thermodynamic analysis of closed systems at constant T and V
- Statistical mechanics → partition function calculation
5. Gibbs Free Energy (G)
5.1 Definition
Gibbs free energy is defined as:
G=H−TS=U+PV−TSG = H – TS = U + PV – TSG=H−TS=U+PV−TS
- Natural variables: G=G(T,P)G = G(T, P)G=G(T,P)
- Differential form:
dG=dH−TdS−SdT=VdP−SdTdG = dH – TdS – SdT = VdP – SdTdG=dH−TdS−SdT=VdP−SdT
5.2 Physical Significance
- Represents maximum non-expansion work obtainable from a system at constant temperature and pressure
- Spontaneous processes:
ΔG<0\Delta G < 0ΔG<0
Applications:
- Predict chemical reaction spontaneity
- Phase transitions (melting, boiling)
- Electrochemical cells
6. Relationships Between Potentials
| Potential | Formula | Natural Variables | Work Function |
|---|---|---|---|
| Internal Energy (U) | U | S, V | dU = TdS – PdV |
| Enthalpy (H) | H = U + PV | S, P | dH = TdS + VdP |
| Helmholtz (A) | A = U – TS | T, V | dA = -SdT – PdV |
| Gibbs (G) | G = H – TS | T, P | dG = -SdT + VdP |
- Legendre transformations used to derive A and G from U
- Allows use of natural variables convenient for different processes
7. Maxwell Relations
From the thermodynamic potentials, we can derive Maxwell relations:
- From U(S,V)U(S,V)U(S,V):
(∂T∂V)S=−(∂P∂S)V\left(\frac{\partial T}{\partial V}\right)_S = – \left(\frac{\partial P}{\partial S}\right)_V(∂V∂T)S=−(∂S∂P)V
- From H(S,P)H(S,P)H(S,P):
(∂T∂P)S=(∂V∂S)P\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P(∂P∂T)S=(∂S∂V)P
- From A(T,V)A(T,V)A(T,V):
(∂S∂V)T=(∂P∂T)V\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V(∂V∂S)T=(∂T∂P)V
- From G(T,P)G(T,P)G(T,P):
(∂S∂P)T=−(∂V∂T)P\left(\frac{\partial S}{\partial P}\right)_T = – \left(\frac{\partial V}{\partial T}\right)_P(∂P∂S)T=−(∂T∂V)P
Applications:
- Derive relations between measurable quantities
- Predict system behavior without detailed microscopic knowledge
8. Applications of Thermodynamic Potentials
- Chemical Reactions
- Gibbs free energy used to predict reaction spontaneity and equilibrium constants
- Phase Transitions
- ΔG = 0 → equilibrium between phases
- Heat Engines
- Helmholtz energy → work at constant volume
- Gibbs energy → work at constant pressure
- Electrochemistry
- Gibbs free energy → maximum electrical work
- Statistical Mechanics
- Partition functions related to A and G
9. Examples
Example 1: Gibbs Free Energy Change
- Reaction: A → B at T = 298 K, P = 1 atm
- ΔH = -100 kJ, ΔS = -200 J/K
ΔG=ΔH−TΔS=−100000−298(−200)=−100000+59600=−40400 J\Delta G = \Delta H – T \Delta S = -100000 – 298(-200) = -100000 + 59600 = -40400 \, \text{J} ΔG=ΔH−TΔS=−100000−298(−200)=−100000+59600=−40400J
- ΔG < 0 → spontaneous at 298 K
Example 2: Maximum Work from Helmholtz Energy
- System at constant T and V, ΔA = -500 J
- Maximum useful work obtainable = 500 J
10. Graphical Representation
- Gibbs Free Energy vs Temperature
- Slope = -S, intercept = H
- Helmholtz Free Energy vs Volume
- Slope = -P, useful for work analysis
- Phase Diagrams
- Lines of equilibrium → ΔG = 0
11. Relations with Equilibrium
- At equilibrium, potentials minimized:
- Constant T, V → A minimum
- Constant T, P → G minimum
- ΔG = 0 → system at equilibrium
12. Summary Table
| Potential | Formula | Natural Variables | Application |
|---|---|---|---|
| U | U | S, V | Isochoric/adiabatic processes |
| H | H = U + PV | S, P | Constant pressure processes |
| A | A = U – TS | T, V | Maximum work at T,V constant |
| G | G = H – TS | T, P | Maximum work at T,P constant, chemical reactions |
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