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Chemical thermodynamics provides the fundamental framework for understanding energy transformations and equilibrium in chemical systems. It's essential for designing chemical processes, predicting reaction feasibility, and optimizing energy usage.

The Laws of Thermodynamics

First Law: Conservation of Energy

The first law states that energy cannot be created or destroyed, only converted from one form to another:

\Delta U = Q - W

Where:

$\Delta U$ = change in internal energy
$Q$ = heat transferred to the system
$W$ = work done by the system

For chemical processes, we often use the enthalpy form:

\Delta H = \Delta U + \Delta (PV)

Second Law: Entropy and Spontaneity

The second law introduces entropy ( $S$ ) as a measure of disorder:

\Delta S_{universe} \geq 0

For a spontaneous process at constant temperature and pressure:

\Delta G = \Delta H - T\Delta S \leq 0

Where $\Delta G$ is the Gibbs free energy change.

Third Law: Absolute Entropy

The entropy of a perfect crystal at absolute zero temperature is zero:

\lim_{T \to 0} S = 0

Phase Equilibria

Phase Rule

Gibbs phase rule determines the number of degrees of freedom in a system:

F = C - P + 2

Where:

$F$ = degrees of freedom
$C$ = number of components
$P$ = number of phases

Vapor-Liquid Equilibrium

For ideal mixtures, Raoult's Law applies:

y_i P = x_i P_i^{sat}

For non-ideal systems, we use activity coefficients:

y_i P = x_i \gamma_i P_i^{sat}

Chemical Potential

The chemical potential ( $\mu_i$ ) is the partial molar Gibbs free energy:

\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}}

At equilibrium, the chemical potential of each component is equal in all phases:

\mu_i^\alpha = \mu_i^\beta = \mu_i^\gamma = \cdots ``` ## Fugacity and Activity For real systems, we define fugacity ($f$) as:

\mu_i = \mu_i^\circ + RT \ln\left(\frac{f_i}{f_i^\circ}\right)

Activity ($a_i$) relates to fugacity:

a_i = \frac{f_i}{f_i^\circ}

--- ## Advanced Thermodynamic Concepts ### Thermodynamic Potentials Chemical thermodynamics uses several key potentials to describe system behavior: **Helmholtz Free Energy ($A$):**

A = U - TS

Useful for constant volume, constant temperature processes. **Gibbs Free Energy ($G$):**

G = H - TS = U + PV - TS

Most important for chemical processes at constant pressure and temperature. **Maxwell Relations:** These relations connect different thermodynamic properties:

\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V

\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P

### Chemical Equilibrium For a general chemical reaction:

aA + bB \rightleftharpoons cC + dD

The equilibrium constant is related to Gibbs free energy:

\Delta G^\circ = -RT \ln K

Where the standard Gibbs free energy change is:

\Delta G^\circ = \sum \nu_i \Delta G_f^\circ

### Non-Ideal Systems For real systems, we use activities instead of concentrations: **Activity Coefficient ($\gamma$):**

a_i = \gamma_i x_i

**Fugacity ($f$):** For gases, fugacity represents "effective pressure":

f_i = \phi_i P

Where $\phi_i$ is the fugacity coefficient. ### Thermodynamic Consistency The Gibbs-Duhem equation ensures thermodynamic consistency:

\sum n_i d\mu_i = -S dT + V dP

## Real-World Application: Refrigeration Cycle A vapor-compression refrigeration cycle demonstrates thermodynamic principles: ### Process Steps: 1. **Compression**: Work input increases pressure and temperature (isentropic compression) 2. **Condensation**: Heat rejection at constant pressure (isobaric heat rejection) 3. **Expansion**: Pressure reduction through throttling (isenthalpic expansion) 4. **Evaporation**: Heat absorption at constant pressure (isobaric heat absorption) ### Thermodynamic Analysis Each process can be analyzed using the first law: **Compressor (1\to2):**

W_{in} = h_2 - h_1

**Condenser (2→3):**

Q_{out} = h_2 - h_3

**Expansion valve (3→4):**

h_3 = h_4 \quad \text{(isenthalpic)}

**Evaporator (4→1):**

Q_{in} = h_1 - h_4

### Energy Balance Example Calculate the coefficient of performance (COP) for a refrigerator: ```python # Refrigerant properties (R134a example) h_evap_in = 250 # kJ/kg (enthalpy at evaporator inlet) h_evap_out = 400 # kJ/kg (enthalpy at evaporator outlet) h_cond_out = 250 # kJ/kg (enthalpy at condenser outlet) # TODO: Calculate COP # COP = (heat absorbed in evaporator) / (work input in compressor) heat_absorbed = 0 work_input = 0 COP = 0 print(f"Heat absorbed: {heat_absorbed} kJ/kg") print(f"Work input: {work_input} kJ/kg") print(f"COP: {COP:.2f}") # Additional analysis: Carnot COP for comparison T_evap = 273 - 10 # -10°C evaporator temperature (K) T_cond = 273 + 40 # 40°C condenser temperature (K) carnot_COP = T_evap / (T_cond - T_evap) print(f"Carnot COP (theoretical maximum): {carnot_COP:.2f}") ``` --- ## Your Challenge: Reaction Feasibility Analysis In this exercise, you'll determine if a chemical reaction is feasible under given conditions. **Goal**: Calculate Gibbs free energy change and equilibrium constant for a reaction. ### Reaction System Consider the water-gas shift reaction: $$ CO + H_2O \rightleftharpoons CO_2 + H_2 $$ Given thermodynamic data at 298 K: - $\Delta H^\circ = -41.2$ kJ/mol - $\Delta S^\circ = -42.1$ J/mol·K ### Key Equations Gibbs free energy:

\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ

Equilibrium constant:

K = e^{-\Delta G^\circ / RT}

Where $R = 8.314$ J/mol·K ```python # Thermodynamic data delta_H = -41.2 # kJ/mol delta_S = -42.1 # J/mol·K temperature = 298 # K R = 8.314 # J/mol·K # TODO: Calculate Gibbs free energy and equilibrium constant # Convert units as needed delta_G = 0 # kJ/mol K_eq = 0 print(f"Gibbs free energy change: {delta_G:.2f} kJ/mol") print(f"Equilibrium constant: {K_eq:.2f}") # Determine feasibility if delta_G < 0: print("Reaction is spontaneous at 298 K") else: print("Reaction is not spontaneous at 298 K") ``` What temperature would make this reaction spontaneous? How does the equilibrium constant change with temperature?

Chemical Thermodynamics