01: First Law of Thermodynamics

1. Statement of the Law

Energy is conserved in all processes. It can change form or transfer between system and surroundings, but cannot be created or destroyed.

$\boxed{\Delta E = Q - W}$

where $E = U + E_k + E_p$ is the total energy of the system.

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2. Mathematical Form (Closed System)

For a differential process: $\delta Q - \delta W = dU + dE_k + dE_p$

Neglecting kinetic and potential energy: $\boxed{\delta Q - \delta W = dU}$

Integrating between states 1 and 2: $Q_{12} - W_{12} = U_2 - U_1$

Sign Convention

• ($Q > 0$): heat into system
• ($W > 0$): work done by system

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3. Forms of Work

Type	Differential Form	Remarks
Boundary (expansion)	( $\delta W_b = P\,dV$ )	Only for quasi-static (reversible)
Shaft work	( $\delta W_s = \tau\,d\theta$ )	Rotation
Electrical	( $\delta W_e = E\,dq$ )	Potential × charge
Surface	( $\delta W_\sigma = \sigma\,dA$ )	Interface creation
Magnetic	( $\delta W_m = H\,dB$ )	Magnetic energy
Flow work	( $\delta W_{flow} = P\,dv$ )	Present in open systems

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4. Energy Balance for Control Volume (Open System)

Start with the Reynolds Transport Theorem:

$\frac{dE_{sys}}{dt} = \frac{d}{dt}\int_{CV} \rho e \, dV + \dot{m}(e + Pv + \tfrac{v^2}{2} + gz)$

Simplify to the steady-flow energy equation:

$\boxed{\dot{Q} - \dot{W} = \dot{m}\left(h_2 - h_1 + \frac{v_2^2 - v_1^2}{2} + g(z_2 - z_1)\right)}$

where $(h = u + Pv)$.

If kinetic and potential changes are negligible: $\dot{Q} - \dot{W} = \dot{m}(h_2 - h_1)$

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5. Specific Forms for Common Devices

Device	Simplification	Equation
Nozzle	Adiabatic, ($W=0$), no height change	( $h_1 + \frac{v_1^2}{2} = h_2 + \frac{v_2^2}{2}$ )
Diffuser	Reverse of nozzle	( $h_2 > h_1$ )
Compressor	Adiabatic, steady	( $\dot{W}_{in} = \dot{m}(h_2 - h_1)$ )
Turbine	Adiabatic, steady	( $\dot{W}_{out} = \dot{m}(h_1 - h_2)$ )
Throttle (valve)	( $Q=W=0$ )	( $h_1 = h_2$ )
Heat exchanger	Negligible work	( $\dot{Q}_A + \dot{Q}_B = 0$ )

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6. Enthalpy

Defined for convenience in open systems: $h = u + Pv$ Differential: $dh = du + PdV + VdP$ At constant pressure heating: $\delta Q_P = dh$

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7. Internal Energy and Enthalpy of Ideal Gases

For an ideal gas, ($u$) and ($h$) depend only on ($T$): $du = c_v dT, \quad dh = c_p dT$ $c_p - c_v = R$

Integrating: $u_2 - u_1 = \int_{T_1}^{T_2} c_v(T)\,dT$ $h_2 - h_1 = \int_{T_1}^{T_2} c_p(T)\,dT$

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8. Polytropic Processes (Closed System)

General work relation: $W_{12} = \int_{V_1}^{V_2} P\,dV$

If ($PV^n = \text{constant}$): $W_{12} = \frac{P_2V_2 - P_1V_1}{1 - n}$ $Q_{12} = \Delta U + W_{12}$ Special cases:

Process	Exponent (n)	Expression
Isothermal	1	( $W = P_1V_1 \ln(V_2/V_1)$ )
Adiabatic (reversible)	( $\gamma = c_p/c_v$ )	( $PV^\gamma = \text{const}$ )
Isochoric	$\infty$	( $W = 0$ )
Isobaric	0	( $W = P(V_2 - V_1)$ )

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9. Energy in Chemical Systems

For chemical reactions, the first law extends to include chemical work (bond rearrangements).

Let ($U$) include chemical internal energy ($U_{chem}$): $\Delta U = Q - W + \sum_i \nu_i \Delta u_{f,i}^0$

At constant pressure: $\boxed{\Delta H_{rxn} = Q_P = \sum_i \nu_i h_{f,i}^0}$

where:

• ( $\nu_i$ ): stoichiometric coefficient (+ for products, − for reactants)
• ( $h_f^0$ ): standard molar enthalpy of formation

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10. Gibbs Energy and Helmholtz Energy

Two derived energy potentials simplify equilibrium and spontaneous process analysis.

Potential	Definition	Natural Variables	Use
Internal Energy (U)	(U)	(S, V)	General system energy
Enthalpy (H)	(U + PV)	(S, P)	Constant pressure heating
Helmholtz Free Energy (A)	(U - TS)	(T, V)	Constant (T,V) processes
Gibbs Free Energy (G)	(U + PV - TS = H - TS)	(T, P)	Constant (T,P) processes

Differentials:

$\begin{aligned} dU &= T\,dS - P\,dV \\ dH &= T\,dS + V\,dP \\ dA &= -S\,dT - P\,dV \\ dG &= -S\,dT + V\,dP \end{aligned}$

At constant ($T,P$): $\Delta G = \Delta H - T\Delta S$

A spontaneous process (in a closed isothermal–isobaric system) requires: $\boxed{\Delta G < 0}$

At equilibrium: ( $\Delta G = 0$ ).

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11. Chemical Potential

For multicomponent systems: $G = \sum_i n_i \mu_i$ and differential: $dG = -S\,dT + V\,dP + \sum_i \mu_i\,dn_i$

At constant ($T,P$): $\mu_i = \left( \frac{\partial G}{\partial n_i} \right)_{T,P,n_{j\ne i}}$ The condition for chemical equilibrium: $\sum_i \nu_i \mu_i = 0$

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12. Enthalpy and Gibbs Energy of Reaction

At constant ($T,P$): $\Delta_r H = \sum_i \nu_i h_i$ $\Delta_r G = \sum_i \nu_i g_i$

and the equilibrium constant: $\boxed{K = e^{-\Delta_r G^\circ / (RT)}}$

This links macroscopic thermodynamics to equilibrium chemistry.

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13. Energy Diagrams

Type	System	Dominant Terms
Adiabatic turbine	Open	( $\dot{W} = \dot{m}(h_1 - h_2)$ )
Electrochemical cell	Closed	( $W_{elec} = -\Delta G$ )
Heat engine	Cyclic	( $W_{net} = Q_{in} - Q_{out}$ )
Chemical reactor	Open	( $Q + W = \Delta H_{rxn}$ )

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14. Summary Equations

Category	Equation	Conditions
Closed system	( $Q - W = \Delta U$ )	General
Steady-flow	( $\dot{Q} - \dot{W} = \dot{m}(h_2 - h_1 + \frac{v^2}{2} + gz)$ )	Open, steady
Adiabatic reversible	( $PV^\gamma = \text{const}$ )	Ideal gas
Enthalpy	( $h = u + Pv$ )	Definition
Gibbs	( $G = H - TS$ )	Definition
Equilibrium	( $\sum \nu_i \mu_i = 0$ )	Reaction equilibrium

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15. References

• Moran & Shapiro, Fundamentals of Engineering Thermodynamics
• Çengel & Boles, Thermodynamics: An Engineering Approach
• Smith, Van Ness & Abbott, Introduction to Chemical Engineering Thermodynamics
• Atkins & de Paula, Physical Chemistry

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