Thermodynamics Summary

First Law, Thermodynamic Processes, Work, Heat, and Internal Energy

Basic Concepts
First Law
Thermodynamic Processes
Applications

1. Basic Thermodynamic Concepts

Thermodynamics is concerned with the work done by a system and the heat it exchanges with its surroundings.

1.1 Work Done in Thermodynamic Processes

When a system is taken quasistatically from equilibrium state i to another equilibrium state f, the total work done by the system is:

\[W = \int_{V_i}^{V_f} p \, dV\]

Key Points:

  • Work is represented by the area under the curve on a P-V diagram
  • If \(V_f > V_i\), work done by the gas is positive
  • If volume decreases, work done by the gas is negative
  • Work depends on the thermodynamic path taken

1.2 Heat Capacity Relations

\[C_p - C_v = R\] \[\frac{C_p}{C_v} = \gamma\]

Where:

  • \(C_p\) = Molar heat capacity at constant pressure
  • \(C_v\) = Molar heat capacity at constant volume
  • R = Universal gas constant
  • γ = Adiabatic exponent

2. First Law of Thermodynamics

First Law of Thermodynamics

The difference \(Q - W\) is the same for all paths between given initial and final equilibrium states, and equals the change in internal energy ΔU of the system.

\[Q - W = \Delta U\] \[Q = \Delta U + W\]

2.1 Interpretation

  • Q = Heat transferred to the system
  • W = Work done by the system
  • ΔU = Change in internal energy of the system

Important Notes:

  • Both Q and W depend on the thermodynamic path
  • ΔU depends only on initial and final states (state function)
  • For an ideal gas, internal energy depends only on temperature
  • ΔU = nC_vΔT for an ideal gas

2.2 Sign Conventions

Quantity Positive Value Negative Value
Heat (Q) Heat added to system Heat removed from system
Work (W) Work done by system Work done on system
Internal Energy (ΔU) Increases Decreases

3. Thermodynamic Processes

Isothermal Process

Definition: Temperature remains constant

\[PV = nRT = \text{constant}\]

Work Done:

\[W = nRT \ln \left| \frac{V_f}{V_i} \right|\]

Internal Energy: ΔU = 0 (since ΔT = 0)

Adiabatic Process

Definition: No heat exchange with surroundings (Q = 0)

\[PV^\gamma = \text{constant}\]

Work Done:

\[W = \int_{V_i}^{V_f} p \, dV\] \[W = \frac{nR\Delta T}{1-\gamma}\]

First Law: ΔU = -W

3.1 Comparison of Processes

Process Heat (Q) Work (W) Internal Energy (ΔU)
Isothermal Q = W \(nRT \ln(V_f/V_i)\) 0
Adiabatic 0 -ΔU nC_vΔT
Isochoric nC_vΔT 0 nC_vΔT
Isobaric nC_pΔT PΔV nC_vΔT

4. Applications of First Law

4.1 Thermodynamic Cycles

The First Law is applied to analyze various thermodynamic cycles where a system undergoes a series of processes and returns to its initial state.

Example Cycle: Carnot Cycle (Ideal Gas)

1 → 2: Isothermal Expansion
\[\Delta U = 0\] \[W_1 = Q_1 = nRT \ln \frac{V_2}{V_1} \quad \text{(positive)}\]
2 → 3: Adiabatic Expansion
\[Q = 0\] \[W_2 = -\Delta U = \frac{nR\Delta T}{1-\gamma}\]
3 → 4: Isothermal Compression
\[\Delta U = 0\] \[W_3 = Q_2 = nRT \ln \left( \frac{V_4}{V_3} \right) \quad \text{(negative)}\]
4 → 1: Adiabatic Compression
\[Q = 0\] \[W_4 = -\Delta U = \frac{nR\Delta T}{1-\gamma}\]

4.2 Work Calculation in Different Processes

Isothermal Process Work Calculation:

For an isothermal process, work is calculated as:

\[W = \int_{V_i}^{V_f} p \, dV = nRT \int_{V_i}^{V_f} \frac{dV}{V}\] \[W = nRT \ln \left| \frac{V_f}{V_i} \right|\]

Adiabatic Process Work Calculation:

For an adiabatic process, work is related to temperature change:

\[W = \frac{nR(T_i - T_f)}{\gamma - 1}\]

Since Q = 0, all work done comes from internal energy change.

4.3 Important Relationships

  • For any cyclic process: ΔU = 0 (system returns to initial state)
  • Net work done in a cycle = Area enclosed by the cycle on P-V diagram
  • Efficiency of heat engine = Work output / Heat input
  • For adiabatic process: \(TV^{\gamma-1} = \text{constant}\) and \(P^{1-\gamma}T^\gamma = \text{constant}\)