# Lecture 22: The First Law of Thermodynamics

## Physics I — Mechanics & Thermodynamics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Define internal energy, work, and heat in a thermodynamic context
2. State and apply the first law of thermodynamics
3. Calculate work done by or on a gas during expansion and compression
4. Analyze thermodynamic processes: isothermal, isobaric, isochoric, and adiabatic
5. Interpret and use P-V diagrams to calculate work and identify processes
6. Apply the ideal gas law in thermodynamic calculations

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## Lecture Content

### I. Thermodynamic Systems and State Variables

A **thermodynamic system** is a defined quantity of matter or region of space under study, and everything outside it constitutes the **surroundings**. **State variables** describe the current state of the system: pressure (P), volume (V), temperature (T), number of moles (n), and internal energy (U). These depend only on the current state, not on how the system arrived there. By contrast, **process variables** such as heat (Q) and work (W) depend on the path and are not state variables.

The **ideal gas law** serves as the equation of state relating P, V, and T: PV = nRT, where R = 8.314 J/(mol K) is the universal gas constant and n is the number of moles.

### II. Internal Energy

**Internal energy** (U) is the total microscopic energy of a system, encompassing the kinetic and potential energy of all its molecules. For an ideal gas, internal energy depends only on temperature: U = n C_V T, where C_V is the molar heat capacity at constant volume. For a monatomic ideal gas, C_V = (3/2)R, giving U = (3/2)nRT. For a diatomic ideal gas, C_V = (5/2)R, so U = (5/2)nRT.

The change in internal energy is Delta U = n C_V Delta T. Because internal energy is a **state function**, Delta U depends only on the initial and final temperatures, regardless of the process that connects them.

### III. Work Done by a Gas

When a gas expands or is compressed, it exchanges work with the surroundings: W = integral from V_i to V_f of P dV. For expansion (V_f > V_i), the gas does positive work on its surroundings. For compression (V_f < V_i), the surroundings do work on the gas, and W is negative. Graphically, work equals the **area under the P-V curve**. Because different processes between the same endpoints trace different curves, work depends on the path, not just on the initial and final states.

### IV. The First Law of Thermodynamics

The first law is a statement of **conservation of energy** applied to thermodynamic systems: **Delta U = Q - W**, where Delta U is the change in internal energy, Q is the heat added to the system (positive when heat flows in), and W is the work done by the system (positive when the gas expands). Some texts use an alternative sign convention, Delta U = Q + W, where W represents work done on the system.

The first law tells us that the change in internal energy equals the net energy input: heat flowing in minus work flowing out. For a cyclic process, the system returns to its initial state, so Delta U = 0 and Q = W.

<image>A schematic of a gas in a cylinder with a movable piston. Arrows show heat Q flowing in from a hot reservoir at the bottom, the piston being pushed up as the gas expands (work W = integral P dV done by the gas), and the internal energy U of the gas increasing or decreasing. The first law equation Delta U = Q - W is written prominently. A bar diagram shows the energy balance: Q entering the system is split between Delta U (stored) and W (output).</image>

### V. Thermodynamic Processes

An **isobaric process** occurs at constant pressure. The work done is W = P Delta V = P (V_f - V_i), the heat exchanged is Q = n C_P Delta T (where C_P = C_V + R), and the change in internal energy is Delta U = n C_V Delta T. On a P-V diagram, this appears as a horizontal line.

An **isochoric (isovolumetric) process** occurs at constant volume. Since there is no volume change, W = 0, and all heat goes directly into changing internal energy: Q = n C_V Delta T = Delta U. On a P-V diagram, this is a vertical line.

An **isothermal process** occurs at constant temperature. For an ideal gas whose internal energy depends only on temperature, Delta U = 0, so Q = W. The work done is W = nRT ln(V_f / V_i). On a P-V diagram, this traces a hyperbola (PV = constant).

An **adiabatic process** involves no heat exchange (Q = 0), so Delta U = -W, and any work comes entirely from the internal energy. The relationship PV^gamma = constant holds (where gamma = C_P/C_V), as does TV^(gamma-1) = constant. A gas cools during adiabatic expansion and heats during adiabatic compression. On a P-V diagram, the adiabatic curve is steeper than the isothermal curve.

<image>A P-V diagram showing four processes starting from the same initial state (P_i, V_i): (1) isobaric expansion (horizontal line to the right), (2) isothermal expansion (hyperbolic curve, labeled PV = const), (3) adiabatic expansion (steeper curve below the isothermal, labeled PV^gamma = const), (4) isochoric process (vertical line downward). Each process is drawn in a different color. The area under each expansion curve (representing work) is shaded, clearly showing that isobaric work > isothermal work > adiabatic work for the same final volume. Equations for W are listed beside each curve.</image>

### VI. P-V Diagrams and Cyclic Processes

A P-V diagram is a powerful graphical tool for thermodynamic analysis. Each point represents a unique equilibrium state characterized by P, V, and T. The area under a process curve equals the work done during that process.

For a **cyclic process** that forms a closed loop on the P-V diagram, the net work equals the **area enclosed** by the loop. A clockwise loop represents net positive work done by the system, corresponding to an engine. A counterclockwise loop represents net work done on the system, corresponding to a refrigerator or heat pump. Since the system returns to its initial state, Delta U = 0, and the net work equals the net heat: W_net = Q_net.

### VII. Molar Heat Capacities of Ideal Gases

The molar heat capacity at constant volume, **C_V**, takes characteristic values depending on molecular complexity. For monatomic gases (He, Ne, Ar), C_V = (3/2)R = 12.5 J/(mol K). For diatomic gases (N_2, O_2, H_2) at moderate temperatures, C_V = (5/2)R = 20.8 J/(mol K). For polyatomic gases, C_V is approximately 3R = 24.9 J/(mol K).

The molar heat capacity at constant pressure, **C_P**, is always larger: C_P = C_V + R. This gives C_P = (5/2)R for monatomic and (7/2)R for diatomic gases. The **ratio of heat capacities** gamma = C_P / C_V equals 5/3 = 1.67 for monatomic gases and 7/5 = 1.40 for diatomic gases.

The reason C_P exceeds C_V is that at constant pressure, part of the added heat goes into expansion work (P Delta V = nR Delta T), leaving less energy available for raising the temperature. Therefore, more heat input is needed for the same temperature change. The equipartition theorem provides a microscopic explanation: each degree of freedom contributes (1/2)R to C_V.

<image>A diagram showing degrees of freedom for gas molecules. Left: a monatomic gas particle (single sphere) with 3 translational degrees of freedom (arrows along x, y, z), giving C_V = (3/2)R. Center: a diatomic molecule (two spheres connected by a bond) with 3 translational + 2 rotational degrees of freedom, giving C_V = (5/2)R. The two rotation axes perpendicular to the bond axis are shown. Right: a table summarizing C_V, C_P, and gamma for monatomic, diatomic, and polyatomic gases.</image>