Premed · Premed · Physics 1
Lecture 22: The First Law of Thermodynamics
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define internal energy, work, and heat in a thermodynamic context
- State and apply the first law of thermodynamics
- Calculate work done by or on a gas during expansion and compression
- Analyze thermodynamic processes: isothermal, isobaric, isochoric, and adiabatic
- Interpret and use P-V diagrams to calculate work and identify processes
- Apply the ideal gas law in thermodynamic calculations
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>
