The Timpson Lyceum · University Physics I (2025)

First Law of Thermodynamics - Lecture 19 (University Physics I)

Overview

This lecture introduces the first law of thermodynamics, the broad statement of energy conservation that governs systems of gas (here, ideal gases almost exclusively). It defines the internal energy of a gas, shows how to compute the work done by a gas as the area under a curve on a pV diagram, and ties heat, work, and internal energy together through $\Delta U = Q - W$. With that machinery in place, it walks through the four canonical processes a gas can undergo — isothermal, isobaric, isochoric (constant volume), and adiabatic — deriving the work and heat for each, and explores why work and heat are not state variables while internal energy is. It closes with the molar heat capacities $C_V$ and $C_P$, their relation $C_P = C_V + R$, the ratio $\gamma = C_P/C_V$, and the adiabatic relations between pressure, volume, and temperature.

Key Concepts

Internal energy ($U$)

• The sum of all the microscopic kinetic and potential energies of a gas's particles — translational and rotational kinetic energy plus vibrational/intermolecular potential energy. It has nothing to do with chemical energy or rest-mass energy.
• For an ideal gas, $U$ is a function of temperature only. Same temperature (with the same gas and same number of moles) means same internal energy, regardless of how you got there.

The first law of thermodynamics

• $\Delta U = Q - W$, where $Q$ is the heat added to the system and $W$ is the work done by the system. In differential form, $dU = dQ - dW$.
• Sign convention (used throughout): $Q > 0$ when heat flows into the system; $W > 0$ when the system does work on its surroundings (e.g. a gas expanding). Adding heat raises $U$; the system doing work lowers $U$.
• Heat and work are two distinct ways to change a system's energy. You can raise a gas's temperature without adding heat by doing work on it (rapidly compressing it heats it up — no flame, no hot plate). "Temperature up" means internal energy up, not "heat added."
• Heat $\ne$ temperature. Temperature measures average kinetic energy / internal energy; heat is a transfer of energy.

Work done by a gas

• Starting from $W = \int F\,dx$ in one dimension, substitute $F = pA$ (pressure $\times$ piston area). Since $A\,dx = dV$, the work done by an ideal gas is $W = \int_{V_1}^{V_2} p\,dV$
• Geometric meaning: on a pV diagram, work is the area under the curve from $V_1$ to $V_2$. Integrating right-to-left (a compression) gives negative work. A free expansion into a vacuum (no opposing pressure) does zero work.

State vs. path variables

• Internal energy is a state variable: $\Delta U$ depends only on the start and end states. Over any closed cycle, $\Delta U = 0$ (you return to the same temperature). Hence for a full cycle $Q = W$.
• Work and heat are NOT state variables. You cannot ask "how much work/heat is in a system" — they depend on the path taken, not the state.

The four processes (each computes work via $W=\int p\,dV$)

• Isothermal ($T$ constant): the process follows a $T$-line ($p = nRT/V$). Since $\Delta T = 0$, $\Delta U = 0$, so $Q = W$. $W = nRT\,\ln\!\frac{V_2}{V_1} = nRT\,\ln\!\frac{p_1}{p_2}$ (using $p_1V_1 = p_2V_2$, so $V_2/V_1 = p_1/p_2$).
• Isobaric ($p$ constant): $W = p\,(V_2 - V_1)$ — the area of a rectangle under the curve.
• Isochoric / constant volume ($V$ constant): no displacement, so $W = 0$. All added heat goes into internal energy: $\Delta U = Q = nC_V\,\Delta T$.
• Adiabatic ($Q = 0$): no heat exchanged. Then $\Delta U = -W$, so the work is $W = -nC_V\,\Delta T = nC_V\,(T_1 - T_2)$.

Adiabatic relations

• From $dU + dW = 0$ with $dU = nC_V\,dT$ and $dW = p\,dV$, integrating yields the path constraints (with $\gamma - 1 = R/C_V$):

• These let you find a missing temperature or pressure from the volumes when the variables you need aren't given directly.

Heat capacities of a gas

• Heat capacity = heat needed per unit temperature change. For a gas it depends on the constraint, so we define two molar values.
• Constant volume: $Q = nC_V\,\Delta T$ and $\Delta U = nC_V\,\Delta T$. From counting degrees of freedom ($\tfrac12 R$ each): monatomic $C_V = \tfrac32 R$, diatomic $C_V = \tfrac52 R$, polyatomic $\approx 3R$.
• Constant pressure: $Q = nC_P\,\Delta T$. $C_P > C_V$ because some added heat goes into the work of expansion. The first law gives the Mayer relation: $C_P = C_V + R$
• Adiabatic index: $\gamma = \dfrac{C_P}{C_V}$. Monatomic $\gamma = \tfrac53 \approx 1.67$; diatomic $\gamma = \tfrac75 = 1.4$. Given $\gamma$ (often what tables list), recover $C_V = \dfrac{R}{\gamma - 1}$ and $C_P = C_V + R$.

Worked Example

Vaporizing water (constant pressure). One gram ($1\text{ cm}^3$) of water at its boiling point is converted entirely to steam at atmospheric pressure $p = 1.013\times10^5\text{ Pa}$. Steam occupies $1671\times$ the liquid volume, so $\Delta V = 1670\text{ cm}^3 = 1670\times10^{-6}\text{ m}^3$. The heat of vaporization is $L_v = 2.256\times10^6\text{ J/kg}$.

Part A — work done by the expanding water (isobaric, so $W = p\,\Delta V$):

Part B — change in internal energy. The heat added is $Q = mL_v$, then apply the first law $\Delta U = Q - W$:

Almost all of the added heat becomes internal energy; only about 169 J goes into pushing back the atmosphere.

Summary

• The first law is $\Delta U = Q - W$ with $Q$ = heat into the system and $W$ = work done by the gas; in differential form $dU = dQ - dW$.
• Internal energy of an ideal gas depends only on temperature; over any closed cycle $\Delta U = 0$, forcing $Q = W$.
• Work done by a gas is $W = \int p\,dV$ — the area under the pV curve. Work and heat are path-dependent; only $U$ is a state variable.
• The four processes: isothermal ($W = nRT\ln(V_2/V_1)$, $Q = W$), isobaric ($W = p\,\Delta V$), isochoric ($W = 0$, $Q = \Delta U = nC_V\Delta T$), and adiabatic ($Q = 0$, $W = -nC_V\Delta T$, with $pV^{\gamma}$ and $TV^{\gamma-1}$ constant).
• Heat capacities satisfy $C_P = C_V + R$ ($C_P$ is larger because expansion does work) and $\gamma = C_P/C_V$, with $C_V = R/(\gamma-1)$.
• Degrees of freedom set $C_V$: $\tfrac32 R$ monatomic ($\gamma=\tfrac53$), $\tfrac52 R$ diatomic ($\gamma=1.4$), $\approx 3R$ polyatomic.