Thermodynamics

A thermodynamic system is a specified macroscopic quantity of matter bounded by an arbitrary closed surface separating it from the external surroundings. Systems are classified by the permeability of their boundaries:

• Open System: Exchanges both matter and energy (heat/work) with surroundings (e.g., boiling water in an open beaker).
• Closed System: Exchanges energy but NOT matter with surroundings (e.g., gas enclosed in a cylinder fitted with a sealed piston).
• Isolated System: Exchanges neither matter nor energy with surroundings (e.g., hot liquid in an ideally insulated vacuum flask).

Diathermic Wall: An impermeable boundary that readily permits heat flow (thermal conductor like copper). Adiabatic Wall: An impermeable boundary that completely prevents heat exchange (ideal thermal insulator like asbestos or vacuum jacket).

The thermodynamic state of an equilibrium system is completely described by macroscopic parameters called state variables:

• Intensive Variables: Independent of the size or mass of the system (e.g., Pressure $P$, Temperature $T$, Density $\rho$).
• Extensive Variables: Directly proportional to the size or mass of the system (e.g., Volume $V$, Internal Energy $U$, Entropy $S$, Mass $M$).

The functional relationship connecting state variables in equilibrium is the Equation of State. For $n$ moles of an ideal gas:

where $R = 8.314\,\text{J}/(\text{mol}\cdot\text{K})$ is the Universal Gas Constant.

Two thermodynamic systems are said to be in thermal equilibrium if, when placed in thermal contact across a diathermic wall, no net exchange of heat occurs between them and their macroscopic thermodynamic coordinates remain constant in time.

Physical Significance: The Zeroth Law implies the existence of a scalar state property called Temperature ($T$). Temperature is the physical property whose equality between two systems determines whether they are in mutual thermal equilibrium. This law provides the theoretical foundation for thermometry: system $C$ serves as a thermometer to calibrate systems $A$ and $B$.

The Internal Energy ($U$) of a thermodynamic system is the total energy possessed by its constituent molecules, comprising molecular kinetic energies (translational, rotational, vibrational) and intermolecular potential energies:

For an ideal gas, intermolecular attractive forces are negligible ($E_P = 0$), so internal energy is purely kinetic and depends exclusively on temperature (Joule's Law):

State Function: $U$ depends solely on the current equilibrium state ($P, V, T$) of the system and is independent of the past path taken to reach that state. In any cyclic process returning to the initial state, the net change in internal energy is strictly zero: $\oint dU = 0$.

When a gas enclosed in a cylinder of cross-sectional area $A$ pushes a frictionless piston outward by distance $dx$ against external pressure $P$, the incremental mechanical work done by the system is:

Total work done during a quasi-static expansion from initial volume $V_1$ to final volume $V_2$:

Sign Convention in Physics:

• $W > 0$ (Positive): Work done by the gas on surroundings during expansion ($dV > 0$).
• $W < 0$ (Negative): Work done on the gas by external agents during compression ($dV < 0$).
• $Q > 0$ (Positive): Heat absorbed or added to the system.
• $Q < 0$ (Negative): Heat released or rejected by the system.

The First Law is the formulation of the Law of Conservation of Energy applied to thermodynamic systems:

In differential form for an infinitesimal quasi-static process:

Key Insight: Heat ($dQ$) and work ($dW$) are path-dependent functions (inexact differentials), but their difference $dU = dQ - dW$ is a path-independent exact differential representing the change in a fundamental state function ($U$).

A quasi-static process is an idealized infinitely slow succession of equilibrium states, allowing state variables to remain well-defined throughout.

Differentiating the process equations with respect to volume $V$:

• Isothermal Curve: $P V = \text{const} \implies P\,dV + V\,dP = 0 \implies \left(\frac{dP}{dV}\right)_{\text{iso}} = -\frac{P}{V}$
• Adiabatic Curve: $P V^\gamma = \text{const} \implies V^\gamma\,dP + \gamma P V^{\gamma-1}\,dV = 0 \implies \left(\frac{dP}{dV}\right)_{\text{adi}} = -\gamma \frac{P}{V}$

Using the ideal gas equation $P = nRT/V$, Poisson's adiabatic relations can be expressed in three equivalent formulations:

Molar Heat Capacity ($C$): The amount of heat required to raise the temperature of 1 mole of a substance by 1 Kelvin ($1\,^\circ\text{C}$):

• Molar Heat Capacity at Constant Volume ($C_v$): Since $dV = 0 \implies dW = 0$, all added heat increases internal energy: $C_v = \frac{1}{n}\left(\frac{dU}{dT}\right)_V$.
• Molar Heat Capacity at Constant Pressure ($C_p$): The added heat must not only raise the temperature (increase $U$) but also perform external mechanical work ($P\,dV$) as the gas expands against constant pressure: $C_p = \frac{1}{n}\left(\frac{dQ}{dT}\right)_P$. Hence $C_p > C_v$.

Consider 1 mole ($n = 1$) of an ideal gas. By the First Law of Thermodynamics:

At constant volume: $dV = 0 \implies dQ_v = dU = C_v\,dT$.

Since the internal energy of an ideal gas depends solely on temperature, the expression $dU = C_v\,dT$ applies to any process where temperature changes by $dT$.

At constant pressure: $dQ_p = C_p\,dT$. Substituting into the First Law:

From the ideal gas equation for 1 mole, $P V = R T$. Differentiating at constant pressure $P$:

Substituting $P\,dV = R\,dT$ into the First Law expression:

Dividing throughout by $dT$ yields Mayer's Relation:

Law of Equipartition of Energy: For any dynamical system in thermal equilibrium at absolute temperature $T$, the average kinetic energy associated with each independent degree of freedom is $\frac{1}{2} k_B T$, where $k_B$ is Boltzmann's constant.

For 1 mole of gas ($N_A k_B = R$) with $f$ degrees of freedom:

The First Law enforces total energy conservation but imposes no directionality on physical processes. For instance, when a bullet stops inside a wooden target, mechanical kinetic energy converts entirely into heat. Yet the First Law cannot explain why heat from the block never spontaneously reassembles into kinetic energy to fire the bullet backward. The Second Law of Thermodynamics specifies the permissible direction of spontaneous thermodynamic processes.

Equivalence of the Two Statements: A violation of the Kelvin-Planck statement directly leads to a violation of the Clausius statement, and vice-versa. They are two manifestations of the same underlying physical law.

A Heat Engine is a cyclic thermodynamic device that converts thermal energy into mechanical work:

1. Absorbs heat $Q_1$ from a high-temperature heat source at absolute temperature $T_1$.
2. Performs external mechanical work $W$.
3. Rejects remaining waste heat $Q_2$ to a low-temperature heat sink at absolute temperature $T_2$.

By energy conservation over a complete cycle ($\Delta U = 0$): $W = Q_1 - Q_2$. The Thermal Efficiency ($\eta$) is:

A Refrigerator operates as a reversed heat engine. Working in a cycle, it absorbs heat $Q_2$ from a cold refrigerated interior at $T_2$, consumes external mechanical/electrical work $W$, and discharges heat $Q_1 = Q_2 + W$ to the warmer surroundings at $T_1$.

The performance of a refrigerator is characterized by its Coefficient of Performance (COP, $\beta$):

Relationship between Heat Engine efficiency $\eta$ and Refrigerator COP $\beta$:

In 1824, Nicolas Léonard Sadi Carnot conceived an idealized, perfectly reversible cyclic heat engine operating between a source at $T_1$ and a sink at $T_2$. The working substance (an ideal gas) undergoes four successive reversible operations on a $P\text{--}V$ diagram:

1. Step 1 $\to$ 2: Reversible Isothermal Expansion (at Source Temperature $T_1$):
   The cylinder is placed on the source at $T_1$. The gas expands slowly from $(P_1, V_1)$ to $(P_2, V_2)$. Since $\Delta T = 0 \implies \Delta U = 0$, heat absorbed from source equals work done:$$Q_1 = W_{1\to 2} = n R T_1 \ln\left(\frac{V_2}{V_1}\right)$$
2. Step 2 $\to$ 3: Reversible Adiabatic Expansion (from $T_1$ to Sink Temperature $T_2$):
   The cylinder is placed on an insulating stand ($dQ = 0$). Gas expands from $(P_2, V_2)$ to $(P_3, V_3)$, cooling from $T_1$ to $T_2$:$$W_{2\to 3} = \frac{n R (T_1 - T_2)}{\gamma - 1}, \quad \text{with } T_1 V_2^{\gamma-1} = T_2 V_3^{\gamma-1}$$
3. Step 3 $\to$ 4: Reversible Isothermal Compression (at Sink Temperature $T_2$):
   The cylinder is placed on the sink at $T_2$. Gas is slowly compressed from $(P_3, V_3)$ to $(P_4, V_4)$, rejecting heat $Q_2$:$$Q_2 = -W_{3\to 4} = n R T_2 \ln\left(\frac{V_3}{V_4}\right)$$
4. Step 4 $\to$ 1: Reversible Adiabatic Compression (from $T_2$ back to $T_1$):
   The cylinder is placed back on the insulating stand ($dQ = 0$). Gas is compressed from $(P_4, V_4)$ back to original state $(P_1, V_1)$, heating up from $T_2$ to $T_1$:$$W_{4\to 1} = -\frac{n R (T_1 - T_2)}{\gamma - 1}, \quad \text{with } T_2 V_4^{\gamma-1} = T_1 V_1^{\gamma-1}$$

Net work done per cycle: $W = W_{1\to 2} + W_{2\to 3} + W_{3\to 4} + W_{4\to 1}$. Since $W_{2\to 3} = -W_{4\to 1}$, the adiabatic works cancel identically:

From the adiabatic expansion and compression equations:

Dividing the two relations yields the volume ratio equivalence:

Therefore, the ratio of rejected heat to absorbed heat simplifies to:

Carnot's Theorem:

1. No real engine operating between two given constant thermal reservoirs can have a higher efficiency than a reversible Carnot engine operating between the same two reservoirs ($\eta_{\text{real}} \le \eta_{\text{Carnot}}$).
2. All reversible engines operating between the same two temperatures have the identical efficiency, completely independent of the nature of the working substance.

Since $\eta = 1 - T_2/T_1$ depends solely on temperatures, Lord Kelvin utilized Carnot's relation to establish the Absolute Thermodynamic Temperature Scale, which is independent of any specific thermometric substance.