Introduction

This chapter delves into the laws governing thermal energy and the conversion between work and heat. It revisits concepts like heat and temperature, contrasting the historical "caloric" theory with the modern understanding of heat as a form of energy. The work of Benjamin Thomson (Count Rumford) in demonstrating the conversion of work to heat through cannon boring is highlighted. Thermodynamics is defined as a macroscopic science that deals with bulk systems and their properties like pressure, volume, and temperature, without delving into the molecular constitution of matter. It is distinguished from mechanics, which focuses on the motion of systems as a whole, by its focus on the internal macroscopic state of a body.

Thermal Equilibrium

In thermodynamics, a system is considered to be in a state of thermodynamic equilibrium if its macroscopic variables, such as pressure, volume, and temperature, do not change with time. This equilibrium state is influenced by the surroundings and the nature of the wall separating the system from its surroundings. An adiabatic wall prevents the flow of energy (heat), while a diathermic wall allows heat to flow. When two systems are separated by a diathermic wall, their macroscopic variables will change until they reach a state where there is no net energy flow, indicating they are in thermal equilibrium. This state is characterized by equal temperatures.

Zeroth Law Of Thermodynamics

The Zeroth Law of Thermodynamics states that if two systems, A and B, are separately in thermal equilibrium with a third system, C, then A and B are also in thermal equilibrium with each other. This law is fundamental because it establishes the existence of a physical quantity that is common to all systems in thermal equilibrium. This quantity is identified as temperature (T). If systems A and B are both in thermal equilibrium with system C, then their temperatures are equal (T_A = T_C and T_B = T_C), implying T_A = T_B. This law provides the basis for constructing temperature scales and for thermometry.

Heat, Internal Energy And Work

Temperature is a measure of the "hotness" of a body and dictates the direction of heat flow between bodies in thermal contact. Heat flows from a higher temperature to a lower temperature until thermal equilibrium is reached.

Internal energy (U) of a system is the sum of the kinetic and potential energies of its molecular constituents. It is a state variable, meaning its value depends only on the current state of the system, not on how that state was achieved. For an ideal gas, where intermolecular forces are negligible, internal energy is primarily the sum of the kinetic energies associated with the random translational, rotational, and vibrational motions of its molecules. The overall kinetic energy of the system's center of mass is not included in internal energy.

Internal energy can be changed by two modes of energy transfer:

• Heat (ΔQ): Energy transfer due to a temperature difference between the system and its surroundings.
• Work (ΔW): Energy transfer accomplished through other means, such as the mechanical action of a piston.

It is crucial to distinguish between internal energy (a state variable) and heat or work (modes of energy transfer). Statements like "a gas in a given state has a certain amount of heat" are meaningless; only statements about heat supplied or work done are meaningful.

First Law Of Thermodynamics

The First Law of Thermodynamics is a statement of the conservation of energy applied to thermodynamic systems. It relates the heat supplied to a system, the work done by the system, and the change in its internal energy:

$$ \Delta Q = \Delta U + \Delta W $$

where:

• $\Delta Q$ is the heat supplied to the system.
• $\Delta U$ is the change in internal energy of the system.
• $\Delta W$ is the work done by the system on the surroundings.

This equation can also be written as $\Delta Q - \Delta W = \Delta U$. While $\Delta U$ is path-independent (depends only on the initial and final states), $\Delta Q$ and $\Delta W$ generally depend on the path taken during the process. The combination $\Delta Q - \Delta W$, however, is path-independent.

For a gas in a cylinder with a movable piston, if the pressure P is constant, the work done by the system is $\Delta W = P \Delta V$, where $\Delta V$ is the change in volume. Thus, the First Law becomes $\Delta Q = \Delta U + P \Delta V$.

An example of its application is the phase transition of water to steam. For 1 g of water, the heat absorbed (latent heat) is 2256 J. The work done against atmospheric pressure during this expansion is approximately 169.2 J. The change in internal energy is then 2256 J - 169.2 J = 2086.8 J, indicating that most of the heat supplied goes into increasing the internal energy.

Specific Heat Capacity

Heat capacity (S) is the amount of heat required to change the temperature of a substance by a unit amount ($\Delta Q / \Delta T$).

Specific heat capacity (s) is the heat capacity per unit mass:

$$ s = \frac{1}{m} \frac{\Delta Q}{\Delta T} $$

Its SI unit is J kg^-1 K^-1.

Molar specific heat capacity (C) is the heat capacity per mole:

$$ C = \frac{1}{\mu} \frac{\Delta Q}{\Delta T} $$

Its SI unit is J mol^-1 K^-1.

For solids at room temperature, the law of equipartition of energy predicts a molar specific heat capacity of 3R, which generally agrees with experimental values (except for carbon). This is derived from the average energy of atoms in a solid being 3k_BT, leading to a total energy of 3N k_BT = 3RT for a mole.

The specific heat capacity of water varies slightly with temperature. The historical definition of calorie was based on the heat required to raise the temperature of 1 g of water by 1°C, but a more precise definition uses the range 14.5°C to 15.5°C. In SI units, the specific heat capacity of water is 4186 J kg^-1 K^-1.

For gases, two types of specific heat capacities are defined:

• Specific heat capacity at constant volume (C_v)
• Specific heat capacity at constant pressure (C_p)

For an ideal gas, these are related by Mayer's relation:

$$ C_p - C_v = R $$

This relation is derived using the First Law of Thermodynamics and the ideal gas law (PV = μRT).

The relationship between specific heat capacities and temperature for an ideal gas can be summarized as:

• At constant volume: $\Delta Q = \mu C_v \Delta T$, leading to $C_v = \frac{1}{\mu} \left(\frac{\partial U}{\partial T}\right)_V$. For an ideal gas, U depends only on T, so $C_v = \frac{1}{\mu} \frac{dU}{dT}$.
• At constant pressure: $\Delta Q = \mu C_p \Delta T$. Using $\Delta Q = \Delta U + P \Delta V$ and the ideal gas law, we get $C_p = \left(\frac{\partial U}{\partial T}\right)_P + R$.

Thermodynamic State Variables And Equation Of State

An equilibrium state of a thermodynamic system is completely defined by a set of macroscopic variables called state variables. For a gas, these typically include pressure (P), volume (V), and temperature (T). These variables are not necessarily independent; their relationship is described by an equation of state. For an ideal gas, this is the ideal gas law: PV = μRT.

State variables can be classified as:

• Extensive variables: Indicate the "size" of the system and change when the system is divided (e.g., internal energy U, volume V, total mass M).
• Intensive variables: Do not change when the system is divided (e.g., pressure P, temperature T, density ρ).

Thermodynamic equations should be consistent with this classification; for example, in $\Delta Q = \Delta U + P \Delta V$, both sides are extensive quantities.

Systems not in equilibrium (e.g., during rapid expansion or combustion) cannot be described by state variables because properties like pressure and temperature may not be uniform throughout.

Thermodynamic Processes

A thermodynamic process describes the change of a system from one state to another. Key types of processes include:

Quasi-Static Process

A quasi-static process is an idealized process that occurs so slowly that the system remains in thermodynamic equilibrium with its surroundings at every infinitesimal stage. This means the differences in pressure and temperature between the system and its surroundings are infinitesimally small. Real-world processes that are sufficiently slow approximate quasi-static processes.

Isothermal Process

An isothermal process occurs at a constant temperature (T = constant). For an ideal gas, this means PV = constant (Boyle's Law). During an isothermal expansion of an ideal gas, the heat absorbed ($\Delta Q$) is equal to the work done ($\Delta W$) by the gas. For an ideal gas, $\Delta U = 0$ in an isothermal process.

Work done in an isothermal expansion from V₁ to V₂ is:

$$ W = \mu RT \ln \left(\frac{V_2}{V_1}\right) $$

Adiabatic Process

An adiabatic process occurs with no heat exchange between the system and its surroundings ($\Delta Q = 0$). From the First Law ($\Delta Q = \Delta U + \Delta W$), any work done by the system ($\Delta W > 0$) leads to a decrease in its internal energy ($\Delta U < 0$), and consequently, a decrease in temperature (for an ideal gas). For an ideal gas, the relationship between pressure and volume in an adiabatic process is:

$$ PV^\gamma = \text{constant} $$

where $\gamma = C_p / C_v$ is the adiabatic index.

Work done in an adiabatic process from state (P₁, V₁, T₁) to (P₂, V₂, T₂) is:

$$ W = \frac{\mu R (T_1 - T_2)}{\gamma - 1} $$

If work is done by the gas (W > 0), then T₂ < T₁ (cooling). If work is done on the gas (W < 0), then T₂ > T₁ (heating).

Isochoric Process

An isochoric process occurs at constant volume (V = constant). In this case, no work is done by the gas ($\Delta W = 0$). Any heat absorbed ($\Delta Q$) directly increases the internal energy ($\Delta U$), and thus the temperature.

Isobaric Process

An isobaric process occurs at constant pressure (P = constant). The work done by the gas is $\Delta W = P \Delta V = \mu R \Delta T$. Heat absorbed is used for both increasing internal energy and doing work.

Cyclic Process

In a cyclic process, the system returns to its initial state. Since internal energy is a state variable, the net change in internal energy for a cyclic process is zero ($\Delta U = 0$). Consequently, the total heat absorbed by the system equals the total work done by the system ($\Delta Q = \Delta W$).

Second Law Of Thermodynamics

The Second Law of Thermodynamics places fundamental limitations on the conversion of heat into work and the direction of spontaneous processes. It states that certain processes, though consistent with the First Law (conservation of energy), are impossible in nature.

Two common statements of the Second Law are:

• Kelvin-Planck Statement: It is impossible to construct a device that operates in a cycle and produces no effect other than the absorption of heat from a single reservoir and the conversion of this heat into an equivalent amount of work. (This implies that no heat engine can have 100% efficiency.)
• Clausius Statement: It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a colder body to a hotter body. (This implies that refrigerators or heat pumps require work input.)

These statements are equivalent and highlight that natural processes tend towards increasing disorder (entropy) and that perfect efficiency in converting heat to work or transferring heat against a temperature gradient is unattainable.

Reversible And Irreversible Processes

A reversible process is an idealized process that can be reversed such that both the system and its surroundings are returned to their original states, with no net change elsewhere in the universe. Reversible processes are necessarily quasi-static and free from dissipative effects like friction or viscosity.

Most natural processes are irreversible. This irreversibility arises from:

• Processes that move the system through non-equilibrium states.
• Dissipative effects like friction, viscosity, and inelastic collisions.

For example, the free expansion of a gas, the combustion of fuel, and the diffusion of a gas are irreversible processes. Dissipative effects mean that some energy is always lost as heat, reducing the efficiency of processes.

The concept of reversibility is crucial because reversible processes represent the theoretical maximum efficiency for heat engines and the maximum coefficient of performance for refrigerators operating between two temperatures.

Carnot Engine

A Carnot engine is a theoretical, ideal reversible engine operating between two heat reservoirs at temperatures T₁ (hot) and T₂ (cold). It operates on a Carnot cycle, which consists of four reversible processes:

1. Isothermal Expansion: The working substance (e.g., an ideal gas) absorbs heat Q₁ from the hot reservoir at T₁ and expands, doing work W_1→2.
2. Adiabatic Expansion: The gas expands further without heat exchange ($\Delta Q = 0$), and its temperature drops from T₁ to T₂, doing work W_2→3.
3. Isothermal Compression: The working substance releases heat Q₂ to the cold reservoir at T₂ and is compressed isothermally, with work W_3→4 done on the gas.
4. Adiabatic Compression: The gas is compressed further without heat exchange ($\Delta Q = 0$), its temperature rises from T₂ back to T₁, with work W_4→1 done on the gas.

The efficiency ($\eta$) of a heat engine is defined as the ratio of the net work done (W) to the heat absorbed from the hot reservoir (Q₁):

$$ \eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1} $$

For a Carnot engine operating with an ideal gas, it can be shown that the ratio of heats is equal to the ratio of absolute temperatures:

$$ \frac{Q_2}{Q_1} = \frac{T_2}{T_1} $$

Therefore, the efficiency of a Carnot engine is:

$$ \eta_{\text{Carnot}} = 1 - \frac{T_2}{T_1} $$

This formula shows that the maximum possible efficiency of any heat engine operating between two temperatures T₁ and T₂ is the Carnot efficiency, and this efficiency is independent of the nature of the working substance. This relationship also forms the basis for defining a truly universal thermodynamic temperature scale.

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