Basics and Laws of Thermodynamics

Introduction

Thermodynamics is a macroscopic science that studies the relationships between heat, work, temperature, and energy. It deals with large-scale properties of matter, such as pressure, volume, and temperature, without needing to know the details of the behaviour of individual atoms and molecules (unlike the kinetic theory of gases). It is based on a few fundamental laws that are derived from observations of macroscopic systems.

The central focus of thermodynamics is the transformation of energy, particularly the conversion of heat into work and vice versa. This is crucial for understanding and designing energy conversion devices like engines, refrigerators, and power plants.

Thermodynamics provides a framework to determine the possible outcomes of energy transformations and the efficiency with which these transformations can occur. It establishes limitations on how energy can be used and conserved.

Key concepts in thermodynamics include:

• System and Surroundings: A thermodynamic system is the specific part of the universe under consideration. The surroundings are everything else interacting with the system. Systems can be open (exchange energy and matter), closed (exchange energy but not matter), or isolated (exchange neither energy nor matter).
• State of a System: The condition of a system is described by its thermodynamic state variables.
• Thermodynamic Processes: Changes in the state of a system.

The laws of thermodynamics are universal principles that apply to all forms of energy and all macroscopic systems. They are among the most fundamental laws of physics.

Thermal Equilibrium

The concept of thermal equilibrium is central to defining temperature and is the basis for the Zeroth Law of Thermodynamics.

Definition of Thermal Equilibrium

Two systems are said to be in thermal equilibrium with each other if there is no net flow of heat between them when they are in thermal contact.

Thermal contact allows energy to be transferred between systems. When two systems at different temperatures are placed in thermal contact, heat will flow from the hotter system to the colder system. This heat flow will continue until both systems reach the same temperature. Once they reach the same temperature, they are in thermal equilibrium, and the net heat flow between them ceases.

The state of thermal equilibrium is reached when the macroscopic properties of the systems that are allowed to change (like temperature) no longer change with time.

The term "equilibrium" in thermodynamics is broader than just thermal equilibrium. A system is in thermodynamic equilibrium if it is in thermal, mechanical, and chemical equilibrium. Thermal equilibrium means uniform temperature throughout the system and no temperature gradients. Mechanical equilibrium means no unbalanced forces or pressure gradients. Chemical equilibrium means no net chemical reactions are occurring.

Zeroth Law Of Thermodynamics

The Zeroth Law of Thermodynamics provides the foundation for defining temperature and using thermometers. It establishes the transitivity of thermal equilibrium.

Statement of the Zeroth Law

The Zeroth Law states:

If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

Let A, B, and C be three thermodynamic systems. If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then the Zeroth Law states that systems A and B are also in thermal equilibrium with each other.

This law seems almost self-evident based on our experience with temperature, but it is not derivable from the other laws of thermodynamics and is logically necessary for defining temperature as a property that can be measured and compared.

Significance and Role in Defining Temperature

The Zeroth Law allows us to state that there is a property common to all systems that are in thermal equilibrium. This property is called temperature.

The law implies that if system A and system C have the same temperature (because they are in thermal equilibrium), and system B and system C have the same temperature, then system A and system B must have the same temperature (because they are in thermal equilibrium). Temperature is the property that determines whether two systems will be in thermal equilibrium if placed in thermal contact.

The Zeroth Law justifies the use of a thermometer. A thermometer acts as the "third system" (system C). If we place a thermometer in thermal contact with system A and it reaches thermal equilibrium (shown by a stable reading), the thermometer is at the same temperature as A. If we then place the same thermometer in thermal contact with system B and it shows the same stable reading, it means B is also at the same temperature as the thermometer, and therefore, by the Zeroth Law, A and B are at the same temperature.

In essence, the Zeroth Law confirms that temperature is a consistent property that can be measured and used to determine whether two systems are in thermal equilibrium.

Heat, Internal Energy And Work

These three concepts are fundamental to the First Law of Thermodynamics and describe how energy is transferred and stored within a thermodynamic system.

Internal Energy (U or E)

Internal energy ($U$ or $E$) is the total energy contained within a thermodynamic system. It is the sum of the kinetic energy of the random motion of its constituent particles (translational, rotational, vibrational) and the potential energy associated with the forces between these particles (intermolecular forces, chemical bonds, nuclear forces). Internal energy is a property of the system and depends on its state (e.g., pressure, volume, temperature). It is often treated as a function of temperature for ideal gases, but for real substances, it also depends on volume or pressure.

Internal energy is a state function or point function, meaning its value depends only on the current state of the system, not on how the system reached that state. The change in internal energy ($\Delta U$) between two states depends only on the initial and final states, not on the path taken during the process. $U_{final} - U_{initial} = \Delta U$.

The SI unit of internal energy is the Joule (J).

Heat (Q)

Heat ($Q$) is the transfer of thermal energy between a system and its surroundings due to a temperature difference. Heat is energy in transit. It is not stored within the system; once transferred, it contributes to the system's internal energy or is used to do work.

Heat is not a state function. The amount of heat transferred during a process depends on the specific path taken from the initial to the final state.

Conventionally, heat absorbed by the system is taken as positive ($+Q$), and heat released by the system is taken as negative ($-Q$).

The SI unit of heat is the Joule (J).

Work (W)

Work ($W$) in thermodynamics is the transfer of energy between a system and its surroundings that is not due to a temperature difference. Thermodynamic work is often associated with changes in volume or pressure, such as a gas expanding against a piston or a force moving a boundary.

Work is also not a state function. The amount of work done during a process depends on the specific path taken from the initial to the final state.

Conventionally, work done by the system on the surroundings is taken as positive ($+W$), and work done on the system by the surroundings is taken as negative ($-W$). This convention is commonly used in physics and thermodynamics, though chemistry texts often use the opposite convention for work done by the system.

For a gas expanding or compressing at pressure $P$ against a piston, the work done by the gas during an infinitesimal volume change $dV$ is $dW = P \, dV$. The total work done during a finite volume change from $V_1$ to $V_2$ is $W = \int_{V_1}^{V_2} P \, dV$. This integral depends on how pressure changes with volume during the process (the path).

The SI unit of work is the Joule (J).

Energy Transfer Mechanisms

Heat and work are the two primary ways energy is transferred into or out of a thermodynamic system. Heat is driven by temperature difference, while work is driven by other forces (like pressure, mechanical force, electrical force, etc.) that cause displacement or movement of the system boundaries or contents in a non-random manner.

Both heat and work are processes of energy transfer, leading to changes in the system's internal energy or its other properties.

First Law Of Thermodynamics ($ \Delta U = Q - W $)

The First Law of Thermodynamics is essentially a restatement of the principle of conservation of energy applied to thermodynamic systems. It relates the change in internal energy of a system to the heat added to it and the work done by it.

Statement of the First Law

The First Law states:

The change in the internal energy ($\Delta U$) of a closed thermodynamic system is equal to the heat energy added to the system ($Q$) minus the work done by the system on its surroundings ($W$).

Mathematically, this is expressed as:

$ \Delta U = Q - W $

where:

• $\Delta U = U_{final} - U_{initial}$ is the change in the internal energy of the system.
• $Q$ is the net heat added to the system during the process.
• $W$ is the net work done by the system on its surroundings during the process.

Using the convention where work done on the system is negative (e.g., compression), the formula can also be written as $\Delta U = Q + W_{on\,system}$, where $W_{on\,system} = -W$. Both formulations are equivalent as long as the sign conventions are consistent.

For an infinitesimal process, the First Law is written in differential form:

$ dU = dQ - dW $

Note the use of $d$ instead of $\Delta$ and the bar ($d$) over $Q$ and $W$ to indicate that $dQ$ and $dW$ are infinitesimal amounts of heat and work transfer during a small part of the process, and that heat and work are path-dependent (not exact differentials of state functions, unlike $dU$). $dU$ is the change in internal energy, which is an exact differential because U is a state function.

Implications of the First Law

• Conservation of Energy: The law implies that energy is conserved. Energy can be transferred as heat or work, and these transfers change the internal energy of the system. For an isolated system ($Q=0, W=0$), $\Delta U = 0$, so the internal energy of an isolated system is constant.
• Perpetual Motion Machines of the First Kind are Impossible: A perpetual motion machine of the first kind is a hypothetical device that would produce work indefinitely without a corresponding input of energy (heat or other forms). The First Law of Thermodynamics states that this is impossible, as energy must be conserved. To produce work, energy must be supplied to the system or extracted from its internal energy.

The First Law is a powerful tool for analysing energy transformations in various thermodynamic processes and cycles.

Applying the First Law to Different Processes

The First Law can be applied to different types of thermodynamic processes:

• Isobaric Process (Constant Pressure): Work done $W = \int_{V_1}^{V_2} P \, dV = P(V_2 - V_1) = P\Delta V$. First Law: $\Delta U = Q - P\Delta V$.
• Isochoric Process (Constant Volume): Volume is constant, so $dV = 0$. Work done $W = \int P \, dV = 0$. First Law: $\Delta U = Q$. For a constant volume process, the heat added to the system is equal to the increase in its internal energy.
• Isothermal Process (Constant Temperature): Temperature is constant, so for an ideal gas, $U$ depends only on $T$, meaning $\Delta U = 0$. First Law: $0 = Q - W$, so $Q = W$. For an isothermal process with an ideal gas, any heat added to the system is entirely converted into work done by the system, and vice-versa.
• Adiabatic Process (No Heat Exchange): No heat is transferred into or out of the system, so $Q = 0$. First Law: $\Delta U = -W$. Work done by the system comes at the expense of its internal energy (cooling), and work done on the system increases its internal energy (heating).
• Cyclic Process: A process that starts and ends in the same state. Since internal energy is a state function, the change in internal energy over a complete cycle is zero: $\Delta U_{cycle} = 0$. First Law for a cycle: $0 = Q_{net} - W_{net}$, so $Q_{net} = W_{net}$. The net heat added to the system over a cycle equals the net work done by the system over the cycle. This principle is fundamental to understanding heat engines and refrigerators.

Answer:

Specific Heat Capacity

Specific heat capacity, as discussed earlier, quantifies the amount of heat required to change the temperature of a unit mass (or mole) of a substance by one degree. In thermodynamics, specific heat capacity is directly related to the change in internal energy and the work done during a process where heat is exchanged and temperature changes.

From the First Law, $Q = \Delta U + W$. For a process involving a temperature change $\Delta T$, the average specific heat capacity $c$ is defined as $c = \frac{Q}{m\Delta T}$ (or molar specific heat $C = \frac{Q}{n\Delta T}$). So, $Q = mc\Delta T$.

$ mc\Delta T = \Delta U + W $

$ c = \frac{\Delta U + W}{m\Delta T} $

This shows that the specific heat capacity depends on how much of the added heat goes into increasing internal energy and how much is used to do work.

For solids and liquids, the volume change during heating is generally small, so the work done against external pressure is usually negligible ($W \approx 0$ in most cases, especially at constant volume). In such cases, $Q \approx \Delta U$. The specific heat capacity for solids and liquids is thus primarily related to the change in internal energy with temperature: $Q = \Delta U = mc\Delta T$, so $c \approx \frac{\Delta U}{m\Delta T}$.

For gases, however, the volume can change significantly, and the work done is often not negligible. This is why, for gases, specific heat capacity is defined for specific processes:

• Specific Heat at Constant Volume ($c_v$ or $C_V$ for molar): For a constant volume process ($W=0$), $Q_V = \Delta U$. So, $Q_V = mc_v\Delta T = \Delta U$, or $c_v = \frac{1}{m} \left(\frac{\Delta U}{\Delta T}\right)_V$. This is the specific heat associated purely with the change in internal energy with temperature at constant volume.
• Specific Heat at Constant Pressure ($c_p$ or $C_P$ for molar): For a constant pressure process, $Q_P = \Delta U + W = \Delta U + P\Delta V$. So, $Q_P = mc_p\Delta T = \Delta U + P\Delta V$, or $c_p = \frac{1}{m} \frac{\Delta U + P\Delta V}{\Delta T}$. The specific heat at constant pressure is higher than at constant volume because extra heat is needed to do the work of expansion.

The difference between $c_p$ and $c_v$ for a gas is related to the work done during constant pressure expansion. For an ideal gas, $PV = nRT$, $P\Delta V = nR\Delta T$. Also, $\Delta U = nC_V\Delta T$ for an ideal gas regardless of the process. So, $Q_P = nC_V\Delta T + nR\Delta T = n(C_V + R)\Delta T$. Since $Q_P = nC_P\Delta T$, we get $C_P = C_V + R$, or $C_P - C_V = R$. This relationship between molar specific heats is a direct consequence of the First Law of Thermodynamics and the ideal gas law.

The values of specific heat capacities are crucial in calorimetry calculations and in analysing energy transfers involving temperature changes.

Thermodynamic State Variables And Equation Of State

To describe the state of a thermodynamic system, we use a set of macroscopic properties called thermodynamic state variables. These variables describe the equilibrium condition of the system.

State Variables (or State Functions)

State variables are properties of a system whose value depends only on the current state of the system, not on the history or path taken to reach that state. When the state of a system changes, the change in a state variable depends only on the initial and final states.

Common state variables include:

• Pressure ($P$)
• Volume ($V$)
• Temperature ($T$)
• Internal Energy ($U$)
• Enthalpy ($H = U + PV$)
• Entropy ($S$)
• Gibbs Free Energy ($G = H - TS$)

Work ($W$) and Heat ($Q$) are not state variables. They are path-dependent processes that describe how energy is transferred between states. $\int dU$ is independent of path, but $\int dQ$ and $\int dW$ are path-dependent.

Extensive and Intensive Variables

State variables can be classified as extensive or intensive:

• Extensive variables: Properties that depend on the amount of matter in the system (e.g., mass, volume, internal energy, heat capacity, enthalpy, entropy). If you combine two identical systems, the extensive variables add up.
• Intensive variables: Properties that do not depend on the amount of matter in the system (e.g., pressure, temperature, density, specific heat capacity, molar heat capacity). Intensive variables can be used to describe a system without knowing its size.

An extensive variable can be converted into an intensive variable by dividing it by an extensive variable like mass or volume (e.g., mass/volume = density, volume/mass = specific volume, internal energy/mass = specific internal energy).

Thermodynamic Equilibrium State

A system is in a state of thermodynamic equilibrium when its macroscopic properties are constant throughout the system and do not change with time. These properties are described by the state variables. For a simple, single-phase, pure substance system, two independent intensive state variables are sufficient to determine the state (e.g., P and T, or P and V/m, or T and V/m).

Equation of State

An equation of state is a functional relationship between the state variables of a substance that describes the equilibrium state of the substance. It allows us to determine the value of one state variable if the values of other independent state variables are known.

The most famous and simplest example is the ideal-gas equation of state:

$ PV = nRT $

or $ PV = N k_B T $

This equation relates the pressure ($P$), volume ($V$), temperature ($T$), and amount of gas ($n$ or $N$) for an ideal gas. Given any three of these, the fourth can be determined. For a fixed amount of gas, $P, V, T$ are related, and knowing any two determines the third.

Real gases do not obey the ideal-gas law perfectly, especially at high pressures and low temperatures where intermolecular forces and molecular volume become significant. More complex equations of state have been developed for real gases, such as the van der Waals equation:

$ \left(P + \frac{an^2}{V^2}\right) (V - nb) = nRT $

where $a$ and $b$ are constants specific to the gas, accounting for intermolecular attractions and the finite volume of molecules, respectively. Other equations of state exist for various substances in different phases.

Equations of state are empirical (derived from experiments) or semi-empirical, though theoretical models like KTG can provide insights into their form.