Thermodynamics: First Law, Work, and Heat Capacity

Classical Thermodynamics

Introduction and Foundational Concepts

• Early thermodynamic theories were formulated based on macroscopic features such as temperature, volume, and heat, as the underlying atomic and electron structures were not yet understood.

• Initially, heat was conceptualized as a fluid called caloric, which was believed to flow between substances.

• This caloric theory was later superseded by the kinetic theory of gases, which provided a more accurate atomic-level understanding of heat.

• Despite the inaccuracy of the caloric fluid picture, many of the ideas and equations developed during that time proved to be correct within the classical physics range.

• This series of lectures will adopt a classical physics approach to thermodynamics, focusing on macroscopic variables and processes, even though these concepts can be derived from statistical mechanics, which is more complex.

• The primary topics will include heat, work, the First Law of Thermodynamics, and later, entropy.

• Occasional references to more fundamental concepts like the equipartition theorem from kinetic theory will be made.

• The overarching principle underlying these ideas is the conservation of energy.

The First Law of Thermodynamics

Internal Energy (E)

• Consider a system, such as a gas contained within a cylinder with a movable piston.

• The gas at a given temperature T possesses kinetic energy due to the motion of its particles.

• This kinetic energy, along with potential energy, is collectively termed internal energy (E) (sometimes also called thermal energy or internal thermal energy).

• For a monatomic gas, internal energy is primarily kinetic, assuming no interactions between gas particles or other chemical reactions.

• For diatomic or polyatomic gases, internal energy can also include potential energy associated with the bonds between atoms within the molecules (e.g., vibrational potentials).

• In most standard calculations, internal energy can often be considered as primarily kinetic energy, as temperature is the principal driver of a gas's energy content.

• Gravitational potential energy of the entire container is generally neglected unless the vertical displacement is substantial (e.g., taking the container up a mountain or into space), as it doesn't typically affect the internal state of the gas itself.

• Assumptions for the system: The number of particles (n or N) is fixed, meaning no gas enters or leaves the sealed container.

Changing Internal Energy

• There are two fundamental ways to change the total internal energy ($E$) of the system:

  1. Adding or Subtracting Heat (Q):

     • If heat is added to the system (e.g., placing it on a stove), it raises the gas's temperature, causing particles to move faster and increasing their kinetic/internal energy.

     • Convention: Q > 0 dicates heat added to the system, Q < 0 indicates heat subtracted (cooling).

     • In a sealed container with no work done, all added heat transforms into internal energy ($Q = ext{change in } E$), exemplifying energy conservation.

  2. Doing Work (W) on the system:

     • Compressing the gas (e.g., pushing down the piston) increases the density of particles and their collision frequency with walls and the piston.

     • This causes the atoms to move faster, increasing their kinetic energy and thus the gas's internal energy, leading to a temperature rise.

     • The work done goes directly into the internal energy ($W = ext{change in } E$) if no heat exchange occurs.

• Combinations: Both heat transfer and work can occur simultaneously, resulting in a combined change in internal energy.

Formulation of the First Law

• The change in internal energy ($ext{d}E$) is the sum of the infinitesimal heat added ($ext{d}Q$) and the infinitesimal work done ($ext{d}W$) on the system.

• First Law of Thermodynamics (Differential Form): $dE = dQ + dW$.

• For a finite process: $ext{Delta}E = Q + W$.

• This law is a statement of energy conservation and applies universally to any system, not just ideal gases.

• It implies that internal energy cannot be changed without either adding/subtracting heat or doing/extracting work.

Work in Thermodynamics

• Work (W) and Heat (Q) are not state variables (also known as path functions or process-dependent variables).

  • Their values depend on the specific path or process taken between two states.

  • For example, the amount of heat needed to raise temperature by $10^ ext{o}C$ depends on whether volume or pressure is kept constant.

  • Similarly, the work done depends on how quickly or slowly a piston is moved, or what other conditions (like constant pressure or volume) are maintained.

• Internal Energy (E), however, IS a state variable (or point function).

  • The total amount of internal energy depends only on the current state of the system (e.g., temperature, pressure, volume), not on the history or path taken to reach that state.

  • The change in internal energy ($ext{Delta}E$) between two states is independent of the process.

Deriving the Work Equation

• Consider a gas in a cylinder with a piston, where a force $F$ is applied over a small distance $dx$.

• The infinitesimal work $dW$ is defined as $dW = F dx$ (assuming force and displacement are in the same direction).

• Given that pressure (P) is defined as $P = F/A$ (where $A$ is the area of the piston), the force can be written as $F = PA$.

• Substituting $F$: $dW = PA dx$.

• The product of the area $A$ and the infinitesimal displacement $dx$ corresponds to the infinitesimal change in volume $dV$ ($A dx = dV$).

• The sign convention: When work is done on the gas (compressing it), the volume decreases (dV < 0), but the work done on the gas should be positive.

• Therefore, the work done on the gas is defined as: $dW = -P dV$.

• The total work done for a finite process is found by integrating this expression:
  $W = - ext{integral}_{V_{initial}}^{V_{final}} P dV$.

• If the volume is constant ($dV=0$), then no work is done ($W=0$).

• For calculations, pressure $P$ often needs to be expressed as a function of volume $V$ (e.g., using the ideal gas law) if it's not constant.

• Sign Convention Caution: Some textbooks define work as work done by the gas instead of on the gas. In such cases, the sign of $W$ in the first law is opposite ($dE = dQ - dW_{by}$), and the internal definition of $dW_{by}$ would be $PdV$ (without the negative sign).

  • It is crucial to be careful about the specific definition and sign convention used in any given problem or textbook.

First Law with Work Definition

• Substituting $dW = -PdV$ into the First Law yields:
  $dE = dQ - PdV$.

• Practical application: Car engines utilize the mechanical work done by expanding gases to move pistons, which then rotate wheels, demonstrating a crucial real-world application of work being done by/on a gas.

Heat Capacity at Constant Volume ($C_v$)

• We will now focus on the heat (Q) variable and how to determine the amount of heat required to change a substance's temperature, leading to the concept of heat capacity.

• This concept is known as calorimetry in chemistry.

• Consider a simple situation: An ideal gas (or a solid/liquid, provided volume changes are negligible) inside a fixed, insulated container at constant volume.

Derivation for Constant Volume

• If the volume (V) remains constant, then the change in volume $dV = 0$.

• Consequently, the work done $dW = -PdV = 0$.

• Under these conditions, the First Law of Thermodynamics simplifies to: $dE = dQ$.

• This means that all the heat added to the system directly contributes to changing its internal energy.

• Internal energy can be changed by:

  1. Changing the temperature of the substance (our current focus).

  2. Changing the phase of the substance (e.g., melting ice into water), which will be discussed later.

• Example: Heating liquid water from $20^ ext{o}C$ to $70^ ext{o}C$ without boiling involves only a temperature change.

Definition of Heat Capacity

• Heat capacity is defined as the amount of heat required to change the temperature of a substance by one degree Celsius or Kelvin.

• Mathematically, the heat capacity at constant volume ($C_v$) is defined as:
  $C_v = rac{dQ}{dT}_V$

• The subscript $V$ explicitly denotes that this definition applies for processes occurring at constant volume.

• It is crucial to specify