Unit 4: First Law of Thermodynamics

Introduction to Thermodynamics

Thermodynamics is a branch of physics that deals with heat and its relation to other forms of energy and work. It describes how thermal energy is converted to and from other forms of energy and how it affects matter. At its core, thermodynamics is about energy transformations and the fundamental laws that govern these transformations. This chapter will introduce the foundational concepts, including different types of thermodynamic systems, the concept of work in the context of volume changes, and the crucial First Law of Thermodynamics, which is a statement of the conservation of energy.

1. Thermodynamic Systems

To study energy transformations, we must first define the boundaries of our study. A thermodynamic system is the specific part of the universe that is under consideration, while the surroundings encompass everything else outside this system that can interact with it. The boundary separating the system from its surroundings can be real or imaginary, rigid or flexible, and can permit or prevent the exchange of matter and energy.

Types of Thermodynamic Systems:

• Open System: An open system is characterized by its ability to exchange both matter and energy with its surroundings. A common example is a boiling pot of water without a lid, where steam (matter) escapes, and heat (energy) is exchanged with the environment. Living organisms are also excellent examples of open systems, constantly taking in nutrients and energy while releasing waste and heat.
• Closed System: A closed system allows for the exchange of energy (in the form of heat or work) with its surroundings, but it does not allow the exchange of matter. Imagine a sealed pressure cooker: heat can enter or leave, but no steam escapes. Another example is a gas enclosed in a cylinder with a movable piston; the gas can do work on the piston or heat can be added/removed, but the amount of gas remains constant.
• Isolated System: An isolated system is the most restrictive type, as it exchanges neither matter nor energy with its surroundings. Such a system is an idealization, as perfect isolation is difficult to achieve in practice. A well-insulated thermos flask containing hot coffee, for a short period, approximates an isolated system, minimizing both heat loss and matter exchange. The entire universe is often considered the ultimate isolated system.

2. Work Done During Volume Change

In thermodynamics, work is a form of energy transfer that is not due to a temperature difference. When a gas expands or contracts, it performs work or has work performed on it. This work is directly related to the change in the system's volume against an external pressure.

Mathematical Representation:

The infinitesimal amount of work dW done by a system during an infinitesimal change in volume dV against an external pressure P is given by:

For a finite change in volume from V1 to V2, the total work W done is the integral of P dV:

Where:

• W is the work done by the system.
• P is the external pressure.
• dV is the infinitesimal change in volume.

On a Pressure-Volume (P-V) diagram, the work done during a process is represented by the area under the curve of the process path. This highlights that work is a path-dependent function, meaning its value depends on the specific path taken between the initial and final states, not just the states themselves.

Sign Convention for Work:

• Expansion: Work Done by System (Positive W): When a system expands (V2 > V1), it pushes against its surroundings, doing work. In physics, this work done by the system is typically considered positive. For example, a gas expanding and pushing a piston upwards performs positive work.
• Compression: Work Done on System (Negative W): When a system is compressed (V2 < V1), the surroundings do work on the system. This work done on the system is considered negative. For instance, if an external force pushes a piston down, compressing a gas, negative work is done by the system (or positive work is done on the system).

It's crucial to be consistent with the sign convention. In this chapter, we will adhere to the convention where W is the work done by the system.

3. Heat, Work, Internal Energy and First Law

The First Law of Thermodynamics is a fundamental principle that expresses the conservation of energy. It relates the change in a system's internal energy to the heat added to the system and the work done by the system.

Heat (Q):

Heat is defined as the transfer of thermal energy between a system and its surroundings due to a temperature difference. Heat always flows spontaneously from a region of higher temperature to a region of lower temperature. It is a form of energy transfer, not a property possessed by a system.

• Positive Q: Heat added to the system.
• Negative Q: Heat removed from the system.

Internal Energy (U):

The internal energy of a thermodynamic system refers to the total energy contained within the system due to the random motion and interactions of its constituent molecules. For an ideal gas, internal energy primarily consists of the total kinetic energy of its molecules (translational, rotational, vibrational). For real substances, it also includes potential energy associated with intermolecular forces. Internal energy is a state function, meaning its value depends only on the current state of the system (e.g., temperature, pressure, volume), not on how that state was reached.

ΔU = Ufinal - Uinitial

Where:

• ΔU is the change in internal energy.
• Ufinal is the final internal energy.
• Uinitial is the initial internal energy.

The First Law of Thermodynamics:

The First Law of Thermodynamics states that the change in the internal energy of a closed thermodynamic system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings.

ΔU = Q - W

Where:

• ΔU is the change in the internal energy of the system.
• Q is the heat added to the system.
• W is the work done by the system.

This law is essentially a restatement of the principle of conservation of energy, adapted for thermodynamic systems. It implies that energy cannot be created or destroyed, only transformed from one form to another or transferred between a system and its surroundings.

Consider the implications:

• If Q is positive and W is zero (e.g., heating a gas at constant volume), then ΔU = Q, meaning all added heat increases internal energy.
• If W is positive and Q is zero (e.g., adiabatic expansion), then ΔU = -W, meaning the work done by the system comes at the expense of its internal energy (it cools down).
• If ΔU is zero (e.g., isothermal process), then Q = W, meaning all heat added is converted into work done by the system.

4. Thermodynamic Processes

A thermodynamic process is a change in the state of a thermodynamic system from an initial state to a final state. These processes are often characterized by holding one or more thermodynamic variables (like pressure, volume, or temperature) constant.

Types of Thermodynamic Processes:

• Adiabatic Process:
  • Definition: A process in which no heat is exchanged between the system and its surroundings (Q = 0). This occurs either when the system is perfectly insulated or when the process happens very rapidly, preventing significant heat transfer.
  • First Law: ΔU = -W. The change in internal energy is solely due to the work done. If the system does work, its internal energy decreases (and vice versa).
  • Equation: For an ideal gas, PVγ = constant, where γ is the adiabatic index.
• Isochoric Process (Constant Volume):
  • Definition: A process in which the volume of the system remains constant (ΔV = 0).
  • Work Done: Since ΔV = 0, the work done W = ∫ P dV = 0. No work is done by or on the system due to volume change.
  • First Law: ΔU = Q. All heat added to or removed from the system directly changes its internal energy.
  • Example: Heating a gas in a rigid, sealed container.
• Isothermal Process (Constant Temperature):
  • Definition: A process in which the temperature of the system remains constant (ΔT = 0). This typically requires slow processes and good thermal contact with a heat reservoir.
  • Internal Energy: For an ideal gas, internal energy depends only on temperature. Thus, for an isothermal process, ΔU = 0.
  • First Law: Q = W. Any heat added to the system is entirely converted into work done by the system, and vice versa.
  • Equation: For an ideal gas, PV = constant (Boyle's Law).
• Isobaric Process (Constant Pressure):
  • Definition: A process in which the pressure of the system remains constant (ΔP = 0).
  • Work Done: Since pressure is constant, the integral for work simplifies to W = P ΔV.
  • First Law: ΔU = Q - P ΔV.
  • Example: A gas expanding in a cylinder with a freely moving piston under atmospheric pressure.

5. Heat Capacities

Heat capacity is a measure of the amount of heat energy required to raise the temperature of a substance by a certain amount. For gases, heat capacity can be defined under different conditions, most notably at constant volume and constant pressure, which leads to different values.

Heat Capacity at Constant Volume (Cv):

Cv is the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius (or Kelvin) while keeping its volume constant. Since volume is constant, no work is done (W = 0). From the First Law (ΔU = Q - W), it follows that ΔU = Q. Therefore, at constant volume, all the heat supplied goes into increasing the internal energy of the gas.

Qv = n Cv ΔT

Where:

• Qv is heat exchanged at constant volume.
• n is the number of moles.
• Cv is the molar heat capacity at constant volume.
• ΔT is the change in temperature.

Heat Capacity at Constant Pressure (Cp):

Cp is the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius (or Kelvin) while keeping its pressure constant. In this case, as the gas expands (if heated), it does work on its surroundings (W = P ΔV). Therefore, the heat supplied not only increases the internal energy but also provides the energy for the work done against the constant external pressure.

Qp = n Cp ΔT

Where:

• Qp is heat exchanged at constant pressure.
• n is the number of moles.
• Cp is the molar heat capacity at constant pressure.
• ΔT is the change in temperature.

It is always observed that Cp > Cv for gases because at constant pressure, some of the added heat energy is used to perform work of expansion, in addition to increasing the internal energy. At constant volume, all added heat goes directly into increasing internal energy.

Mayer's Relation:

For an ideal gas, there's a direct relationship between Cp and Cv, known as Mayer's relation:

Cp - Cv = R

Where R is the universal gas constant (approximately 8.314 J/(mol·K)).

This relation can be derived from the First Law. For an isobaric process, Qp = ΔU + W. We know Qp = n Cp ΔT and W = P ΔV. Also, for an ideal gas, ΔU = n Cv ΔT. So, substituting these into the First Law:

n Cp ΔT = n Cv ΔT + P ΔV

From the ideal gas law, PV = nRT. For a constant pressure process, P ΔV = nR ΔT. Substituting this:

n Cp ΔT = n Cv ΔT + nR ΔT

Dividing by n ΔT (assuming n ≠ 0 and ΔT ≠ 0):

Cp = Cv + R

Or, Cp - Cv = R.

Adiabatic Index (γ):

The ratio of the heat capacities at constant pressure and constant volume is called the adiabatic index or heat capacity ratio:

γ = Cp / Cv

The value of γ depends on the molecular structure of the gas (e.g., monatomic, diatomic, polyatomic) and is crucial in describing adiabatic processes.

6. Isothermal and Adiabatic Processes

These two processes are fundamental in thermodynamics and represent idealizations of how gases behave under specific conditions. They are often compared due to their distinct characteristics, especially concerning temperature change and heat exchange.

Isothermal Process:

An isothermal process occurs at a constant temperature (T = constant). For an ideal gas, this means that its internal energy ΔU = 0. According to the First Law (ΔU = Q - W), if ΔU = 0, then Q = W. This implies that any heat absorbed by the system is entirely converted into work done by the system, and vice versa.

• Equation of State: For an ideal gas, Boyle's Law applies: PV = constant.
• Work Done: Consider a gas undergoing a reversible isothermal expansion from volume V1 to V2.

  W = ∫ P dV

  From PV = nRT, we have P = nRT/V. Since T is constant:

  W = ∫V1V2 (nRT/V) dV

  W = nRT ∫V1V2 (1/V) dV

  W = nRT [ln V]V1V2

  W = nRT ln(V2/V1)

  Since PV = constant, P1V1 = P2V2, which means V2/V1 = P1/P2. So, work can also be expressed as:

  W = nRT ln(P1/P2)

Adiabatic Process:

An adiabatic process is one where no heat is exchanged between the system and its surroundings (Q = 0). This means the system is either perfectly insulated or the process occurs too quickly for significant heat transfer to take place. From the First Law, ΔU = -W. If the system expands (does positive work), its internal energy decreases, leading to a drop in temperature. If the system is compressed (negative work by system, positive work on system), its internal energy increases, leading to a rise in temperature.

• Equation of State: For an ideal gas, the adiabatic relation is:

  PVγ = constant

  Where γ = Cp/Cv is the adiabatic index.

  Other forms of the adiabatic relation, derived using PV = nRT, are:

  • TVγ-1 = constant
  • P1-γTγ = constant
• Work Done: For a reversible adiabatic process, the work done is:

  W = ∫ P dV

  Since PVγ = K (constant), P = K/Vγ.

  W = ∫V1V2 (K/Vγ) dV = K ∫V1V2 V-γ dV

  W = K [V1-γ / (1-γ)]V1V2

  W = (K / (1-γ)) [V21-γ - V11-γ]

  Since K = P1V1γ = P2V2γ, we can substitute K:

  W = (1 / (1-γ)) [P2V2γV21-γ - P1V1γV11-γ]

  W = (P2V2 - P1V1) / (1-γ)
  or W = (P1V1 - P2V2) / (γ - 1)

  Using PV = nRT, this can also be written as:

  W = nR(T1 - T2) / (γ - 1)

Comparison on P-V Diagram:

When plotted on a P-V diagram, both isothermal and adiabatic curves are hyperbolas, but their slopes differ significantly:

• The slope of an isothermal curve (PV = constant) is dP/dV = -P/V.
• The slope of an adiabatic curve (PVγ = constant) is dP/dV = -γ(P/V).

Since γ > 1 (for all real gases), the absolute value of the slope of an adiabatic curve is always greater than that of an isothermal curve at any given point. This means that an adiabatic curve is steeper than an isothermal curve on a P-V diagram. This steeper slope for adiabatic expansion means that for the same volume change, the pressure drops more significantly (and temperature drops) compared to an isothermal expansion where temperature is kept constant.