Heat and Thermodynamics: First and Second Laws

Heat and 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. This field is crucial for understanding everything from how car engines work to the processes occurring within stars.

4. First Law of Thermodynamics

The First Law of Thermodynamics is a statement of the conservation of energy, specifically adapted for thermodynamic systems. It states that energy cannot be created or destroyed, only transferred or changed from one form to another. In thermodynamic terms, it relates the change in internal energy of a system to the heat added to the system and the work done by the system.

4.1 Thermodynamic Systems

A thermodynamic system is a precisely defined portion of the universe that we choose to study. The surroundings are everything outside the system. The boundary separates the system from its surroundings. Systems can be classified into three types:

• Open System: Exchanges both energy (heat and work) and matter with its surroundings. Example: A boiling pot of water without a lid.
• Closed System: Exchanges energy but not matter with its surroundings. Example: A sealed can of soda.
• Isolated System: Exchanges neither energy nor matter with its surroundings. Example: An ideal thermos flask.

4.2 Work Done During Volume Change

In thermodynamics, work is often associated with the expansion or compression of a gas. When a gas expands, it does work on its surroundings; when it is compressed, work is done on it by the surroundings.

For a quasi-static (very slow) process, the work (W) done by a gas expanding or compressing is given by:

W = ∫ P dV

where P is the pressure and dV is the infinitesimal change in volume. If the volume increases, dV is positive, and work done by the gas is positive. If the volume decreases, dV is negative, and work done by the gas is negative (meaning work is done on the gas).

4.3 Heat and Work; Internal Energy and First Law of Thermodynamics

Internal Energy (U): This is the total energy contained within a thermodynamic system, including the kinetic energy of its molecules (translational, rotational, vibrational) and the potential energy associated with intermolecular forces. For an ideal gas, internal energy is solely dependent on temperature.

Heat (Q): Energy transferred between a system and its surroundings due to a temperature difference.

Work (W): Energy transferred between a system and its surroundings by means other than temperature difference (e.g., mechanical work).

The First Law of Thermodynamics is expressed as:

ΔU = Q - W

where ΔU is the change in internal energy of the system, Q is the heat added to the system, and W is the work done by the system on its surroundings. An alternative convention, ΔU = Q + W, is also common, where W represents the work done on the system.

4.4 Thermodynamic Processes: Adiabatic, Isochoric, Isothermal and Isobaric

Thermodynamic processes describe how a system changes from one state to another. These processes are often characterized by holding one thermodynamic variable constant:

• Adiabatic Process: No heat exchange with the surroundings (Q=0). Occurs very rapidly or in a well-insulated system. The First Law becomes ΔU = -W. Example: Rapid expansion of gas in a burst tire.
• Isochoric Process (or Isovolumetric): Volume remains constant (ΔV=0). Since W = ∫ P dV, no work is done (W=0). The First Law becomes ΔU = Q. Example: Heating a gas in a rigid container.
• Isothermal Process: Temperature remains constant (ΔT=0). For an ideal gas, since U depends only on T, ΔU=0. The First Law becomes Q = W. Example: A gas expanding very slowly in contact with a large heat reservoir.
• Isobaric Process: Pressure remains constant (ΔP=0). Work done is simply W = PΔV. The First Law becomes ΔU = Q - PΔV. Example: Heating water in an open pot (at atmospheric pressure).

4.5 Heat Capacities of an Ideal Gas at Constant Pressure and Volume and Relation Between Them

Heat Capacity at Constant Volume (C_V): The amount of heat required to raise the temperature of a substance by one degree Celsius (or Kelvin) while keeping its volume constant. For an ideal gas, dU = C_VdT.

Heat Capacity at Constant Pressure (C_P): The amount of heat required to raise the temperature of a substance by one degree Celsius (or Kelvin) while keeping its pressure constant. In this case, some energy is also used to do work against the constant pressure as the gas expands.

For an ideal gas, the molar heat capacities are related by Mayer's Relation:

C_P - C_V = R

where R is the ideal gas constant (approximately 8.314 J/(mol·K)). This relation highlights that C_P is always greater than C_V because at constant pressure, some of the added heat goes into doing work of expansion, in addition to increasing internal energy.

4.6 Isothermal and Adiabatic Processes for an Ideal Gas

Isothermal Process: For an ideal gas, PV = nRT (constant T, n, R means PV = constant). Thus, P ∝ 1/V. The P-V diagram for an isothermal process is a hyperbola.

P₁V₁ = P₂V₂

Work done during an isothermal process for n moles of ideal gas:

W = nRT ln(V_f/V_i)

Adiabatic Process: For an ideal gas, the relationship between pressure and volume is given by:

PV^γ = constant

where γ (gamma) is the adiabatic index or heat capacity ratio, γ = C_P/C_V. For monatomic ideal gases, γ ≈ 1.67; for diatomic ideal gases, γ ≈ 1.4. Adiabatic curves are steeper than isothermal curves on a P-V diagram.

Also, T V^γ-1 = constant and T^γ P^1-γ = constant

Work done during an adiabatic process for n moles of ideal gas:

W = (P_iV_i - P_fV_f) / (γ - 1) = nR(T_i - T_f) / (γ - 1)

5. Second Law of Thermodynamics

While the First Law tells us that energy is conserved, it doesn't tell us about the direction of energy flow or why certain processes occur spontaneously while others do not. This is where the Second Law of Thermodynamics comes into play, introducing the concept of entropy and limiting the efficiency of energy conversion.

5.1 Thermodynamic Systems and Direction of Thermodynamic Processes

The Second Law explains why heat flows spontaneously from hot to cold, but never the other way around spontaneously. It indicates that natural processes are irreversible and tend towards a state of greater disorder.

For example, a hot cup of coffee will cool down to room temperature, but a cold cup of coffee will not spontaneously heat up from room temperature without external intervention.

5.2 Second Law of Thermodynamics

The Second Law of Thermodynamics can be stated in several equivalent ways:

• Clausius Statement: It is impossible to construct a device which operates in a cycle and produces no effect other than the transfer of heat from a colder body to a hotter body. (Implies heat doesn't spontaneously flow from cold to hot).
• Kelvin-Planck Statement: It is impossible to construct a device which operates in a cycle and produces no effect other than the extraction of heat from a single thermal reservoir and the performance of an equivalent amount of work. (Implies no heat engine can be 100% efficient).

5.3 Heat Engines

A heat engine is a device that converts thermal energy into mechanical work. It operates in a cycle, taking heat from a high-temperature reservoir (Q_H), converting some of it into useful work (W), and expelling the remaining heat to a low-temperature reservoir (Q_C).

The efficiency (η) of a heat engine is defined as the ratio of the work done to the heat absorbed from the hot reservoir:

η = W / Q_H = (Q_H - Q_C) / Q_H = 1 - (Q_C / Q_H)

According to the Kelvin-Planck statement, η can never be 1 (100%).

5.4 Internal Combustion Engines: Otto cycle, Diesel cycle; Carnot cycle

Carnot Cycle: This is an idealized, reversible thermodynamic cycle that represents the most efficient possible heat engine operating between two given temperatures. It consists of two isothermal and two adiabatic processes. The efficiency of a Carnot engine is:

η_Carnot = 1 - (T_C / T_H)

where T_C and T_H are the absolute temperatures of the cold and hot reservoirs, respectively. No real engine can achieve Carnot efficiency.

Otto Cycle: An idealized thermodynamic cycle that describes the functioning of a typical spark-ignition internal combustion engine (like in gasoline cars). It consists of two adiabatic and two isochoric processes.

Diesel Cycle: An idealized thermodynamic cycle that describes the functioning of a compression-ignition internal combustion engine (diesel engine). It consists of two adiabatic, one isochoric, and one isobaric process.

5.5 Refrigerator

A refrigerator (or heat pump) is essentially a heat engine operating in reverse. It uses external work (W) to transfer heat from a cold reservoir (Q_C) to a hot reservoir (Q_H), thereby cooling the cold reservoir. It is impossible for a refrigerator to operate without work input.

The performance of a refrigerator is measured by its Coefficient of Performance (COP):

COP_refrigerator = Q_C / W = Q_C / (Q_H - Q_C)

For a heat pump, the goal is to heat the hot reservoir, so:

COP_{heat pump} = Q_H / W = Q_H / (Q_H - Q_C)

5.6 Entropy and Disorder (Introduction Only)

Entropy (S): A measure of the disorder or randomness of a system. The Second Law of Thermodynamics, in terms of entropy, states that the total entropy of an isolated system can only increase over time, or remain constant in ideal reversible processes; it can never decrease.

ΔS ≥ 0 (for an isolated system)

For a reversible process, the change in entropy is defined as:

dS = δQ_rev / T

The universe as a whole is considered an isolated system, and its entropy is continuously increasing, leading to the idea of the "heat death" of the universe, where all energy is uniformly distributed, and no further work can be done.

Understanding these fundamental laws of thermodynamics is essential for various scientific and engineering applications, from designing more efficient power plants to comprehending biological processes.