3.4 Applications of the First Law

The First Law of Thermodynamics is all about energy conservation. It tells us that energy can't be created or destroyed, only transformed. This law helps us understand how energy changes in different processes, like heating, cooling, and doing work.

When we apply the First Law, we can solve real-world problems. We can calculate energy changes in engines, refrigerators, and chemical reactions. This helps us design more efficient machines and understand how energy flows in natural systems.

Energy Changes in Thermodynamic Processes

First Law of Thermodynamics

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• The first law of thermodynamics states that the change in internal energy of a system (ΔU) is equal to the heat added to the system (Q) minus the work done by the system (W): ΔU = Q - W
• It is a statement of the conservation of energy principle which states that energy cannot be created or destroyed, only converted from one form to another
• The change in internal energy (ΔU) of a system can be determined by measuring the heat exchange (Q) and work done (W) during a thermodynamic process

Components of the First Law

• Internal energy (U) is the sum of the kinetic and potential energies of the particles in a system
  • It is a state function, meaning its value depends only on the current state of the system and not on the path taken to reach that state
• Heat (Q) is the energy transferred between a system and its surroundings due to a temperature difference
  • It is positive when heat is added to the system and negative when heat is removed from the system
• Work (W) is the energy transferred between a system and its surroundings due to a force acting through a distance
  • It is positive when work is done by the system on its surroundings and negative when work is done on the system by its surroundings

Solving Thermodynamic Problems

Types of Thermodynamic Processes

• Isothermal processes occur at constant temperature, and the internal energy change (ΔU) is zero
  • The heat added to the system (Q) is equal to the work done by the system (W): Q = W
  • For an ideal gas undergoing an isothermal process, the work done can be calculated using the equation: W = nRTln(V2/V1), where n is the number of moles, R is the gas constant, T is the temperature, and V1 and V2 are the initial and final volumes, respectively
• Adiabatic processes occur without heat exchange between the system and its surroundings (Q = 0)
  • The work done by the system (W) is equal to the change in internal energy (ΔU): W = -ΔU
  • For an ideal gas undergoing an adiabatic process, the work done can be calculated using the equation: W = (nR/(γ-1))(T1-T2), where γ is the ratio of the heat capacities at constant pressure and volume (Cp/Cv)
• Cyclic processes involve a series of thermodynamic processes that return the system to its initial state
  • The net change in internal energy (ΔU) over a complete cycle is zero, and the net heat added (Q) is equal to the net work done (W): Q = W

Calculating Work and Heat

• The work done by a system during a thermodynamic process can be calculated by integrating the pressure-volume (PV) curve: W = ∫PdV
• The heat exchanged during a process can be determined using the first law of thermodynamics: Q = ΔU + W
• Examples of calculating work and heat:
  • Isothermal expansion of an ideal gas from 1 L to 2 L at 300 K: W = nRTln(V2/V1) = (1 mol)(8.314 J/mol·K)(300 K)ln(2/1) = 1,723 J
  • Adiabatic compression of an ideal gas from 2 L to 1 L with γ = 1.4: W = (nR/(γ-1))(T1-T2) = (1 mol)(8.314 J/mol·K)/(1.4-1))(300 K - 476 K) = -1,045 J

Efficiency of Heat Engines and Refrigerators

Heat Engines

• A heat engine is a device that converts heat into work by operating in a cyclic process between a high-temperature reservoir (TH) and a low-temperature reservoir (TL)
• The efficiency (e) of a heat engine is defined as the ratio of the work output (W) to the heat input from the high-temperature reservoir (QH): e = W/QH
• The maximum theoretical efficiency of a heat engine operating between two reservoirs is given by the Carnot efficiency: eC = 1 - (TL/TH), where TL and TH are the absolute temperatures of the low and high-temperature reservoirs, respectively
• Example: A heat engine operates between a high-temperature reservoir at 500 K and a low-temperature reservoir at 300 K. The maximum theoretical efficiency is: eC = 1 - (300 K/500 K) = 0.4 or 40%

Refrigerators and Heat Pumps

• A refrigerator is a device that transfers heat from a low-temperature reservoir (TL) to a high-temperature reservoir (TH) by consuming work (W)
  • The coefficient of performance (COP) of a refrigerator is defined as the ratio of the heat removed from the low-temperature reservoir (QL) to the work input (W): COPR = QL/W
• A heat pump is a device that transfers heat from a low-temperature reservoir (TL) to a high-temperature reservoir (TH) by consuming work (W), similar to a refrigerator but with the goal of heating the high-temperature reservoir
  • The coefficient of performance (COP) of a heat pump is defined as the ratio of the heat delivered to the high-temperature reservoir (QH) to the work input (W): COPHP = QH/W
• Example: A refrigerator removes 1,000 J of heat from a cold reservoir and consumes 250 J of work. The coefficient of performance is: COPR = QL/W = 1,000 J/250 J = 4

Entropy and Spontaneous Processes

Limitations of the First Law

• The first law of thermodynamics does not provide information about the direction of spontaneous processes or the equilibrium state of a system
• It does not account for the irreversibility of real processes, which always involve an increase in entropy and a decrease in the available energy for performing work

Entropy and the Second Law

• Entropy (S) is a measure of the disorder or randomness of a system
  • It is a state function that increases during spontaneous processes and remains constant at equilibrium
• The second law of thermodynamics states that the total entropy of an isolated system always increases during a spontaneous process and remains constant at equilibrium
• Spontaneous processes are those that occur naturally without external intervention, and they always proceed in a direction that increases the total entropy of the system and its surroundings
• The change in entropy (ΔS) of a system during a reversible process can be calculated using the equation: ΔS = Q/T, where Q is the heat exchanged and T is the absolute temperature

Exergy and Available Energy

• The concept of exergy, or available energy, combines the first and second laws of thermodynamics to describe the maximum useful work that can be obtained from a system as it reaches equilibrium with its surroundings
• Example: A system at 400 K exchanges 1,000 J of heat with its surroundings at 300 K. The change in entropy is: ΔS = Q/T = 1,000 J/400 K = 2.5 J/K, indicating an increase in entropy and a decrease in available energy for performing work