The First Law of Thermodynamics is all about energy conservation. Internal energy, enthalpy, and specific heats are key players in this law, helping us understand how energy moves and changes in systems.

These concepts are crucial for figuring out energy changes in processes. They're the building blocks for understanding heat transfer, work done, and temperature changes in everything from engines to chemical reactions.

Internal energy, enthalpy, and specific heats

Defining and explaining key concepts

• Internal energy represents the total energy of a system, including kinetic and potential energies of the particles, as well as the chemical and nuclear energies
  • Kinetic energy is associated with the motion of particles (translational, rotational, and vibrational)
  • Potential energy is related to the position of particles and their interactions (intermolecular forces, chemical bonds, and nuclear forces)
• Enthalpy is a state function defined as the sum of the internal energy and the product of pressure and volume ($H = U + PV$)
  • Enthalpy represents the total heat content of a system
  • As a state function, enthalpy depends only on the initial and final states of the system, not on the path taken
• Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by one degree
  • Specific heat is a measure of a substance's ability to store thermal energy
  • Specific heat at constant volume ($C_v$) is the heat capacity when the volume is held constant
  • Specific heat at constant pressure ($C_p$) is the heat capacity when the pressure is held constant
  • The relationship between $C_p$ and $C_v$ is given by $C_p - C_v = R$, where $R$ is the universal gas constant

Relationship between specific heats and molecular structure

• Monatomic gases (helium, neon) have lower specific heats compared to diatomic (nitrogen, oxygen) and polyatomic gases (carbon dioxide, methane)
  • Monatomic gases only have translational kinetic energy, while diatomic and polyatomic gases also have rotational and vibrational energy modes
• Solids generally have lower specific heats than liquids due to the more restricted motion of particles in solids
• Substances with higher molecular weights tend to have lower specific heats per unit mass
  • This is because the heat is distributed among a larger number of particles, resulting in a smaller temperature change for a given amount of heat

Internal energy and enthalpy changes

First law of thermodynamics and heat-work interactions

• The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system ($ΔU = Q - W$)
  • Heat ($Q$) is the energy transferred due to a temperature difference between the system and its surroundings
  • Work ($W$) is the energy transferred when a force acts through a distance (examples: expansion work, electrical work, shaft work)
• For a constant pressure process, the change in enthalpy is equal to the heat added to the system ($ΔH = Q$)
  • In this case, the system does expansion work against the constant external pressure, and the heat added accounts for both the change in internal energy and the work done
• In an adiabatic process (no heat exchange, $Q = 0$), the change in internal energy is equal to the negative of the work done ($ΔU = -W$)
  • For an adiabatic expansion, the system does work, and its internal energy decreases
  • For an adiabatic compression, work is done on the system, and its internal energy increases
• For an isothermal process (constant temperature), the change in internal energy is zero ($ΔU = 0$), and the heat added is equal to the work done ($Q = W$)
  • In an isothermal expansion, the system does work, and heat is added to maintain constant temperature
  • In an isothermal compression, work is done on the system, and heat is removed to maintain constant temperature

Reversible and irreversible processes

• A reversible process is a quasi-static process that can be reversed without any net change in the system and its surroundings
  • Reversible processes are ideal and serve as a benchmark for real processes
  • Examples: slow, frictionless expansion or compression, heat transfer between two reservoirs with an infinitesimal temperature difference
• An irreversible process is a process that cannot be reversed without a net change in the system and its surroundings
  • Most real processes are irreversible due to factors such as friction, turbulence, and finite temperature gradients
  • Examples: rapid expansion or compression, heat transfer between reservoirs with a large temperature difference, mixing of gases

Calculating internal energy and enthalpy changes

Using specific heat values

• The change in internal energy can be calculated using the formula $ΔU = m × C_v × ΔT$, where $m$ is the mass, $C_v$ is the specific heat at constant volume, and $ΔT$ is the change in temperature
  • This formula assumes that the specific heat is constant over the temperature range considered
  • For an ideal gas, $ΔU$ depends only on the change in temperature and is independent of pressure
• The change in enthalpy can be calculated using the formula $ΔH = m × C_p × ΔT$, where $C_p$ is the specific heat at constant pressure
  • This formula is applicable to both solids and liquids, as well as gases at constant pressure
  • For an ideal gas, the change in enthalpy is independent of pressure and only depends on temperature change: $ΔH = n × C_p × ΔT$, where $n$ is the number of moles

Applying Hess's law

• Hess's law states that the total enthalpy change for a reaction is independent of the route taken from reactants to products
  • This law is based on the conservation of energy and the state function nature of enthalpy
  • Hess's law allows the calculation of enthalpy changes for reactions that are difficult to measure directly by combining known enthalpy changes of other reactions
• To apply Hess's law, break down the desired reaction into a series of steps with known enthalpy changes
  • If a reaction is reversed, the sign of its enthalpy change is reversed
  • If a reaction is multiplied by a factor, its enthalpy change is multiplied by the same factor
  • The total enthalpy change is the sum of the enthalpy changes of the individual steps

Temperature and pressure effects on internal energy and enthalpy

Ideal gas behavior

• For an ideal gas, internal energy is a function of temperature only
  • As temperature increases, the kinetic energy of the particles increases, leading to an increase in internal energy
  • The relationship between internal energy and temperature is given by $U = \frac{3}{2}nRT$ for a monatomic ideal gas, where $n$ is the number of moles and $R$ is the universal gas constant
• Enthalpy is a function of both temperature and pressure for an ideal gas
  • Increasing temperature at constant pressure leads to an increase in enthalpy, as both internal energy and the $PV$ term increase
  • Increasing pressure at constant temperature also increases enthalpy due to the $PV$ term
  • The change in enthalpy with respect to pressure at constant temperature is given by $(\frac{∂H}{∂P})_T = V$

Real gas behavior and the Joule-Thomson effect

• In real gases, intermolecular forces and molecular size affect the relationship between temperature, pressure, and enthalpy
  • Attractive intermolecular forces (van der Waals forces) cause real gases to have lower internal energy and enthalpy than ideal gases at the same temperature and pressure
  • Molecular size leads to a reduction in the available volume and an increase in pressure, which increases enthalpy
• The Joule-Thomson effect describes the change in temperature of a real gas when it expands adiabatically from high to low pressure
  • The Joule-Thomson coefficient $μ_\text{JT} = (\frac{∂T}{∂P})_H$ determines whether the gas cools or warms upon expansion
  • For most gases at room temperature, $μ_\text{JT}$ is positive, meaning the gas cools upon expansion (examples: nitrogen, oxygen, carbon dioxide)
  • For some gases (hydrogen, helium) at room temperature, $μ_\text{JT}$ is negative, meaning the gas warms upon expansion
  • The Joule-Thomson effect is the basis for many industrial cooling processes, such as liquefaction of gases and refrigeration cycles