Table of Contents
Internal Energy, Entropy and Enthalpy
1. Introduction
Internal energy, entropy and enthalpy are important thermodynamic properties used to understand energy transformations, heat transfer, chemical reactions and the behaviour of physical systems.
These properties help explain:
- How energy is stored within matter.
- How energy is distributed among microscopic particles.
- Why natural processes have a preferred direction.
- How much heat is absorbed or released during processes at constant pressure.
- Whether a physical or chemical process is thermodynamically favourable.
Although these three quantities are related, they describe different aspects of a thermodynamic system.
2. Internal Energy
Internal energy is the total microscopic energy contained within a thermodynamic system.
It includes the microscopic kinetic and potential energies associated with the particles of the system.
2.1 Components of Internal Energy
Internal energy may include:
Microscopic kinetic energy
- Translational motion of particles.
- Rotational motion of molecules.
- Vibrational motion of atoms and molecules.
Microscopic potential energy
- Intermolecular forces.
- Chemical bonding.
- Interatomic interactions.
- Other microscopic interactions within the system.
Internal energy does not include the macroscopic kinetic energy of the entire system moving as a whole or its macroscopic gravitational potential energy.
2.2 Symbol and SI Unit
Internal energy is represented by:
[
U
]
Its SI unit is the joule (J).
The change in internal energy is represented by:
[
\Delta U=U_2-U_1
]
Where:
- (U_1) = initial internal energy
- (U_2) = final internal energy
2.3 Internal Energy and Temperature
Temperature is related to the microscopic kinetic energy of particles.
For an ideal gas, internal energy depends only on absolute temperature.
For a monatomic ideal gas:
[
U=\frac{3}{2}nRT
]
Where:
- (n) = number of moles
- (R) = universal gas constant
- (T) = absolute temperature
For a fixed amount of monatomic ideal gas:
[
\Delta U=\frac{3}{2}nR\Delta T
]
For other ideal gases, the internal energy depends on their active degrees of freedom and may be expressed as:
[
U=\frac{f}{2}nRT
]
under the classical equipartition approximation, where (f) is the number of active quadratic degrees of freedom.
2.4 Internal Energy of Real Substances
For real substances, internal energy depends on more than temperature alone.
It may also depend on:
- Pressure.
- Volume.
- Physical state.
- Molecular interactions.
- Chemical composition.
For example, when ice melts into water at its melting point, internal energy increases even though temperature remains approximately constant. The change is associated with the alteration of molecular structure and interactions.
3. Internal Energy and the First Law
The First Law of Thermodynamics relates the change in internal energy to heat transfer and work.
Using the convention that work done by the system is positive:
[
\Delta U=Q-W
]
Where:
- (\Delta U) = change in internal energy
- (Q) = heat supplied to the system
- (W) = work done by the system
3.1 When Internal Energy Increases
Internal energy increases when the net energy transferred into the system is positive.
Examples include:
- Heating a gas in a rigid container.
- Compressing a gas through work done on it.
- Heating water.
- Melting ice.
3.2 When Internal Energy Decreases
Internal energy may decrease when:
- A system loses heat.
- A gas expands and performs work without sufficient heat input.
- A hot object cools.
- Steam condenses and releases energy.
4. Entropy
Entropy is a thermodynamic state property that describes the dispersal of energy and the number of microscopic arrangements, or microstates, consistent with a system’s macroscopic condition.
It is one of the central concepts of the Second Law of Thermodynamics.
Entropy is commonly represented by:
[
S
]
Its SI unit is:
[
\text{J K}^{-1}
]
4.1 Statistical Interpretation of Entropy
In statistical mechanics, entropy is related to the number of accessible microscopic arrangements of a system.
Boltzmann’s entropy equation is:
[
S=k_B\ln\Omega
]
Where:
- (S) = entropy
- (k_B) = Boltzmann constant
- (\Omega) = number of accessible microstates
A larger number of accessible microstates corresponds to a higher entropy, under the assumptions of this statistical description.
4.2 Entropy and Energy Dispersal
Entropy is often explained in terms of energy dispersal.
For example, when a hot metal block is placed in contact with cooler water, thermal energy spreads from the hotter object to the cooler one.
The total entropy of the combined isolated system increases during this irreversible process.
Entropy should not be interpreted simply as disorder. The statistical definition is more precise and applies to a wide range of physical systems.
5. Entropy Change
For a reversible transfer of heat at absolute temperature (T):
[
dS=\frac{\delta Q_{\text{rev}}}{T}
]
For a reversible process occurring at constant temperature:
[
\Delta S=\frac{Q_{\text{rev}}}{T}
]
Where:
- (\Delta S) = entropy change
- (Q_{\text{rev}}) = reversible heat transfer
- (T) = absolute temperature in kelvin
This relationship is useful for calculating entropy changes during phase transitions and other thermodynamic processes.
5.1 Entropy of an Isolated System
The Second Law states:
[
\Delta S_{\text{total}}\geq0
]
For an isolated system:
- A reversible process has zero total entropy change.
- An irreversible process produces a positive total entropy change.
- The total entropy cannot decrease.
5.2 Entropy of the Universe
For the universe considered as an isolated system, the total entropy increases in irreversible processes and remains constant in an ideal reversible process.
This provides the thermodynamic basis for the natural direction of many processes.
6. Examples of Entropy Changes
6.1 Melting of Ice
When ice melts at its melting point, it absorbs energy and changes into liquid water.
The entropy of the water increases because the liquid state has a greater number of accessible molecular arrangements.
6.2 Freezing of Water
When water freezes, its entropy decreases as the molecular structure becomes more organised.
However, the surroundings receive the released energy. The total entropy of the system and surroundings must satisfy the Second Law.
6.3 Expansion of a Gas
When a gas expands freely into a larger volume, its entropy increases because the molecules have access to a greater number of possible positions and microscopic arrangements.
6.4 Mixing of Gases
When different gases mix, the number of accessible microscopic arrangements generally increases, producing an entropy increase.
7. Enthalpy
Enthalpy is a thermodynamic state function defined as the sum of a system’s internal energy and the product of its pressure and volume.
It is represented by:
[
H=U+PV
]
Where:
- (H) = enthalpy
- (U) = internal energy
- (P) = pressure
- (V) = volume
Its SI unit is the joule (J).
Enthalpy is particularly useful in analysing processes involving flowing fluids, chemical reactions and heat transfer at constant pressure.
8. Enthalpy Change
The enthalpy change is the difference between the final and initial enthalpy.
[
\Delta H=H_2-H_1
]
For a chemical reaction:
[
\Delta H=H_{\text{products}}-H_{\text{reactants}}
]
8.1 Enthalpy at Constant Pressure
For a process occurring at constant pressure, with only pressure-volume work and under the usual thermodynamic sign convention:
[
Q_p=\Delta H
]
This means that the heat transferred to a system at constant pressure equals its enthalpy change under these conditions.
8.2 Importance of Enthalpy
Enthalpy is useful because many chemical and physical processes take place under approximately constant atmospheric pressure.
Examples include:
- Combustion reactions.
- Neutralisation reactions.
- Dissolution processes.
- Heating and cooling of flowing fluids.
- Steam generation.
- Industrial chemical manufacturing.
9. Exothermic and Endothermic Processes
Enthalpy change helps classify chemical reactions and other processes.
9.1 Exothermic Process
An exothermic process releases heat to its surroundings.
Its enthalpy change is negative:
[
\Delta H<0
]
Examples include:
- Combustion of fuels.
- Many neutralisation reactions.
- Freezing of water.
- Condensation of steam.
The products of an exothermic chemical reaction have lower enthalpy than the reactants under the same reference conditions.
9.2 Endothermic Process
An endothermic process absorbs heat from its surroundings.
Its enthalpy change is positive:
[
\Delta H>0
]
Examples include:
- Melting of ice.
- Vaporisation of water.
- Thermal decomposition reactions.
- Certain dissolution processes.
The products of an endothermic chemical reaction have higher enthalpy than the reactants under the same reference conditions.
10. Enthalpy and Internal Energy
Enthalpy and internal energy are related but not identical.
[
H=U+PV
]
For changes between two states:
[
\Delta H=\Delta U+\Delta(PV)
]
For an ideal gas:
[
PV=nRT
]
For a fixed amount of ideal gas:
[
\Delta H=\Delta U+nR\Delta T
]
This relationship assumes that the amount of gas remains constant and that the ideal-gas model applies.
For an ideal gas, both internal energy and enthalpy depend only on temperature.
11. Specific Enthalpy
Specific enthalpy is the enthalpy per unit mass.
It is commonly represented by:
[
h=\frac{H}{m}
]
Where:
- (h) = specific enthalpy
- (H) = total enthalpy
- (m) = mass
Its SI unit is:
[
\text{J kg}^{-1}
]
In engineering, specific enthalpy is particularly important in analysing steam turbines, boilers, condensers, heat exchangers and refrigeration systems.
12. Standard Enthalpy of Formation
The standard enthalpy of formation is the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states under specified standard conditions.
It is represented by:
[
\Delta H_f^\circ
]
For an element in its standard reference state, the standard enthalpy of formation is defined as zero.
12.1 Hess’s Law
Hess’s law states that the total enthalpy change of a reaction is independent of the route by which the reaction occurs, provided the initial and final states are the same.
This follows because enthalpy is a state function.
For a reaction:
\sum \nu\Delta H_f^\circ(\text{reactants})
]
Where (\nu) represents the stoichiometric coefficients.
Hess’s law is useful when a reaction’s enthalpy change cannot be measured directly.
13. Comparison of Internal Energy, Entropy and Enthalpy
| Property | Symbol | Meaning | SI unit |
|---|---|---|---|
| Internal energy | (U) | Total microscopic energy of a system | J |
| Entropy | (S) | Thermodynamic measure related to energy dispersal and accessible microstates | J K(^{-1}) |
| Enthalpy | (H) | Internal energy plus pressure-volume product | J |
13.1 Key Differences
- Internal energy describes microscopic energy stored in a system.
- Entropy describes the statistical and thermodynamic distribution of energy and accessible microstates.
- Enthalpy is especially useful for analysing constant-pressure processes.
All three are state functions, meaning their changes depend on the initial and final states rather than the path followed.
14. Applications in Science and Engineering
14.1 Chemical Industry
Enthalpy changes help engineers calculate the heating and cooling requirements of chemical reactions.
14.2 Power Generation
Internal energy and enthalpy are used to analyse steam generation, turbine expansion and condensation.
14.3 Refrigeration
Entropy and enthalpy are important in evaluating refrigerant cycles and their performance.
14.4 Material Science
Changes in internal energy and entropy help explain melting, crystallisation and other phase transformations.
14.5 Environmental Science
Entropy and energy dispersal are important in understanding atmospheric processes, energy conversion and environmental systems.
15. Worked Examples
Example 1: Change in Internal Energy
A system receives 800 J of heat and performs 300 J of work. Calculate the change in internal energy.
Using:
[
\Delta U=Q-W
]
[
\Delta U=800-300
]
[
\boxed{\Delta U=500\ \text{J}}
]
The internal energy increases by 500 J.
Example 2: Entropy Change During Melting
Calculate the entropy change when 1 kg of ice melts at 0°C.
Take:
[
L_f=3.34\times10^5\ \text{J kg}^{-1}
]
[
T=273.15\ \text{K}
]
The heat absorbed is:
[
Q=mL_f
]
[
Q=1\times3.34\times10^5
]
For reversible melting at the melting point:
[
\Delta S=\frac{Q_{\text{rev}}}{T}
]
[
\Delta S=\frac{334000}{273.15}
]
[
\boxed{\Delta S\approx1223\ \text{J K}^{-1}}
]
The entropy of the ice-water system increases during melting.
Example 3: Enthalpy Change of an Ideal Gas
One mole of an ideal gas is heated from 300 K to 400 K. Calculate the change in enthalpy if its molar heat capacity at constant pressure is 29 J mol(^{-1}) K(^{-1}).
Using:
[
\Delta H=nC_p\Delta T
]
[
\Delta H=1\times29\times(400-300)
]
[
\boxed{\Delta H=2900\ \text{J}}
]
The enthalpy increases by 2.9 kJ.
16. Common Examination Mistakes
- Treating internal energy and enthalpy as identical.
- Assuming entropy means disorder only.
- Confusing the entropy change of a system with the total entropy change of the system and surroundings.
- Forgetting that entropy is measured in J K(^{-1}).
- Forgetting to use absolute temperature in entropy calculations.
- Assuming every exothermic process is automatically spontaneous under all conditions.
- Assuming enthalpy change always equals heat transfer regardless of process conditions.
- Forgetting that enthalpy is a state function.
36 Key Takeaways for Exams
- Internal energy is the total microscopic energy of a thermodynamic system.
- Internal energy includes microscopic kinetic and potential energy.
- Internal energy is represented by (U).
- The SI unit of internal energy is the joule.
- Internal energy is a state function.
- The First Law relates internal energy change to heat and work.
- Using work done by the system as positive, (\Delta U=Q-W).
- For a monatomic ideal gas, (U=\frac{3}{2}nRT).
- For an ideal gas, internal energy depends only on temperature.
- Real substances may have internal energy changes associated with phase and molecular interactions.
- Entropy is a thermodynamic state property.
- Entropy is represented by (S).
- The SI unit of entropy is J K(^{-1}).
- Boltzmann’s entropy equation is (S=k_B\ln\Omega).
- The number of accessible microstates is represented by (\Omega).
- Entropy is related to energy dispersal and microscopic possibilities.
- Entropy should not be interpreted merely as disorder.
- The entropy change for reversible heat transfer is (dS=\delta Q_{\text{rev}}/T).
- Entropy calculations require absolute temperature in kelvin.
- The total entropy of an isolated system cannot decrease.
- An irreversible process produces positive total entropy.
- A reversible process has zero total entropy change for the system and surroundings combined.
- Enthalpy is defined as (H=U+PV).
- Enthalpy is represented by (H).
- The SI unit of enthalpy is the joule.
- Enthalpy is a state function.
- Enthalpy change is (H_2-H_1).
- At constant pressure, heat transfer equals enthalpy change under the usual assumptions of pressure-volume work only.
- An exothermic process has a negative enthalpy change.
- An endothermic process has a positive enthalpy change.
- The standard enthalpy of formation is the enthalpy change when one mole of a compound forms from its elements in their standard states.
- Hess’s law states that reaction enthalpy is independent of the reaction pathway.
- For a fixed amount of ideal gas, (\Delta H=\Delta U+nR\Delta T).
- Specific enthalpy is enthalpy per unit mass.
- Internal energy, entropy and enthalpy are different thermodynamic properties.
- These properties are essential in chemical engineering, power generation, refrigeration and energy analysis.