Introduction to Thermodynamics - First Law and Internal Energy

First Law of Thermodynamics

Definitions:

• Energy: The ability to do work.
• Heat: The transfer of energy between two systems at different temperatures.
• Work: The transfer of energy from one system to another due to a force acting over a distance.
• Internal Energy: The sum of all the kinetic and potential energy of the particles within a system.

Mathematical Formulation: The first law of thermodynamics states that the total energy of an isolated system remains constant. This can be expressed as: $$\text{ΔE}_\text{int}=\text{Q}+\text{W}$$

• ∆Eint: Change in internal energy
• Q: Heat added to the system
• W: Work done by the system

Sign Conventions:

• Heat added to the system is positive, while heat lost by the system is negative.
• Work done by the system is negative, while work done on the system is positive.

Internal Energy

Definition:

Internal energy is a state function that depends only on the current state of the system and not on the path taken to reach that state.

Relation to Other Thermodynamic Variables:

• Internal energy is related to pressure (P), volume (V), and temperature (T) by the equation of state: $$\text{E}_\text{int} =\text{f(P, V, T)}$$
• For an ideal gas, the internal energy depends only on temperature. $$ \text{E}_\text{int} =\frac{3}{2}\text{nRT}$$ where n is the number of moles of gas and R is the ideal gas constant.

Energy Transfer Mechanisms:

• Internal energy can be transferred between systems by heat flow and work.
• Heat flow occurs when two systems at different temperatures come into contact. Heat flows from the hotter system to the colder system.
• Work is done when a force acts over a distance. Work can be done on or by a system.

Internal Energy Changes in Thermodynamic Processes

• Isobaric: $$\text{ΔE}\text{int} = \text{Q}\text{p}-\text{PΔV}$$
• Isochoric: $$\text{ΔE}\text{int} =\text{Q}\text{v}$$
• Isothermal: $$\text{ΔE}\text{int} =\text{W}\text{rev}$$
• Adiabatic: $$\text{ΔE}\text{int} = -\text{W}\text{rev}$$

Specific Heat Capacity

Definition

The specific heat capacity of a substance is the amount of heat required to raise the temperature of one gram of that substance by one degree Celsius.

Types of specific heat capacities:

• Specific heat capacity at constant pressure (Cp): This is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius at constant pressure. $$C_\text{p} =\left(\frac{\partial \text{Q}\text{p}}\partial\text{T}\right)\text{p}$$
• Specific heat capacity at constant volume (Cv): This is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius at constant volume. $$C_\text{v} = \left(\frac{\partial \text{Q}\text{v}}\partial\text{T}\right)\text{V}$$

Dulong-Petit Law:

The Dulong-Petit law states that the specific heat capacity of a solid element is approximately 25 J/mol K at room temperature. This law is useful for estimating the specific heat capacity of solid substances.

Adiabatic Processes:

Definition: An adiabatic process is a process in which there is no heat transfer between the system and its surroundings.

Relationship Between Pressure and Volume: For an adiabatic process, the pressure and volume of a gas are related by the equation:

$$\text{PV}^\gamma = \text{constant}$$ where γ is the ratio of specific heat capacities (Cp/Cv).

Calorimetry:

Techniques:

• Bomb Calorimetry: This technique measures the heat of combustion of a substance by burning it in a closed vessel filled with oxygen.
• Constant Volume Calorimetry: This technique measures the heat of a reaction by placing the reactants in a closed vessel and measuring the temperature change.

Determining Enthalpy Change: The enthalpy change for a chemical reaction can be calculated using the following equation: $$\Delta \text{H} =\text{Q}_\text{p}$$

Hess’s Law: Hess’s law states that the enthalpy change of a chemical reaction is the same whether the reaction occurs in one step or in a series of steps. This law can be used to calculate the enthalpy change for complex reactions.

Applications in Engineering and Physical Sciences

Engineering Applications:

• Designing efficient energy systems
• Refrigeration cycles
• Heat engines
• Power plants

Physical Sciences Applications:

• Understanding the behavior of thermodynamic systems
• Predicting the properties of gases and liquids
• Analyzing chemical reactions

By studying the subtopics within the topic of Introduction to Thermodynamics - First Law and Internal Energy, students can gain a strong foundation for understanding the fundamental principles of thermodynamics and their applications in various fields of engineering and physical sciences.