Energy balance for closed systems is a crucial concept in thermodynamics. It applies the first law to systems that don't exchange matter with their surroundings, focusing on energy transfers through heat and work.

Understanding this topic helps you analyze various processes in closed systems. You'll learn to use energy balance equations, relate thermodynamic properties, and solve problems involving different types of work and heat transfer.

Thermodynamics of Closed Systems

First Law of Thermodynamics

• States energy cannot be created or destroyed, only converted from one form to another
• For a closed system, the change in total energy (ΔE) equals net heat transfer into the system (Q) minus net work done by the system (W): ΔE = Q - W
• Total energy of a closed system consists of internal energy (U), kinetic energy (KE), and potential energy (PE): E = U + KE + PE
• In the absence of kinetic and potential energy changes, the first law simplifies to ΔU = Q - W for a closed system

Application to Closed Systems

• Closed systems do not exchange matter with their surroundings, only energy in the form of heat and work
• Energy balance for a closed system undergoing a process: ΔU = Q - W, where ΔU is the change in internal energy, Q is net heat transfer, and W is net work done
• Examples of closed systems include a sealed piston-cylinder device, a rigid tank with no inlets or outlets, and a pressure cooker

Energy Balance Equations for Closed Systems

Process-Specific Energy Balance Equations

• Isochoric (constant volume) process: W = 0, energy balance equation becomes ΔU = Q
• Isobaric (constant pressure) process: W = P(V₂ - V₁), energy balance equation becomes ΔU = Q - P(V₂ - V₁)
• Isothermal (constant temperature) process: ΔU = 0, energy balance equation becomes Q = W
• Adiabatic process (no heat transfer, Q = 0): energy balance equation becomes ΔU = -W

Relating Thermodynamic Properties

• Ideal gas law (PV = nRT) relates pressure, volume, temperature, and amount of substance for an ideal gas
• Specific heat capacities (Cv and Cp) relate temperature changes to internal energy and heat transfer
  • Cv is the molar specific heat capacity at constant volume: ΔU = nCvΔT
  • Cp is the molar specific heat capacity at constant pressure: ΔH = nCpΔT, where ΔH is the change in enthalpy

Problem Solving in Closed Systems

Problem-Solving Approach

1. Identify the system, the process, and relevant energy terms (heat, work, internal energy change)
2. Apply the appropriate energy balance equation based on the process and given information
3. Use thermodynamic relations (ideal gas law, specific heat capacities) to relate changes in properties
4. Determine the type of work (boundary work, shaft work) and calculate using appropriate formulas
5. Solve for the unknown quantity using the energy balance equation and given information

Types of Work in Closed Systems

• Boundary work (or PV work) occurs when the system expands or contracts against an external pressure: W = P(V₂ - V₁)
• Shaft work involves the transfer of energy through a rotating shaft, such as in a turbine or compressor
• Other types of work include electrical work, spring work, and gravitational work

Example Problems

1. A closed system undergoes an isobaric process at 2 atm, expanding from 1 L to 3 L. If 500 J of heat is added to the system, determine the change in internal energy.
2. An ideal gas in a piston-cylinder device undergoes an isothermal compression from 5 L to 2 L at 300 K. Calculate the heat transfer and work done during the process.

Internal Energy and Energy Balance Analysis

Internal Energy as a State Function

• Internal energy (U) is the sum of the kinetic and potential energies of the particles within a system
• Includes translational, rotational, vibrational, and intermolecular energies
• As a state function, internal energy depends only on the current state, not the path taken to reach that state

Relationship between Internal Energy and Temperature

• For an ideal gas, the change in internal energy is proportional to the change in temperature: ΔU = nCvΔT
  • n is the number of moles
  • Cv is the molar specific heat capacity at constant volume
  • ΔT is the temperature change
• The specific heat capacity relates the amount of heat required to change the temperature of a substance

Role of Internal Energy in Energy Balance Analysis

• Changes in internal energy (ΔU) are caused by heat transfer (Q) and work done (W) on or by the system
• The first law of thermodynamics relates these quantities: ΔU = Q - W
• In energy balance analysis, internal energy is a key component in determining the heat transfer and work interactions between a system and its surroundings during a process