For The Following Processes Calculate The Change in Internal Energy

This guide explains how to calculate the change in internal energy for various thermodynamic processes. Internal energy is a fundamental concept in thermodynamics that represents the total energy of a system's molecules, including kinetic and potential energy. The change in internal energy can be calculated using the first law of thermodynamics, which relates heat added to a system, work done by the system, and the change in internal energy.

What is internal energy?

Internal energy (U) is a thermodynamic property that represents the total energy of all the molecules in a system. It includes both kinetic energy (from molecular motion) and potential energy (from molecular interactions). Internal energy is an extensive property, meaning its value depends on the amount of substance present.

The change in internal energy (ΔU) is calculated using the first law of thermodynamics:

First Law of Thermodynamics

ΔU = Q - W

Where:

ΔU = change in internal energy
Q = heat added to the system
W = work done by the system

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.

Calculating the change in internal energy

To calculate the change in internal energy, you need to know the heat added to the system and the work done by the system. The units for internal energy are typically joules (J) in the International System of Units (SI).

The change in internal energy can be positive or negative:

If ΔU is positive, the system has gained energy.
If ΔU is negative, the system has lost energy.

For ideal gases, the change in internal energy can also be calculated using the molar heat capacity at constant volume (Cv):

Change in Internal Energy for Ideal Gases

ΔU = n × Cv × ΔT

Where:

n = number of moles of gas
Cv = molar heat capacity at constant volume
ΔT = change in temperature

Different thermodynamic processes

Thermodynamic processes describe how a system changes from one equilibrium state to another. The change in internal energy depends on the type of process:

Isothermal Process

An isothermal process occurs at constant temperature. For an ideal gas, the change in internal energy is zero because the temperature remains constant.

Isothermal Process

ΔU = 0

Adiabatic Process

An adiabatic process occurs without transfer of heat. The change in internal energy is equal to the work done by the system.

Adiabatic Process

ΔU = -W

Isobaric Process

An isobaric process occurs at constant pressure. The change in internal energy can be calculated using the molar heat capacity at constant pressure (Cp).

Isobaric Process

ΔU = n × Cp × ΔT

Isochoric Process

An isochoric process occurs at constant volume. The change in internal energy is equal to the heat added to the system.

Isochoric Process

ΔU = Q

Worked examples

Example 1: Isothermal Process

For an isothermal process, the change in internal energy is zero because the temperature remains constant.

Example Calculation

Given:

Process: Isothermal
Initial state: P₁ = 1 atm, V₁ = 22.4 L, T = 300 K
Final state: P₂ = 2 atm, V₂ = 11.2 L, T = 300 K

Calculation:

ΔU = 0 J (since temperature is constant)

Example 2: Adiabatic Process

For an adiabatic process, the change in internal energy is equal to the work done by the system.

Example Calculation

Given:

Process: Adiabatic
Initial state: P₁ = 1 atm, V₁ = 10 L, T₁ = 300 K
Final state: P₂ = 2 atm, V₂ = 5 L, T₂ = 600 K
Work done by the system: W = 200 J

Calculation:

ΔU = -W = -200 J

FAQ

What is the difference between internal energy and enthalpy?

Internal energy (U) is the total energy of a system's molecules, while enthalpy (H) is the sum of internal energy and the product of pressure and volume (PV). Enthalpy is often more useful in processes involving heat transfer at constant pressure.

How does the change in internal energy relate to temperature?

For an ideal gas, the change in internal energy is directly proportional to the change in temperature when the volume is held constant. This is described by the equation ΔU = n × Cv × ΔT.

Can the change in internal energy be negative?

Yes, the change in internal energy can be negative if the system loses energy. This occurs when the work done by the system is greater than the heat added to the system.