Internal Energy, Heat, and Work

Introduction

Key Concepts

Internal Energy

Internal energy ($U$) refers to the total microscopic kinetic and potential energy of all the particles within a system. It encompasses the energy associated with molecular motion, intermolecular forces, and chemical bonds. Internal energy is a state function, meaning it depends only on the current state of the system, not on how it reached that state.

**Mathematical Representation:** $$ U = \sum_{i} \frac{1}{2} m_i v_i^2 + \sum_{i

**Types of Internal Energy:**

• Kinetic Energy: Energy due to the motion of particles within the system.
• Potential Energy: Energy due to the positions or arrangements of particles, including chemical bonds.

**Example:** Consider a gas confined in a cylinder. The internal energy of the gas is proportional to its temperature, indicating that as temperature increases, the kinetic energy of the gas molecules increases, thereby increasing the internal energy.

Heat

Heat ($Q$) is a form of energy transfer between systems or objects with different temperatures. It flows spontaneously from regions of higher temperature to lower temperature until thermal equilibrium is achieved. Heat is not a property of a system but a process of energy transfer.

**Units of Heat:** Heat is measured in joules (J) in the International System of Units (SI).

**Modes of Heat Transfer:**

• Conduction: Transfer of heat through direct contact between particles within a substance.
• Convection: Transfer of heat through fluid movement caused by temperature-induced density differences.
• Radiation: Transfer of heat through electromagnetic waves without the involvement of particles.

**Specific Heat Capacity ($c$):** The amount of heat required to raise the temperature of one gram of a substance by one degree Celsius.

**Equation:** $$ Q = mc\Delta T $$ where $m$ is mass, $c$ is specific heat capacity, and $\Delta T$ is the change in temperature.

**Example:** Heating 200 g of water (with $c = 4.18 \, \text{J/g°C}$) from 25°C to 75°C requires: $$ Q = 200 \, \text{g} \times 4.18 \, \text{J/g°C} \times (75°C - 25°C) = 200 \times 4.18 \times 50 = 41,800 \, \text{J} $$

Work

Work ($W$) in thermodynamics refers to the energy transfer resulting from a force acting through a distance. In the context of gases, work is often associated with expansion or compression when a gas changes its volume.

**Types of Work:**

• Expansion Work: When a gas expands against an external pressure, performing work on the surroundings.
• Compression Work: When work is done on a gas to compress it, increasing its internal energy.

**Mathematical Representation:** $$ W = -P\Delta V $$ where $P$ is the external pressure and $\Delta V$ is the change in volume. The negative sign indicates that work done by the system is considered negative.

**Example:** A piston compresses a gas from 2.0 liters to 1.5 liters against a constant external pressure of 5 atm. The work done on the gas is: $$ W = -P\Delta V = -5 \, \text{atm} \times (1.5 \, \text{L} - 2.0 \, \text{L}) = -5 \times (-0.5) = 2.5 \, \text{L.atm} $$ Converting to joules (1 L.atm ≈ 101.325 J): $$ W \approx 2.5 \times 101.325 = 253.31 \, \text{J} $$

The First Law of Thermodynamics

The First Law of Thermodynamics, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed in an isolated system. Instead, energy can only be transformed from one form to another. Mathematically, this law is expressed as: $$ \Delta U = Q + W $$ where $\Delta U$ is the change in internal energy of the system, $Q$ is the heat added to the system, and $W$ is the work done on the system.

**Implications of the First Law:**

• Energy transfer into a system increases its internal energy.
• Energy transfer out of a system decreases its internal energy.
• The internal energy change accounts for both heat transfer and work done.

**Example:** A gas in a piston absorbs 500 J of heat ($Q = +500 \, \text{J}$) and has 200 J of work done on it ($W = +200 \, \text{J}$). The change in internal energy is: $$ \Delta U = Q + W = 500 \, \text{J} + 200 \, \text{J} = 700 \, \text{J} $$

**Sign Convention:**

• Heat ($Q$): Positive when heat is added to the system; negative when heat is removed.
• Work ($W$): Positive when work is done on the system; negative when work is done by the system.

Internal Energy in Different Processes

Isothermal Process:

In an isothermal process, the temperature remains constant ($\Delta T = 0$), implying that the change in internal energy is zero ($\Delta U = 0$). Therefore, the heat added to the system is equal to the work done by the system. $$ Q = -W $$

Adiabatic Process:

An adiabatic process occurs without any heat transfer ($Q = 0$). The change in internal energy is solely due to work done on or by the system. $$ \Delta U = W $$

Isochoric Process:

In an isochoric (constant volume) process, there is no work done ($W = 0$) because the volume does not change. The change in internal energy is entirely due to heat transfer. $$ \Delta U = Q $$

Molar Specific Heat Capacities

Specific heat capacities at constant volume ($C_V$) and constant pressure ($C_P$) are critical in understanding how substances respond to heat transfer under different conditions.

At Constant Volume:

$$ Q = nC_V\Delta T $$ where $n$ is the number of moles.

At Constant Pressure:

$$ Q = nC_P\Delta T $$

**Relation Between $C_P$ and $C_V$:** $$ C_P = C_V + R $$ where $R$ is the universal gas constant ($8.314 \, \text{J/mol.K}$).

Applications of the First Law

The First Law of Thermodynamics has numerous applications in various fields:

• Heat Engines: Devices that convert heat into work using cyclic processes.
• Refrigerators and Heat Pumps: Systems that transfer heat from cooler areas to warmer areas using work input.
• Chemical Reactions: Analysis of energy changes during reactions involving heat and work.

Challenges in Understanding Energy Transfers

Grasping the nuances of energy transfers via heat and work can be challenging due to:

• Sign Conventions: Differentiating between work done by and on the system.
• Process Paths: Understanding how different thermodynamic processes affect internal energy.
• Calculations Involving Multiple Forms of Energy: Accurately accounting for all energy transfers in complex systems.

Comparison Table

Summary and Key Takeaways