2nd Law

Heat will not spontaneously flow from a colder body to a hotter body. Entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

This law establishes the concept of entropy (S) as a measure of the disorder or randomness in a system. It implies that natural processes tend to move toward states of higher entropy.

Thermal Reservoir

Aka. heat reservoir. A body that can either absorb or release heat at constant temperature. Isothermal heat transfer device.

• If absorbing: it is a heat sink.
• If releasing: it is a heat source.

Kelvin-Planck Statement

It is impossible for any device that operates on a cycle to exchange heat with a single reservoir and produce a net amount of work. Explains the operational constraints of a heat engine.

Carnot Principle

The efficiency of a reversible heat engine operating between two thermal reservoirs is always greater than the the efficiency of an irreversible heat engine operating between the same reservoirs.

Carnot Heat Engine

Reversible heat engines are also known as Carnot engines.

All carnot engines working with the same two thermal reservoirs have the same efficiency.

Carnot Cycle

Has 4 elements. All processes are reversible.

• through the boiler: isothermal heat addition
• through the turbine: adiabatic expansion
• through the condenser: isothermal heat rejection
• through the pump: adiabatic compression

Note

Reversible adiabatic processes are isentropic.

Heat pumps & Refrigerators

Transfers heat from a cooler body to a hotter body, with the aid of work input. Reversed heat machines. Air conditioners work similarly to refrigerators.

Machine	Objective
Heat Pump	Heat up an environment by pumping heat into the concerned environment.
Refrigerator	Cool an environment by pumping heat out from the concerned environment.

Clausius Statement

It is impossible to construct a system which will transfer heat from a cooler body to a hotter body without work being done on the system by surrounding.

Establishes the operational constraints of heat pumps and refrigerators.

Coefficient of Performance

When analyzing the performance of heat pumps and refrigerators, the Coefficient of Performance is used instead of efficiency. COP is always greater than or equal to 1.

$$\text{COP}_\text{HP} = \frac{Q_{\text{H}}}{W_\text{in}} = \frac{1}{1 - \frac{Q_L}{Q_H}}$$ $$\text{COP}_\text{REF} = \frac{Q_{\text{L}}}{W_\text{in}} = \frac{1}{\frac{Q_H}{Q_L} - 1}$$

Note

For a reversible heat engine:

$$\frac{Q_H}{Q_L} = \frac{T_H}{T_L}$$

Here $T_H$ and $T_L$ are the temperatures of the hot and cold reservoirs in Kelvin.

Entropy Formula

For a reversible process:

$$\Delta S = \frac{\delta Q}{T}$$

Where:

• ΔS is the change in entropy
• δQ is the heat transfer
• T is the absolute temperature

For irreversible processes, entropy always increases:

$$\Delta S > \frac{\delta Q}{T}$$

Implications

• Heat engines cannot be 100% efficient
• Refrigerators and heat pumps require work input
• All spontaneous processes are irreversible
• Perfect order is unattainable