Chapter Notes

Introduction to Thermodynamics

Thermodynamics is the branch of physics that studies heat, temperature, and the conversion of heat into other forms of energy, like work. It is a macroscopic science, meaning it deals with large-scale systems (like a container of gas) rather than the individual molecules within them.

Historically, heat was mistakenly thought to be an invisible fluid called "caloric" that flowed from hot objects to cold objects. However, experiments, like those by Count Rumford in 1798 who observed immense heat generated from boring a cannon, showed that heat is actually a form of energy. Work could be converted into heat, and heat could be converted into work (as in a steam engine).

Thermodynamics describes a system using measurable macroscopic variables like pressure, volume, and temperature, without needing to know the position and velocity of every single molecule.

• Mechanics studies the motion of a system as a whole (e.g., a bullet flying through the air).
• Thermodynamics studies the internal state of a system (e.g., the temperature change in the bullet and the wood it hits when it stops). The kinetic energy of the bullet's motion is converted into internal energy (heat).

Thermal Equilibrium

In mechanics, equilibrium means the net force and torque on a system are zero. In thermodynamics, equilibrium has a different meaning. A system is in a state of thermodynamic equilibrium if its macroscopic variables (like pressure, volume, and temperature) are constant over time.

Whether a system is in equilibrium depends on its surroundings and the type of wall separating them. There are two main types of walls:

• Adiabatic Wall: An insulating wall that does not allow any heat (energy) to flow through it.
• Diathermic Wall: A conducting wall that allows heat to flow between systems.

Consider two systems, A and B.

1. If they are separated by an adiabatic wall, they are isolated from each other. Any pressure and volume for A can exist in equilibrium with any pressure and volume for B.
2. If the adiabatic wall is replaced by a diathermic wall, energy will flow between them until their macroscopic variables stop changing. At this point, they have reached thermal equilibrium.

So, what property is the same for two systems in thermal equilibrium? The answer is temperature.

Zeroth Law of Thermodynamics

The Zeroth Law of Thermodynamics provides the formal basis for the concept of temperature. It was formulated long after the First and Second laws, but it is so fundamental that it was named the "Zeroth" law.

The law states:

Two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other.

Let's break this down:

1. Imagine system A is in thermal equilibrium with system C. This means they have the same temperature ($T_A = T_C$).
2. Now, imagine system B is also in thermal equilibrium with system C. This means they also have the same temperature ($T_B = T_C$).
3. The Zeroth Law tells us that if we then bring systems A and B into contact, they will also be in thermal equilibrium with each other, because $T_A = T_B$.

This law establishes that temperature is the fundamental property that determines whether systems are in thermal equilibrium.

Heat, Internal Energy, and Work

Internal Energy (U)

Internal energy (U) is the total energy of all the molecules within a system. It is the sum of the kinetic energies (from random translational, rotational, and vibrational motion) and potential energies (from intermolecular forces) of these molecules.

Internal energy is a state variable, which means its value depends only on the current state of the system (its pressure, volume, and temperature), not on how it got to that state.

Heat (Q) and Work (W)

Heat and work are not properties of a system; they are ways to transfer energy to or from a system, thereby changing its internal energy.

• Heat is energy transferred due to a temperature difference between a system and its surroundings. Heat flows from a hotter body to a colder body.
• Work is energy transferred by other means, such as a piston compressing a gas or a gas expanding and pushing a piston.

First Law of Thermodynamics

The First Law of Thermodynamics is simply the law of conservation of energy applied to a thermodynamic system. It states that the heat supplied to a system is used in two ways: to increase the internal energy of the system and to do work on the surroundings.

The formula for the First Law is: $\Delta Q = \Delta U + \Delta W$ Where:

• $\Delta Q$ = Heat supplied to the system.
• $\Delta U$ = Change in the internal energy of the system.
• $\Delta W$ = Work done by the system on its surroundings.

Since internal energy $\Delta U$ is a state variable, its change depends only on the initial and final states. However, $\Delta Q$ and $\Delta W$ are path-dependent. This means that the combination $\Delta Q - \Delta W$ must be path-independent, as it equals $\Delta U$.

For a gas in a cylinder with a movable piston, the work done at a constant pressure P is given by: $$\Delta W = P \Delta V$$ Where $\Delta V$ is the change in volume. In this case, the First Law becomes: $$\Delta Q = \Delta U + P \Delta V$$

Given

• Mass of water, $m = 1$ g
• Latent heat of vaporization, $L_v = 2256 \text{ J/g}$. This is the heat supplied, so $\Delta Q = 2256 \text{ J}$.
• Initial volume (liquid), $V_l = 1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3$.
• Final volume (vapor), $V_g = 1671 \text{ cm}^3 = 1671 \times 10^{-6} \text{ m}^3$.
• Atmospheric pressure, $P = 1.013 \times 10^5 \text{ Pa}$.

To Find

Change in internal energy, $\Delta U$.

Formula

$\Delta W = P(V_g - V_l)$ $\Delta U = \Delta Q - \Delta W$

Solution

First, calculate the work done by the expanding steam: $\Delta W = (1.013 \times 10^5 \text{ Pa}) \times (1671 \times 10^{-6} \text{ m}^3 - 1 \times 10^{-6} \text{ m}^3)$ $\Delta W = (1.013 \times 10^5) \times (1670 \times 10^{-6}) \approx 169.2 \text{ J}$

Now, use the First Law of Thermodynamics to find the change in internal energy: $\Delta U = 2256 \text{ J} - 169.2 \text{ J} = 2086.8 \text{ J}$

Final Answer The change in internal energy is $2086.8$ J. This shows that most of the heat supplied goes into increasing the internal energy of the molecules, not into doing external work.

Specific Heat Capacity

Heat capacity (S) is the amount of heat required to raise the temperature of a substance by one unit ($\Delta T$). $$S = \frac{\Delta Q}{\Delta T}$$

To create a value that is characteristic of a material, we define:

1. Specific Heat Capacity (s): Heat capacity per unit mass. $$s = \frac{S}{m} = \frac{1}{m} \frac{\Delta Q}{\Delta T}$$ The unit is $\text{J kg}^{-1} \text{K}^{-1}$.

2. Molar Specific Heat Capacity (C): Heat capacity per mole. $$C = \frac{S}{\mu} = \frac{1}{\mu} \frac{\Delta Q}{\Delta T}$$ The unit is $\text{J mol}^{-1} \text{K}^{-1}$.

Molar Specific Heat of Solids

For many solids at room temperature, the molar specific heat capacity is approximately constant. Using the law of equipartition of energy, we can predict this value. For a mole of a solid, the total internal energy is $U = 3RT$. Since solids expand very little, the work done $\Delta W$ is negligible, so $\Delta Q \approx \Delta U$. Therefore, the molar specific heat capacity is: $$C = \frac{\Delta U}{\Delta T} = 3R$$ Since $R \approx 8.314 \text{ J mol}^{-1} \text{K}^{-1}$, this predicts $C \approx 24.9 \text{ J mol}^{-1} \text{K}^{-1}$, which agrees well with experimental values for many solids.

Specific Heat Capacities of Gases

For gases, the heat capacity depends on the conditions under which heat is supplied. We define two types:

• Molar specific heat at constant volume ($C_v$): If the volume is held constant, no work is done ($\Delta W = 0$). All the heat supplied goes into increasing the internal energy. $$C_v = \left( \frac{\Delta Q}{\Delta T} \right)_v = \frac{\Delta U}{\Delta T}$$
• Molar specific heat at constant pressure ($C_p$): If the pressure is held constant, the gas expands and does work. The heat supplied must both increase the internal energy and do work. $$C_p = \left( \frac{\Delta Q}{\Delta T} \right)_p = \frac{\Delta U}{\Delta T} + P \left( \frac{\Delta V}{\Delta T} \right)_p$$

For one mole of an ideal gas, $PV = RT$. At constant pressure, $P \Delta V = R \Delta T$, so $P (\frac{\Delta V}{\Delta T})_p = R$. Substituting this into the equations above gives a simple relation for ideal gases, known as Mayer's relation: $$C_p - C_v = R$$

Thermodynamic State Variables and Equation of State

State variables are macroscopic quantities that describe the equilibrium state of a system, such as Pressure (P), Volume (V), Temperature (T), internal energy (U), and mass (m). The value of a state variable depends only on the state itself, not the path taken to reach it.

An equation of state is a mathematical relationship between state variables. The most common example is the ideal gas law: $$PV = \mu RT$$ where $\mu$ is the number of moles and R is the universal gas constant.

State variables can be classified into two types:

• Extensive variables depend on the size or mass of the system. If you divide the system in half, their value is halved. Examples: Internal energy (U), volume (V), mass (M).
• Intensive variables do not depend on the size of the system. Their value remains the same if you divide the system. Examples: Pressure (P), temperature (T), density ($\rho$).

Thermodynamic Processes

Quasi-static Process

A quasi-static process is an idealized process that happens so slowly that the system remains in thermal and mechanical equilibrium with its surroundings at every single stage. This means the pressure and temperature differences between the system and its surroundings are infinitesimally small throughout the process. Real-world processes can approximate this if they are slow enough.

Types of Thermodynamic Processes

We often study four special types of quasi-static processes:

Isothermal Process

An isothermal process occurs at a constant temperature ($\Delta T = 0$).

• For an ideal gas, this follows Boyle's Law: $PV = \text{constant}$.
• Since the internal energy of an ideal gas depends only on temperature, $\Delta U = 0$.
• From the First Law, $Q = W$. All heat absorbed is converted into work done by the gas.
• The work done by $\mu$ moles of an ideal gas expanding from $V_1$ to $V_2$ is: $$W = \mu R T \ln \left( \frac{V_2}{V_1} \right)$$

Adiabatic Process

An adiabatic process occurs with no heat exchange between the system and its surroundings ($\Delta Q = 0$). This happens in a perfectly insulated system or in a process that is very rapid.

• From the First Law, $\Delta U = -W$. If the gas does work (expands), its internal energy and temperature decrease. If work is done on the gas (compression), its internal energy and temperature increase.
• For an ideal gas, an adiabatic process is described by the equation: $$PV^\gamma = \text{constant}$$ where $\gamma = C_p / C_v$ is the ratio of specific heats.
• The work done by an ideal gas changing from state ($P_1, V_1, T_1$) to ($P_2, V_2, T_2$) is: $$W = \frac{\mu R (T_1 - T_2)}{\gamma - 1}$$

Isochoric Process

An isochoric process occurs at a constant volume ($\Delta V = 0$).

• Since volume does not change, the work done is zero: $W = 0$.
• From the First Law, $\Delta Q = \Delta U$. All heat supplied goes directly into increasing the internal energy.

Isobaric Process

An isobaric process occurs at a constant pressure ($\Delta P = 0$).

• The work done by the gas is $W = P(V_2 - V_1)$.
• The heat supplied goes into both increasing the internal energy and doing work.

Cyclic Process

A cyclic process is one where the system returns to its initial state after a series of changes.

• Since internal energy is a state variable, the net change over a cycle is zero: $\Delta U = 0$.
• From the First Law, the total heat absorbed by the system equals the total work done by the system: $Q = W$.

Second Law of Thermodynamics

The First Law of Thermodynamics (conservation of energy) allows for many processes that we never see in nature. For example, a book on a table could spontaneously cool down and use that energy to jump into the air, conserving total energy. But this never happens. The Second Law of Thermodynamics provides the fundamental principle that forbids such processes. It deals with the direction of natural processes.

The Second Law can be stated in two equivalent ways:

1. Kelvin-Planck Statement:

   No process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat into work. This means a heat engine can never be 100% efficient. Some heat must always be released to a colder reservoir.

2. Clausius Statement:

   No process is possible whose sole result is the transfer of heat from a colder object to a hotter object. This means heat does not spontaneously flow from cold to hot. To make it do so (as in a refrigerator), work must be done.

Reversible and Irreversible Processes

• Irreversible Process: A process that cannot be reversed to bring both the system and its surroundings back to their original states. All spontaneous processes in nature are irreversible. Irreversibility is caused by:

  1. Processes that involve non-equilibrium states (e.g., free expansion of a gas).
  2. Dissipative effects like friction and viscosity, which convert mechanical energy into heat.

• Reversible Process: An idealized process that can be reversed in such a way that both the system and the surroundings return to their initial states with no other changes anywhere. A process is reversible only if it is:

  1. Quasi-static: The system is always in equilibrium.
  2. Non-dissipative: There are no effects like friction.

Reversible processes are important because they set the theoretical limit for the efficiency of heat engines.

Carnot Engine

A Carnot engine is an idealized, reversible heat engine that operates between two reservoirs at different temperatures: a hot reservoir (source) at $T_1$ and a cold reservoir (sink) at $T_2$. It represents the most efficient engine possible operating between these two temperatures.

The Carnot cycle consists of four reversible steps:

1. Isothermal Expansion: The gas expands at constant temperature $T_1$, absorbing heat $Q_1$ from the hot reservoir.
2. Adiabatic Expansion: The gas expands without heat exchange, causing its temperature to drop from $T_1$ to $T_2$.
3. Isothermal Compression: The gas is compressed at constant temperature $T_2$, releasing heat $Q_2$ to the cold reservoir.
4. Adiabatic Compression: The gas is compressed without heat exchange, causing its temperature to rise from $T_2$ back to $T_1$, completing the cycle.

Efficiency of a Carnot Engine

The efficiency ($\eta$) of any heat engine is the ratio of the work done (W) to the heat absorbed from the hot reservoir ($Q_1$). $$\eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1}$$

For a Carnot engine, it can be shown that the ratio of heats is equal to the ratio of the absolute temperatures of the reservoirs: $$\frac{Q_2}{Q_1} = \frac{T_2}{T_1}$$

Therefore, the efficiency of a Carnot engine is: $$\eta = 1 - \frac{T_2}{T_1}$$

This leads to two important conclusions known as Carnot's Theorem:

1. No engine operating between two temperatures can be more efficient than a Carnot engine operating between the same two temperatures.
2. The efficiency of a Carnot engine is independent of the working substance (e.g., ideal gas, water, etc.). It depends only on the temperatures of the hot and cold reservoirs.