State vs. Path Thermodynamic Functions

Thermodynamics is built on two types of functions: state functions and path functions. Understanding the difference between these is essential, as they govern how energy and matter interact in physical and chemical processes.

State Functions

Internal Energy as a State Function

The First Law of Thermodynamics is a cornerstone in understanding how energy is conserved within a system. It states that the internal energy ($U$) of a closed system changes only through heat ($Q$) added to the system and work ($W$) done on or by the system. Mathematically:

The internal energy, $U$, of a system is a measure of the total energy contained within, including kinetic and potential energy at the molecular level. Importantly, internal energy is a state function, meaning its value depends only on the current state of the system, not on the path the system took to reach that state.

Therefore, the change in a state function between two states is path-independent and depends only on the initial and final states. This property of internal energy makes it useful for analyzing systems in cyclic or reversible processes. 

For example, if a system returns to its initial state after any process, the change in internal energy over the cycle is zero, reinforcing that $U$ depends only on the state, not the path.

Enthalpy as a State Function

Enthalpy is another thermodynamic state function that is especially useful in processes occurring at constant pressure, such as many reactions in open containers. It is defined as:

Here, $H$ represents the heat content of the system at constant pressure, where $pV$ work is done by or on the system. Since $U$, $p$, and $V$ are all state functions, their combination ($H$) is also a state function.

In practical terms, enthalpy changes are directly measurable as the heat exchanged in reactions at constant pressure, making $H$ central to studying chemical reactions, phase changes, and other processes in open systems.

A central property of state fucntions is that they can be described by exact differentials, which allows for simiplified mathematical treatment and enable various useful relations.

Path Functions

Heat ($Q$) and Work ($W$)

Unlike state functions, heat ($Q$) and work ($W$) are path functions. This means that their values are dependent on the specific path taken to go from the initial to the final state. In other words, for the same change in internal energy, different amounts of heat and work can be involved depending on the process path.

Consider an example of a gas being compressed in two ways:

• Path 1: The gas is compressed slowly, allowing heat to escape so the temperature remains constant.
• Path 2: The gas is compressed quickly in an insulated container, so no heat escapes and the temperature increases.

Although the initial and final states (volume and pressure) are the same, the heat and work involved differ. Thus, $Q$ and $W$ are not state functions; they depend on the path of the process, meaning they do not have exact differentials. Moreover, unlike state functions, the integrals of work ($W$) and heat ($Q$) over a cyclic process are not zero, since they depend on the specific process path taken:

and

Exact Differentials and Changes in State Functions

In thermodynamics, state functions are described by exact differentials, which means the total change in the function only depends on the initial and final states. For example, the differential form of internal energy $U$ for a gas in terms of volume ($V$) and temperature ($T$) is:

Thus, the change in internal energy can be calculated by considering small changes in $V$ and $T$, regardless of the specific path. This property of state functions is valuable because it allows us to use tabulated values of $U$, $H$, and other state functions, avoiding the need to know the exact process path.

Let's illustrated the power of exact differentioals through the dependace of the properties of ideal gas on its state variables: the amount of substance ($n$), volume ($V$), temperature ($T$), and pressure ($p$). Since these variables are interdependent, we can often express one variable as a function of others. For example:

If $n$ is constant, then:

or equivalently, we can express:

• $T = f(p, V)$
• $V = f(T, p)$

These relationships allow us to use exact differentials to describe changes in state functions.

For example, consider a change in pressure $p$ as a function of $T$ and $V$. Using exact differentials, we have:

This equation represents how a small change in $p$ can be split into contributions from changes in $T$ (at constant $V$) and changes in $V$ (at constant $T$).

To derive useful relationships between partial derivatives, we can apply the chain rule. For instance:

This equation is a result of the chain rule and is very useful in thermodynamics. We can rearrange it to find various partial derivatives:

Using these relationships, we can define specific thermodynamic coefficients. Two important coefficients are:

• Isobaric Expansion Coefficient ($\alpha$): This coefficient describes the fractional change in volume per unit temperature change at constant pressure.

This coefficient is useful for understanding how a substance expands or contracts with temperature at constant pressure.

• Isothermal Compressibility ($\kappa_T$): This coefficient describes the fractional change in volume per unit change in pressure at constant temperature.

Using the definitions and relationships derived above, we arrive at an expression for $dp$ which shows how changes in pressure relate to changes in temperature and volume, depending on the thermodynamic properties of the system:

The use of exact differentials and thermodynamic coefficients like $\alpha$ and $\kappa_T$ enables us to quantify how properties such as pressure, volume, and temperature are interconnected. These tools are fundamental in thermodynamics for describing and predicting system behavior under various conditions.

Exact Differential of Internal Energy, $U$

The internal energy ($U$) of a system depends on the state variables. Assuming a constant amount of substance ($n = \text{constant}$), we can express the exact differential of $U$ as a function of volume ($V$) and temperature ($T$).

Here:

• ${\left( \frac{\partial U}{\partial T} \right)}_V$ represents the rate of change of internal energy with respect to temperature at constant volume.
• ${\left( \frac{\partial U}{\partial V} \right)}_T$ represents the rate of change of internal energy with respect to volume at constant temperature.

Utilizing the well-known relationship and dividing both sides by $dT$ at constant volume, we define the heat capacity at constant volume, $C_v$, as:

Therefore, the change in internal energy for a finite change in temperature at constant volume is:

This shows that for small temperature changes, the internal energy change is proportional to the temperature change at constant volume.

On the other hand, the term ${\left( \frac{\partial U}{\partial V} \right)}_T$ is sometimes called the internal pressure, denoted as $\pi_T$. It represents the rate of change of internal energy with respect to volume at a constant temperature, which reflects the internal interactions within the system such as those in non-ideal gasses.

Exact Differential of Enthalpy, $H$

In thermodynamics, enthalpy ($H$) is a useful state function when dealing with processes at constant pressure. We can express $H$ as a function of pressure ($p$) and temperature ($T$), making it convenient to analyze heat flow in open systems.

Since $H$ is a function of $p$ and $T$, the exact differential of $H$ is given by:

This equation describes how changes in temperature and pressure contribute to changes in enthalpy.

When pressure is constant ($dp = 0$), the differential simplifies to:

Thus, at constant pressure:

Under these conditions, the enthalpy change corresponds to the heat exchanged at constant pressure, $\Delta H = Q_p$

For cases where temperature and pressure vary, we can expand $dH$ using the first law of thermodynamics. At constant temperature ($T = \text{constant}$), the change in enthalpy can be expressed in terms of internal energy ($U$), volume ($V$), and pressure ($p$):

Substituting for $dU$ and using the definitions of heat capacity and internal pressure, we obtain:

At constant temperature ($dT = 0$), $\Delta H = 0$, hence:

dividng by $dp$ yields:

This equation highlights that the change in enthalpy with respect to pressure depends on both volume and the expansion properties of the system.

Relationship Between $c_p$ and $c_v$

The relationship between heat capacities at constant pressure ($c_p$) and constant volume ($c_v$) can also be updated using the exact differential for Internal Energy:

Starting with the first law of thermodynamics:

Rewriting in terms of exact differentials, we have:

Assuming p is constant and $c_v dT = \left( \frac{\partial U}{\partial T} \right)_V dT$ we get:

Simplifying further, we get:

Expressing this in terms of thermal expansion and compressibility coefficients, we get:

where \( \alpha \) is the coefficient of thermal expansion, given by:

and \( \kappa_T \) is the isothermal compressibility, given by:

Further, using the ideal gas equation we derive: 

which implies:

Since the molar volume of the gas \( V_{gas} \) is significantly greater than the molar volume of a solid \( V_{solid} \) or a liquid \( V_{liquid} \), the heat capacities are approximately equal: