Internal Energy and the First Law

The internal energy, \( U \), refers to the total energy contained within a system due to its molecular and atomic composition. This includes kinetic energy (translational, rotational, vibrational) and potential energy resulting from intermolecular forces.

 

The First Law of Thermodynamics

The first law of thermodynamics states that energy is conserved, meaning it cannot be created or destroyed, only transferred. The mathematical formulation of this is:

\[ \Delta U = Q + W \]

Where:

  • \( \Delta U \) is the change in internal energy.
  • \( Q \) is the heat added to the system (positive if heat is absorbed, negative if released).
  • \( W \) is the work done on the system (positive if work is done on the system, negative if done by the system).

Therefore, the internal energy of a system is changed by flow of heat and work across its boundary.

 

Work in Thermodynamic Processes

Work is any action that leads to transfer of energy across the boundary of a system (i.e. between the system and the surrpundings.). Mechanical work happens when a force causes displacement within the system in the direction of the force. For example:

Gravitational work, like lifting a book up to a higher shelf. The system (the book) gains potential energy due to the work done on it by lifting it against gravity. The change in $ \Delta W $ would depend on the book's mass and how far it is lifted.

\[ \Delta W = - mg \Delta h \]

where $ m $ is the mass of the object, $ g$ is the gravitational acceleration (typically $9.8 \, m/s^2 $, and $\Delta h $ represents the change in height of the object (for instance, when it's lifted or lowered).

Displacement work, like displacement from an initial point $x_i $ to a final point $ x_f $ in the direction of the applied force $ \vec{F} $.

\[ \Delta W = \int_{x_i}^{x_f} \vec{F} \cdot d\vec{x} \]

In thermodynamics, we we often deal with work related to expansion and compression of expanding gases. For example, a gas inside a piston, when heated, exerts a force on the piston, causing it to expand. The amount of work done by the gas can be calculated by integrating the pressure applied over the change in volume of the gas.

\[ dW = -p dV \]

where \( p \) is the pressure and \( dV \) is the change in volume. The negative sign indicates that work is done by the system during expansion.

 

Work characteristics:

  • Transitory: Work only appears during the change of state of the system. Once the state stops changing, no further work is done.
  • Net effect: The effect of work is a change in the internal energy (U) of both the system and the surroundings. Energy is either added to the system from the surroundings or transferred from the system to the surroundings.
  • Positive vs. Negative Work:
    • $ \Delta W > 0$: Positive work occurs when the surroundings do work on the system. In this case, energy is added to the system.
    • $\Delta W < 0 $: Negative work happens when the system does work on its surroundings, meaning energy leaves the system.
  • Work is measured or calculated by following changes in either the system or the surroundings. In thermodynamic systems, work is represented mathematically to quantify how much energy is transferred during a process.

 

Reversible and Irreversible Processes

Thermodynamic equilibrium is a lack of tendency to change the state of a system, resulting in an overall oveall rate of diffusion, chemical reactions, etc. of 0. We need to distinguish between the system and the surroounding each being in internal equilibirum (i.e. with itself), and between the system and the surrounding being in equlibirum with each other.

Thermodynamic process can are often discussed as either reversible or irreversible:

  • Reversible processes are quasi-static processes for which the rate of change of the macroscopic variabls (p, V, T) is very small. The system passes through a series of nearly indistinguishable and reversible states. Each step of this process is associated with equilibrium between the previous and the following states, such that allows reversibility of the process.
  • Irreversible processes are processes that occur abruptly (like a pulse of energy imput) with large change of the macroscopic variabls (p, V, T). Therefore, the system passes through trough state with large, irreversible steps.

In this regard, in thermodynamics we need to define process paths, which are a sequence of steps that the system undergoes. Whyle the initial and the final state of a process may be the same, the paths through which the process is guided can be very different with some of them being reversible and other being irreversible paths.

 

Work in a Reversible Process

Reversible processes are a theoretical concept of an ideal process that happens infinitelly slowly and with infinetely small steps, allowing the system to remail in equilibrimum at each step. This allows the system to adapt at each step, and thus results in the maximum amount of work. Theoretically, reversible processes are 100% efficient. Note: reversible processes are a theoretical concept and no real proces can be entirely reversible!

The work done by an ideal gas during reversible compression can be expressed through the displacement of a piston: 

\[ \Delta W_{\text{rev, compress}} = \int_{h_i}^{h_f} p_{\text{ext}} \, Adx \]

where $A$ is the area of the piston and $x$ is the change in hight $h$ of the of the piston. Thereby, the product of the area and the displacement is equivalent to the associate decrease (negative sign) in the volume $Adx = -dV$ for a compression process. Hence:

\[ \Delta W_{\text{rev, compress}} = - \int_{V_i}^{V_f} P_{\text{ext}} \, dV \]

Given that the process is reversible and carried with infinitely small steps at which the system equlibriates with its envirnoment, the external pressure equals the internal pressure ($ p_{\text{int}}  =  p_{\text{ext}} $): 

\[ \Delta W_{\text{rev, compress}} = - \int_{V_i}^{V_f} p_{\text{int}} \, dV \]

Assuming the gas follows the ideal gas law $ p = \frac{nRT}{V} $, the work can be calculated as: 

\[ \Delta W_{\text{rev, compress}} = - nRT \int_{V_i}^{V_f} \frac{dV}{V} \]

Solving this integral leads to a logarithmic dependence of the work on the ratio of the final and initial volumes: 

\[ \Delta W_{\text{rev, compress}} = - nRT \ln \frac{V_f}{V_i} \]

Since during compression $ V_f < V_i $, the result is positive, indicating that work is done on the system.

 

Similar derivation applies for the work done during reversible expansion. However, in this case our inital equation is 

\[ \Delta W_{\text{rev, compress}} = - \int_{h_i}^{h_f} p_{\text{ext}} \, Adx = - \int_{V_i}^{V_f} p_{\text{ext}} dV\]

The negative sign in this equation accounts for the loss of energy of the system (negative work) during the expansion process, here $Adx = +dV$.

Overall, the resulting formula for both expansion and compression work takes the same form. What differs is the wether the work will be negative or positive from the perspective of the system (the gas):

\[ W_{\text{rev}} = -nRT \ln \frac{V_f}{V_i} \]

 

Work in an Irreversible Process

Irreversible processes are associated with abrupt changes that do not allow equilibrium to be established between consecutive states, thereby preventing reversibility. During such a process there are finite, large differences in pressure, temperature, or other variables. There processes are quick and are common in real-world scenarious. The work done in such a process is less thant the work in a reversible process due to energy losses. Irreversible processes are significantly less efficient.

For a constant external pressure \( p_{\text{ext}} \), the work of such a processcan be calcualted based on the final (total) Volume change. This is given by:

\[ \Delta W_{\text{irr}} = - \int_{V_i}^{V_f} P_{\text{ext}} \, dV = -p_{\text{ext}} \Delta V = = -p_{\text{ext}} (V_f - V_i) \]

In real-world systems, most processes are irreversible due to friction, heat loss, and other factors. Reversible processes are useful as idealized benchmarks to measure the maximum possible efficiency.