The First Law of Thermodynamics

The First Law of Thermodynamics is the foundation that links fundamental concepts such as heat, work, and internal energy, establishing that energy is neither created nor destroyed, only transformed. This material explores how this law applies to closed systems, delving into the analysis of thermodynamic work, heat capacities, and the statistical properties of gases. Through a combination of mathematical formulations and physical reasoning, you will discover essential tools for understanding energy processes in complex systems.

Learning Objectives:

By the end of this class, students will be able to:

1. Justify the First Law of Thermodynamics for closed systems, explaining the relationships between heat, work, and internal energy.
2. Analyze the concept of thermodynamic work in compression and expansion processes using differential formulas.
3. Calculate heat capacity under constant volume and pressure conditions, applying thermodynamic constraints.
4. Explain the Maxwell-Boltzmann distribution and the equipartition of energy principle in molecular systems.
5. Demonstrate specific relationships between heat capacities, the adiabatic index, and other thermodynamic properties for ideal gases.

TABLE OF CONTENTS:
Formulation of the First Law of Thermodynamics
Thermodynamic Work
Heat Capacity
Maxwell-Boltzmann Distribution and the Equipartition of Energy
Exercises

Formulation of the First Law of Thermodynamics

The First Law of Thermodynamics
states that:

Internal energy U is a state function because it has a well-defined value for each equilibrium state of the system. The internal energy of the system can be changed by applying heat Q or performing work W; however, work and heat are not state functions. This is because both depend on the process by which energy is added or removed, and once the process is completed, it is impossible to determine the exact amount of heat or work involved in achieving that equilibrium state.

The change in the internal energy of a system can be expressed as:

\Delta U = \Delta Q + \Delta W

Where \Delta Q is the amount of heat supplied, and \Delta W is the amount of work done on the system. By convention, \Delta Q is positive when heat is supplied to the system; if \Delta Q is negative, heat is being removed from the system. Similarly, \Delta W is positive for work done on the system and negative when the system does work on the surroundings.

The relationship between work, heat, and internal energy can also be expressed differentially as:

dU = \delta Q + \delta W.

Here, the symbol \delta is used to denote inexact differentials.

A thermally isolated system is defined as one that cannot exchange heat with its surroundings. In this case, dU = \delta W. This represents the First Law of Thermodynamics applied to an adiabatic system.

Heat Capacity

Let us now
aim to understand in greater detail how the internal energy of a system changes when heat is added. Generally, internal energy is a function of temperature and volume, which allows us to write U=U(T,V). Since energy is an exact differential, the change in U with respect to T and V can be expressed as:

\displaystyle dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV.

Now, using the relations dU=\delta Q + \delta W and \delta W=-PdV, we can reformulate the First Law of Thermodynamics as follows:

\begin{array}{rl} \delta Q &= dU + PdV\\ \\ & \displaystyle =\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV + PdV\\ \\ & \displaystyle =\left(\frac{\partial U}{\partial T}\right)_V dT + \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right]dV \\ \\ \displaystyle \frac{\delta Q}{dT} & \displaystyle =\left(\frac{\partial U}{\partial T}\right)_V + \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right]\frac{dV}{dT}. \end{array}

This is a general relation valid for any change in temperature and volume.

Using this result, we can determine the amount of heat required to produce a temperature change under specific constraints.

Constant Volume Constraint

To analyze what happens under constant volume, recall the definition of heat capacity at constant volume: C_V=(\partial Q/ \partial T)_V. By restricting the analysis to constant volume, we nullify the term dV/dT in the expression for \delta Q/dT. This leads to:

\displaystyle C_V = \left(\frac{\partial U}{\partial T} \right)_V.

Constant Pressure Constraint

If we keep the pressure constant, then:

\displaystyle C_p =\left(\frac{\partial Q}{\partial T}\right)_P=\left(\frac{\partial U}{\partial T}\right)_V + \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right]\left(\frac{\partial V}{\partial T}\right)_p.

Heat Capacity of a Monatomic Gas

When considering
a monatomic gas, the internal energy due to the kinetic energy of its particles is of the form \displaystyle U=\frac{3}{2}Nk_BT. This result is justified by the equipartition of energy principle, which can be studied from a statistical perspective of particle motion.

Maxwell-Boltzmann Distribution and the Equipartition of Energy

Since the energy of a system
is proportional to its Boltzmann Factor e^{-E/(k_BT)}, reasoning from this and considering that the kinetic energy of particles has the form \displaystyle E_{cin}=\frac{1}{2}mv^2, we can infer that the energy associated with particle motion projected along one of the three coordinate axes (let’s focus on the \hat{x} axis for now) will correspond to a velocity distribution g(v_x) proportional to e^{-mv_x^2/(2k_BT)}. In other words:

g(v_x)= A e^{-mv_x^2/(2k_BT)},

where A is a constant to be determined. Since g(v_x) is a distribution function, it must be normalized so that:

\displaystyle\int_{-\infty}^{+\infty} g(v_x)dv_x= 1.

A useful result for analyzing this situation is the Gaussian integral:

\displaystyle\int_{-\infty}^{+\infty} e^{-x^2}dx= \sqrt{\pi}.

From this, we deduce that:

\displaystyle 1= \int_{-\infty}^{+\infty} Ae^{\frac{-mv_x^2}{2k_BT}}dv_x= A\sqrt{\frac{\pi}{m/(2k_BT)}} = A\sqrt{\frac{2\pi k_BT}{m}}.

Thus:

\displaystyle g(v_x) = \sqrt{\frac{m}{2\pi k_BT}}e^{-mv_x^2/(2k_BT)}.

With this result, we can now calculate the average squared velocity projected along the \hat{x} axis, \left\lt v_x^2\right\gt. Its result is:

\displaystyle \left\lt v_x^2\right\gt = \int_{-\infty}^{+\infty} v_x^2 g(v_x) dv_x = \sqrt{\frac{m}{2\pi k_BT}} \int_{-\infty}^{+\infty} v_x^2 e^{-mv_x^2/(2k_BT)} = \frac{k_BT}{m} .

And since the root mean square velocity can be decomposed as \displaystyle \left\lt v^2\right\gt = \left\lt v_x^2\right\gt + \left\lt v_y^2\right\gt + \left\lt v_z^2\right\gt, and each component has the same development and result, the average kinetic energy of the system of particles can be written as:

\displaystyle \left\lt E_{cin}\right\gt =\frac{1}{2}m\left\lt v^2\right\gt = \frac{1}{2}m \cdot 3\frac{k_BT}{m}= \frac{3}{2}k_BT.

This is known as the “equipartition of energy principle.” From this, we can conclude that if the system consists of N particles with an average kinetic energy \displaystyle \left\lt E_{cin}\right\gt, and the system’s total energy is purely kinetic, then not only is the internal energy of the system \displaystyle U=3Nk_BT/2 (as predicted), but it also becomes evident that the internal energy only depends on the system’s temperature, which implies:

\displaystyle \left(\frac{\partial U}{\partial V}\right)_T = 0.

Development for the Ideal Gas

Now, recalling the ideal gas law
, PV=Nk_BT =nRT, solving for the volume gives:

\displaystyle V= \frac{nRT}{P}.

Therefore:

\displaystyle \left(\frac{\partial V}{\partial T} \right)_P = \frac{nR}{P}.

Using the expressions for C_V and C_P, we find:

\begin{array}{rl} C_P - C_V & \displaystyle = \left[\left(\frac{\partial U}{\partial V} \right)_T + P \right]\left(\frac{\partial V}{\partial T} \right)_P = P\cdot \frac{nR}{P} = nR \end{array}

Since \displaystyle C_V=(\partial U / \partial T)_V and U=3Nk_BT/2=3nRT/2, we have:

\displaystyle C_V = \frac{3}{2}nR

And therefore:

C_P = C_V + nR = \displaystyle \frac{3}{2}nR + nR = \frac{5}{2}nR

The Adiabatic Index

A commonly used quantity is the ratio of C_P to C_V, which is given a special name. The adiabatic index \gamma is defined as:

\gamma = \displaystyle \frac{C_P}{C_V}

For ideal gases, the adiabatic index has an exact value:

\gamma = \displaystyle \frac{5}{3}

Exercises

1. Is it always true that dU=C_VdT? Compare the general case with that of ideal gases and justify your answer.
2. Assuming that for an ideal gas it holds that U=C_VT, calculate:
   1. The internal energy per unit mass.
   2. The internal energy per unit volume.
3. One mole of a monatomic ideal gas is confined in a cylinder by a piston and kept at constant temperature T_0 through thermal contact with a reservoir. The gas is slowly expanded from a volume V_1 to another volume V_2, maintaining constant temperature throughout the process.
   1. Does the internal energy of the gas change?
   2. Calculate the work done by the gas and the heat flow into the gas.
4. Show that, for an ideal gas, the following relations hold:

   \displaystyle \frac{R}{C_V} = \gamma - 1

   \displaystyle \frac{R}{C_P} = \frac{\gamma - 1}{\gamma}

   Where C_V and C_P are molar heat capacities.