Question

Consider the development of the Sackur-Tetrode equation for the translational entropy of an ideal gas. a. Evaluate the constant \(C\) in the expression $$ \frac{Z_{r}}{N}=C \frac{T^{5 / 2} M^{3 / 2}}{P} \text {. } $$ where \(T\) is the temperature in \(\mathrm{K}, M\) the molecular weight, and \(P\) the pressure in bars. b. Hence, show that the translational contribution to the entropy is given by $$ \frac{s_{t}}{R}=\frac{5}{2} \ln T+\frac{3}{2} \ln M-\ln P-1.1516 . $$ c. The experimentally measured value of the entropy for vaporous neon at its boiling point \((27.1 \mathrm{~K})\) and a pressure of one bar is \(96.50 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\). To verify the theoretical formulation leading to the Sackur-Tetrode equation, compare your predicted value with the given experimental result. Hint: Keep this problem in mind for future calculations

Short answer

The constant C is derived by equating the partition function ratio to the furnished equation involving T, M, and P, then entropy per mole is calculated using the Sackur-Tetrode equation and compared with the experimental data for neon.

Step-by-step solution

Determine the Constant C

To find the value of the constant C in the given relation \[\frac{Z_r}{N}=C \frac{T^{5 / 2} M^{3 / 2}}{P}\]we will use the general gas constant R and Avogadro's number N_A. The partition function Z_r can be expressed in terms of V, T, and M and then must be divided by N, the number of molecules, and equated to the provided relation. The resulting equation can be solved for C.

Deduce the Translational Entropy

With the determined constant C, plug it into the relation for the entropy per molecule which involves a logarithmic function of the partition function, and then multiply by Avogadro's number to obtain the entropy per mole. Using Stirling's approximation for factorials, the logarithm of partition function per molecule becomes \[s = k\ln\left(C \frac{T^{5 / 2} M^{3 / 2}}{P}\right) + \text{other terms...}\]where 5/2 arises from differentiation with respect to temperature and 3/2 from molecular weight. The constant derived will be plugged in, and simplifying assuming ideal gas behavior will lead to the simplified Sackur-Tetrode equation for translational entropy.

Calculate Theoretical Entropy for Neon

Using the Sackur-Tetrode equation and the values for neon's temperature at boiling point (27.1 K), molecular weight, and pressure (1 bar), we can calculate the theoretical translational entropy \[s_t\] of neon at these conditions. The formula is: \[\frac{s_{t}}{R}=\frac{5}{2} \ln T+\frac{3}{2} \ln M-\ln P-1.1516 \]After calculating the translational entropy per mole, multiply by the gas constant R to achieve units of J/mol*K.

Compare Theoretical and Experimental Entropy Values

The final step is to compare the entropy calculated using the Sackur-Tetrode equation with the experimental value given for neon. This will validate the theoretical model if the values show reasonable agreement, taking into account experimental uncertainties and the assumptions made in the derivation of the Sackur-Tetrode equation.

Key concepts

Statistical Thermodynamics

Statistical thermodynamics is a branch of thermodynamics that merges statistical mechanics with classical thermodynamics. This field aims to explain and predict the macroscopic properties of systems from the microscopic interactions among their particles. It starts with the premise that, in thermodynamic systems, the large number of particles and their interactions lead to a statistical distribution of states.

For instance, when discussing the behavior of gases, statistical thermodynamics examines the various energy states that gas molecules can occupy, and how the population of those states influences the observable properties like pressure, volume, and temperature. The Sackur-Tetrode equation is a product of statistical thermodynamics and provides a way to connect these microscopic states to the macroscopic phenomenon of entropy. This tool allows us to better understand how the thermal energy is distributed among the particles of an ideal gas and the level of disorder or randomness within the system.

Translational Entropy

Translational entropy is a key concept when considering the entropy of a system, particularly gases, measuring the disorder associated with the position of particles. The more ways particles can be arranged without changing the overall energy, the higher the entropy. Translational entropy specifically refers to the freedom of particles to move or 'translate' throughout a space.

In the context of the Sackur-Tetrode equation, the translational entropy (denoted as \(s_t\)) is derived through statistical considerations by calculating the number of available microstates for the gas molecules. It involves fundamental constants, the temperature, molecular weight, and pressure of the gas. Since entropy is a measure of uncertainty or randomness, the translational entropy contributes significantly to our understanding of an ideal gas's overall entropy. The correct understanding of translational entropy allows us to predict how energy distributions change over the degrees of freedom that are available in the translational movement of particles.

Ideal Gas

The ideal gas is a theoretical construct used widely in thermodynamics and statistical mechanics. It assumes a collection of non-interacting particles that have no volume and undergo elastic collisions with each other and the walls of their container. This simplified model makes it possible to derive many useful equations that can describe gas behavior under various conditions.

An ideal gas obeys the Ideal Gas Law, \(PV=nRT\), where \(P\) stands for pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) represents temperature. The Sackur-Tetrode equation, which gives an expression for the translational entropy of an ideal gas, is based on the premise that the gas behaves ideally. Thus, it is important to remember that real gases only approximate ideal behavior under certain conditions, generally at high temperatures and low pressures, where interactions between the molecules are minimized.