Experiment
Topics

Experiments
Ideal Gas
Real Gas
Heat Capacity

Heat Capacities

Many thermodynamic equations of state are mathematical expressions that relate one quantity in terms of another. For example, the difference between the constant pressure and constant volume heat capacities, Cp and Cv, can be related to the expansion coefficient (α) and isothermal compressability, (κT) through the following equation:

This equation, which is derived in Chapter 3, holds for all substances. An important aspect of developing thermodynamic equations of state, is to test them by performing experimental measurments. Devise an experiment (or series of experiments) using the Virtual Substance program to cofirm the validity of this equation.

Notes:

  1. Each quantity measured in this experiment is a derivative quantity, e.g. Cv=(∂U/∂T)V.To get Cv, for example, measure U at a series of T's near the temperature of interest, and find its slope by fitting the data in a program such as Excel. The range of temperatures should span from 90% to 110% of the temperature of interest, and to get reasonable measurments, at least 10 data points should be collected. You will need to measure 4 quantities: the Cp, Cv, the expansion coefficient, and the isothermal compressability.
  2. Perform measurements on Ar assuming first that it is an ideal gas and then using the Lennard-Jones model.
  3. Each group is responsible for measuring data at a different system density, i.e. volume. Find your group in the following table and perform measurments at the two Vm's indicated therin. We will then combine all of the data for the class to identify trends. Perform all measurements at 130 K.
  4. Use systems with 128 particles and periodic boundary conditions. Collect data using a 5 fs time step, and average data over at least 10000 steps. Make sure that your system is equilibrated before making measurments.
  5. The command script language can be used to automate data collection.
  6. For the ideal gas, you should get Cv, Cp, α, and κT values that are very close to the expected ones. If you do not, then you are doing something wrong.

Your report should contain the following elements:

  1. A discussion of your experimental protocols which describes how you go from the simulation output to your "measured quantities". You should present a coherent discussion. A coherent discussion will need to include figures of raw data, e.g. U(T), and a description of how you get from the raw data to the quantity of interest, e.g. Cv. You may want to include some equations.
  2. A presentation of your results. Again, your discussion should be coherent, Include a table summarizing the values needed to confirm the equation of state, and a comparison of the right and left-hand sides of this equation.

Your report should address the following questions:

  1. How does the quantity (Cp-Cv) depend upon system volume for the ideal gas and real gas? Provide a physical explanation for your observations?
  2. How would you expect the (Cp-Cv) curve to change if the "soft-sphere" potential was used instead of the Lennard-Jones potential?

Presentation matters. Be complete, yet concise. Rambling sentences, with no apparent point will not be looked upon favorably. Neither will reams of computer generated printout. Your entire report should be less than 3 pages, single spaced, 1” margins, 11 point font, with embedded figures (3”x3”).