5.9 – Thermo Cheat Sheet

5.9.0 – Learning Objectives

By the end of this section you should be able to:

  1. Have a list of helpful cases in thermodynamics.
  2. Understand what keywords signify in practice problems.
  3. Have an in-depth understanding to certain cases.

5.9.1 – Introduction

The first law of thermodynamics is used in chemical engineering very often. To help, it is important to know when simplifications of the first law can be used to help analyze problems. Recall the equation of the first law:

\[E_{tot} = Q - W\]

This notebook will cover cases when you can add simplifications and the rationale behind the simplifications.

5.9.2 – List of cases

Energy:

  1. If a problem says no change in height and no acceleration kinetic energy \(E_k\) and potential energy \(E_p\) are 0. These are safe assumptions for chemical reactors.
  2. An ideal gas undergoing an isothermal reaction has \(U = 0\).
  3. An ideal gas’ change in internal energy \(\Delta U\) can be written as \(nC_vdT\).
  4. \(\Delta U\) alone, can only be used in a closed system.
  5. Open systems require enthalpy instead of internal energy.
  6. An ideal gas’ change in enthalpy \(\Delta H\) can be written as \(nC_pdT\).

Heat:

  1. If the temperature of the system is equal to the surrounding or is perfectly insulated, the system is adiabatic.
  2. An adiabatic system has \(\Delta Q = 0\)

Work:

  1. If there is no shaft work, no flow work and no electrical generation, work = \(P\Delta V\)

  2. In ideal, isothermal systems \(P\Delta V\) turns into \(nRT \cdot ln \Big( \frac{V_f}{V_i} \Big)\) by the ideal gas law

  3. In ideal, adiabatic systems Work is

    \[\frac{P V^\gamma \Big(V_f^{1-\gamma}-V_i^{1-\gamma} \Big)}{1-\gamma}\]

    where \(\gamma= \frac{C_p}{C_v}\) and \(PV^\gamma\) is often referred as \(K\) a constant.

5.9.3 – Explanation of certain cases

Energy:

  1. Self explanatory
  2. See 3
  3. Recall that internal energy is the vibrational energy held on a molecular level. An ideal gas exhibits elastic collisions by collision theory and have no interactions in between other gas molecules. As temperature rises, the amount of collisions increases proportionally thereby directly raising the internal energy. Here is more help understanding the idea.
  4. Assume a closed system where there is no work done by the system (\(U = Q\)). The key for an open system is that the volume in an open system changes and pressure remains constant with the surrounding. This contradicts the definition of \(U = nC_vdT\) where \(C_v\) is the heat capacity at constant volume. This article also explains the difference.
  5. Enthalpy is the correction for the open system that is also explained in the article above.
  6. Self explanatory.

Heat:

  1. Self explanatory.
  2. Self explanatory.

Work:

  1. \(W_{net} = W_{flow}+W_{shaft}+W_{electrical} + p\Delta V\) This will collapse to the final term.
  2. The pressure term in work \(P\Delta V\) is exchanged for \(\frac{nRT}{V}\) and then integrated where the limits of the integrand are \(V_f\) and \(V_i\).
  3. There is a long and complicated derivation for this equation that was covered in first year physics. It is better to just memorize this fact. Here is a visual representative from hyperphysics.
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