Basic math problems which you should be able to solve before the
start of CH162 Thermodynamics.
Dr.ir. Stefan A. F. Bon
1. The Universal Gas constant, R, has a value of 8.314 J mol-1 K-1. Show that the
following values hold when R is expressed in the units given in Table A.2 below:
2. In the Figure below we present four temperature scales: degrees Celcius/ ºC,
Kelvin/ K, degrees Fahrenheit/ ºF, and Rankine/ R.
(a) Express the temperature in degree Celcius, degrees Fahrenheit and Rankine
as temperature in Kelvin. (b) At what absolute temperature do the Celcius and
the Fahrenheit temperature scales give the same numerical value? What is this
value? (ANSWER: -40)
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3. From a mathematical viewpoint a critical point on a PV diagram (pressurevolume) should obey:
In which P and Vt stand for the total pressure and volume respectively. A
popular equation of state (PVT pressure-volume-temperature) is the Van der
Waals equation:
Proove that for the critical point we get:
Vct = 3nb, Tc = 8a/(27bR), Pc = a/(27b2)
4. Liquid/Vapor saturation pressures Psat is often represented as a function of
temperature by the following equation:
Here, parameters a, b, and c are substance-specific constants. Suppose it is
required to represent Psat by the equivalent equation:
Show how A, B, and C are related to a, b, and c respectively.
5. Using the Flory-Huggins lattice-based theory the free energy of mixing can be
calculated with the following expression:
Where NA and NB are the number of lattice sites occupied by molecules A and B,
A is the volume fraction that molecules A occupy in the lattice, and is the
Flory-Huggins interaction parameter.
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Demonstrate that:
When we equate this second derivative to zero we obtain an expression for the
so-called spinodal (boundary between metastable and unstable regions).
We can calculate the critical composition (critical mole fraction of compound A)
by equating the first derivative of this latter expression to zero:
Show that Acrit is:
6. The heat capacity can be written as a function of temperature. For example:
Proove that we can write the following integral as:
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7. For a three-component gas mixture at equilibrium holds:
The mole fractions can be expressed as a function of the reaction coordinate .
Verify that for K = 0.367 we find = 0.233.
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