The variance v is the number of degrees of freedom of a system consisting of r substances in m phases taking part in n chemical reactions. The variance v is obtained by subtracting n constraints from the number of degrees of freedom f determined by the Gibbs phase rule (6.62),v=f−n= r−m−n+2

Question

The variance v is the number of degrees of freedom of a system consisting of r substances in m phases taking part in n chemical reactions. The variance v is obtained by subtracting n constraints from the number of degrees of freedom f determined by the Gibbs phase rule (6.62),

f = 2 + m(r − 1) − r (m − 1) = r − m + 2 (6.62)

ν = f − n = r − m − n + 2

The pressure p and the temperature T are not fixed. Otherwise, there are additional constraints to fix p and T. Apply this concept to the following situation.

a) Determine the variance v of methane cracking described by the chemical reaction:

[latex] CH_4 (g) \rightleftarrows C(g) + 2 H_2 (g) [/latex].

b) A system consists of three phases and there is one chemical reaction between the substances. The variance is known to be two. Determine the number r of substances in the system.

c) A system is at a fixed temperature and consists of three phases. Its variance is known to be two and there are two chemical reactions between the substances. Determine the number r of substances in the system.

Accepted Answer

a) There are three substances [latex](CH_4, C[/latex] and [latex]H_2)[/latex], i.e. r = 3. There is one gaseous phase, i.e. m = 1. There is one chemical reaction, i.e. n = 1. Thus, the variance ν is,

ν = r − m − n + 2 = 3

b) There are three phases, i.e. m = 3. There is one chemical reaction, i.e. n = 1. The variance is known to be two, i.e. ν = 2. Thus, the number r of substances is,

r = ν + m + n − 2 = 4

c) There are three phases, i.e. m = 3. There are two chemical reactions, i.e. n = 2. The variance is known to be two, i.e. ν= 2. However, there is one additional constraint due to the fact that the temperature is fixed. In this particular case, there is one fewer degree of freedom. Thus, the variance is given by,

ν = r − m − n + 1

which implies that the number r of substances is,

r =ν + m + n − 1 = 6.

Question 8.4: The variance v is the number of degrees of freedom of a syst...

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