The Critical Temperature

The Semenov theory provides an easy method for calculating the critical temperature T_critical. As can be seen from figure 2, the critical temperature occurs when the heat loss line is at a tangent to the heat gain curve . This means that the values of and must be equal. This gives a way of calculation T_critical this is shown below.

At the critical temperature the gradients will also be equal. Therefore differentiating the above equation with respect to temperature results in the following equation.

At the critical temperature for both the above equations to be true then the following must also be true.

For a large value of the activation energy E_A, the value of T_c and the value of T_a become very close and the following result can be stated.

This equation can now be rearranged to give the approximate equation for T_c

Equation 4 - Approximate value equation for the critical temperature

(All variables in this equation are defined in equation 1 and equation 2

The above equation is a good approximation for the calculation of the critical temperature and can be used in most cases. However in some cases it may be better to get a more exact value of the critical temperature. To do this, instead of assuming a large activation energy as before, we rearrange the equation into a quadratic form. This is shown below.

Using the quadratic roots equation the following derivation can be formed

This leads to the following 2 roots

The first root is not likely to occur because it results in very high values for the critical temperature, which will not be physically possible, generally. Therefore that root can be discarded. The second root, however, does provide a value for critical temperature, and it will give a far more accurate result than equation 4.

Examples and questions for both the approximate equation and the more exact value are presented at after the Frank-Kamenetskii section.

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