### Mathematical Issues for Chemists

A brief outline of the mathematical issues faced by chemistry students.

### Reaction Rates

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

### Catalyse That!

Can you work out how to produce the right amount of chemical in a temperature-dependent reaction?

# Gassy Information

##### Stage: 5 Challenge Level:

1) If we do not know whether the alcohol is non-cyclic or cyclic (or whether double bonds are present) this problem is much harder to analyse, as even initial equations cannot be set up. Assuming that the alcohol is non-cyclic and contains only single bonds, we can write the general equation:

$xC_zH_{2z+2}O + x\frac{3z}{2}O_2 \rightarrow xzCO_2 + x(z+1)H_2O$
where $x$ is the number of original moles of the alcohol, and $z$ is the number of carbon atoms that it contains.

From this equation, we can see that the change in number of gaseous moles is:

$\Delta n = x(2z +1) - x(\frac{3z}{2} +1) = \frac{xz}{2}$

By measuring the change in volume, we can use the Ideal gas equation:

$p\Delta V = \Delta n RT$ where pressure and temperature are constant.

This can be rearranged to make the change in moles the subject of the equation:

$\Delta n = \frac{p \Delta V}{RT}$

Equating this to the previous equation involving $\Delta n$ such that this is eliminated, gives:

$\frac{p \Delta V}{RT} = \frac{xz}{2}$

$\therefore z = \frac{ 2p \Delta V}{xRT}$

From this equation, it can be seen that unless the pressure is also known, as well as the number of initial moles of alcohol, the formula of the alcohol cannot be calculated. It can be seen also that if the alcohol were cyclic or had double/triple bonds, these conditions would still need to be satisfied, as so it can be said that in general the formula of the alcohol cannot be calculated.

2) This problem can be again analysed more easily if it clear whether the hydrocarbon is cyclic or not. Assuming initially that it is not:

$xC_zH_{2z+2} + x\left(\frac{3z+1}{2}\right)O_2 \rightarrow xzCO_2 + x(z+1)H_2O$

The change in number of gaseous moles is:

$\Delta n = x\left(\frac{z-1}{2}\right)$

By measuring also the volume change, the Ideal Gas equation can also be used:

$p\Delta V = \Delta nRT$
$\Delta n = \frac{p \Delta V}{RT}$

Equating and rearranging:

$z = \frac{ 2\Delta Vp +RTx}{RTx}$

The gas is now cooled, and the second change in volume, $\Delta V_2$ is measured against a change in temperature, $\Delta T_2$:

$p\Delta V_2 = (x(2z+1))R\Delta T_2$

since the number of moles is given by $x(2z+1)$.

Rearranging to make $x$ the subject:

$x = \frac{p\Delta V_2}{(2z+1)R\Delta T_2}$

Using the equation above to substitute into the equation for z, gives:

$z = \frac{2p\Delta V + \frac{RTp\Delta V_2}{(2z+1)R\Delta T_2}}{\frac{RTp\Delta V_2}{(2z+1)R\Delta T_2}}$

Much rearranging to make z the subject gives:

$z = \frac{2\Delta V \Delta T_2 + T\Delta V_2}{T\Delta V_2 - 4\Delta V \Delta T_2}$

As can be seen, z is expressed fully in terms of parameters which are known. It is therefore possible to calculate what the molecular formula of the hydrocarbon is, provided it is known whether the hydrocarbon is cyclic or not, or how many double/tripled bonds there are. If these are unknown, it is not easily possible to determine what the molecular formula is: however, one method might be to construct equations giving z for a varying number of double and triple bonds, and then seeing which equation gave an integer value for z.

3)It is possible here to set up two simultaneous equations from the Ideal Gas equation:

$pV_1 = nRT_1$ and $pV_2 = nRT_2$

The pressure is unknown, but fixed at the same value for both scenarios. It can therefore be eliminated by dividing the equation by each other:

$\frac {V_1}{V_2} = \frac{T_1}{T_2}$

However, this has also divided through by the number of moles: thus it can be seen that the number of moles is undetermined by this scenario, and so it is never possible to find out its value.

4) In this scenario, the molecular formula of the hydrocarbon is know. For the sake of an example, let us choose octane, $C_8H_{18}$. Since the initial volume of gas of the hydrocarbon is known, provided that the pressure and temperature are known, the number of initial moles can be calculated from the ideal gas equation. Thus, consider the general incomplete combustion of one mole of octane:

$C_8H_{18} + \left(\frac{25 +y}{2}\right)O_2 \rightarrow 9H_20 + yCO + (8-y)CO_2$

If the change in overall volume is known, then the change in total numbers of gaseous moles is known (from the Ideal Gas equation). Since the final number of gaseous moles is always the same (and known), as governed by the formula of the combusted hydrocarbon (eg. for Octane this is 8+9 = 17 moles), the original number of gaseous moles can be calculated by subtracting the change in moles from the final number of moles. Since the initial number of moles of hydrocarbon are known, this means that the original number of moles of oxygen can also be calculated too by subtracting the number of moles of hydrocarbon from the overall number of initial moles.

Although we have only considered a specific amount of a specific hydrocarbon, it can be seen that this approach can be generalised to all other hydrocarbons. A known volume of hydrocarbon will always give the number of moles of that hydrocarbon; a known number of moles of a known hydrocarbon will always combust to give the same number of moles of product. Thus, for all hydrocarbons, it is possible to calculate the original volume of oxygen.