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A new comet (magnitude 18.8) has been discovered beyond the orbit of Jupiter. enter image description here

Comet ISON will get within 0.012 AU of the Sun by the end of November 2013 and ~0.4 AU from of Earth early in January 2014. It may reach very welcome negative magnitudes at the end of November 2013. Forecasts of comet magnitudes have been disappointing in the past. It's clear that comet magnitudes depend on the quantity of dust and ices that are ablated by the Sun's energy, and therefore nucleus size and distance from the Sun and Earth.

My question is, to what extent is a comet's magnitude dependent on the ratio of nucleus dust to ice and on the type of ices?

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This doesn't answer your question about dust and ice, but here's an article about it in Astronomy Now. It suggests that it might reach a magnitude of -16; compare that to -12.92, the maximum brightness of the full Moon. (As with Kohoutek, prepare to be disappointed.) – Keith Thompson Sep 26 '12 at 22:39
You might also be interested in work such as this one: annualreviews.org/doi/full/10.1146/annurev-astro-081309-130811 where the authors discuss a possible taxonomy of comets based on their birthplace. Knowing where a comet was formed gives clues on its likely composition. Empirically one might draw some connections between the composition of known comets and their magnitude. One might then attempt to predict the composition of C/2012 S1 and thence its likely magnitude. There are many uncertainties associated with this however which are likely to render such a scheme unworkable. – Nicholas Sep 28 '12 at 10:01

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The magnitude of any object which doesn't have its own light is dependent on it's albedo i.e. $$\frac{Light \ reflected}{Light \ Received}$$

So, diffrent types of ice have different albedos, so different magnitude. Though, I can't find a relation between ratio of nucleus dust and ice to the comet's magnitude, the albedo/magnitude can be used to find the composition of the surface of comet(it's not pure ice, is it?)

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