Physics Kinetic Theory of Gases question

1) Write an equation for the relationship between the temperature (T) in kelvins and the average molecular speed (v_avg) in a gas (which may contain a numerical constant).

2) Describe the relationship in words between the Temperature (T) and the average molecular speed (v_avg). (i.e. linear, inverse, squared, no relationship at all, etc.)

Any help and explanations you can provide would be awesome! Thanks!

I think all that is really required is:
1) T = A(v_avg)^2 where A is a constant
2) Temperature is proportional to v_avg squared.

This follows from the definition of thermodynamic temperature, T: T is proportional to the average kinetic energy of the particles, which in turn isproportional to v_avg squared.
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But here is a more detailed analysis to answer you concerns.

1)The answer to question 1 is a bit ambiguous.
Often the rms speed is used synonymously with average speed. This is technically incorrect. However it gives (assuming we are dealing with an ideal gas):
(3/2)kT = (1/2)m(v_avg)^2 where m is mass of a single particle (molecule)

This can be reduced to an equation with a single numerical constant, A, say:
T = A(v_avg)^2
where A = m/(3k)

The true average speed depends on the speed distribution (Maxwell-Boltzmann) and is given by
v_avg = sqrt(8RT/(πM)) (M being molar mass) (e.g. see just before halfway down the link)

Introducing a single numerical constant, B, this would become:
T = B(v_avg)^2
where B = πM/(8R) (= πm/8k)

However, v_rms is the ‘best’ value to use, as T is defined in terms of average kinetic energy; and average kinetic energy = (1/2)m(v_rms)^2.

Note that you CANNOT have a relationship between v_avg (however defined) and T unless the constant (A or B) incorporates molar mass or the mass of a molecule.
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2) The relationship is that (for a given gas):