How do Becquerels relate to Grays? I understand that becquerels are SI units for curies, and that curies are a measurement of radioactive decays per second. I understand that gray is the measure of uncharged ionizing rays in a sample matter. What I need to know is under what conditions are ionizing rays emitted. I need this information to convert becquerels to grays. A little sample of this problem is the statistics on the Fukushima disaster only provide information on the fallout in terms of becquerels, including the radioactive isotopes involved; for the assignment that led up to this question, i need statistics on the grays released during the incident. Really the only common ground these two have is radioactive decay, so, if i can get the conditions under which radioactive decay emits ionizing radiation, i can determine an approximation amount of grays in relation to fukushima.
 A: There is no direct relationship, because they measure two different things: The becquerel is a unit for measuring the rate of radioactive decays occurring within a sample of radioactive material - that is, it is a characteristic of a radioactive source. One becquerel refers to an average of one decay per second. Note here that this rate of decay is the total number of nuclei decaying in that one (averaged) second, and not simply the propensity of any individual nucleus to decay: the bigger the material sample, the more nuclei there are to decay for any given propensity, and hence it will rate with more becquerels, just as much as will considering a sample of material whose nuclei have a higher propensity to decay.
Now obviously, since a rate of 1 Bq is very small when you consider how many atoms, and hence atomic nuclei, there are in a typical macroscopic piece of material (on the order of Avogadro's number), we typically use larger versions of the unit such as megabecquerels (MBq) or gigabecquerels (GBq), this last being an average of one billion decays per second within the whole piece of material under consideration. One "curie" is now defined as exactly 37 GBq, and hence also the curie is likewise a unit of rate of radioactive decay.
The unit gray (Gy) is a unit of ionizing radiation energy absorption into a piece of matter - most typically thought of in the context of biological/medical effects of radiation where that the absorbing matter is living tissue, hence this is typically called "dose". One gray corresponds to absorption of one joule (1 J) of ionizing radiation energy into one kilogram of matter.
The two are quite obviously related, but not directly: a radioactive source that undergoes more radioactive decays will pump out more ionizing radiation energy, and hence you can accumulate more Gy standing in front of it than if you were standing in front of a source whose material was one in which fewer decays were occurring. But that is not the only factor: since what is measured in grays involves energy, another important factor is the energy release per each decay, which combined with the rate of decay to give the radioactive power of the source. This is measured in the perhaps more familiar unit of watts (W), just as any other such source. In particular, the radiation power is
$$P_\mathrm{rad} = A \cdot E_\mathrm{decay}$$
where $A$ is the activity, or rate of decays, measured in becquerels, and $E_\mathrm{decay}$ is the amount of energy released by a single atomic nucleus decaying. Typically, these energies are extremely tiny, on the order of picojoules ($10^{-12}$ joules). As a very simple example, if a source has a decay energy of 1 pJ, and an activity level of 1 GBq, then the radiation power generated by that source is 0.001 W or 1 mW (milliwatt).
To get the actual dose, you'd need to know how much of the source's radioactive power is actually absorbed by a target. The same rules apply here as for any other energy absorption problem, such as, say, dealing with thermal radiation, RF (radio-frequency) EM radiation, or sound (acoustic) radiation: if you can treat the source as a point source, you can find the intensity (power per unit area) by the usual inverse square law, and if you know the cross-section that the target being attacked presents to the source, and of course the exposure time, you can then find how much energy it receives, which combined with knowing the absorption characteristics and the target mass, can be used to estimate how many Gy will it absorb. In a realistic, complicated setup, you'd likely just have to measure.
A: Grays is the amount of energy absorbed by something (per mass). Becquerels is a measure of radioactivity - how many decays are there per unit time. When an atom decays it always emits ionizing radiation (It may be $\alpha$ or $\beta$ and most of the time it comes with $\gamma$).
How much energy got absorbed by another material or human being depends on a lot of parameters. For example, how close is the material to the radioactive substance? For how much time did the exposure take place? Geometry considerations and so on and so on...
It isn't direct to tell from one the other, but for specific cases you can take some assumptions and calculate.
