Question: Natural uranium has a half life of 4.5 billion years. Representing with the expression $m(0.5)^{\frac{x}{(4.5\times10^9 \times 364)}}$, where $x$ is in days and $m$ is the initial mass of uranium. In one day, only $4.2 \times 10^{-11} \%$ decays into thorium. It would take around $2.4 \times 10^{13}$ atoms for one alpha particle to be generated. About $\pu{4E-11 mol}$. Assuming $\pu{5 MeV}$ per alpha particle. With $\pu{1.6021773E-13 J}$ in a $\pu{MeV}$. It would take $\pu{2.5E2 mol}$ of uranium or $\pu{6 kg}$ of uranium directly aimed at a point with theoretically $0$ loss of energy to achieve just a single Sievert. Then how is it possible that it is leaking $\pu{10 Sv}$ an hour at Fukushima reactor 1, an approximate of $240$ per day. That would be an equivalent to $1.4$ tons of pure uranium with $0$ theoretical power loss all directly channeled through retroreflectors and a focusing array all with 100% efficiency, channeling all the alpha decay into a single point onto the detector. I just can't wrap my head around this.

TLDR: How come nuclear power plants release so much radiation when uranium is so relatively stable?

Answer: 
The radioactivity associated with nuclear waste does not come from naturally occurring uranium, but from products associated with processing and using uranium. First off, uranium as it occurs in nature is not really nuclear fuel. It must be enriched in the more radioactive but more fissionable (and thus better energy source) uranium-235, leaving behind the more stable uranium-238, or uranium-238 is converted to more useful fuels such as plutonium-239.

Then after the fission process, what is left behind includes isotopes of strontium, caesium, and the aforementioned plutonium, which are the main contributors to the hazardous radioactivity of the waste.

From the [National Resources Council](https://www.nrc.gov/reading-rm/doc-collections/fact-sheets/radwaste.html):

> 
> During the fission process, two things happen to the uranium in the fuel. First, uranium atoms split, creating energy that is used to produce electricity. The fission creates radioactive isotopes of lighter elements such as cesium-137 and strontium-90. These isotopes, called "fission products," account for most of the heat and penetrating radiation in high-level waste. Second, some uranium atoms capture neutrons produced during fission. These atoms form heavier elements such as plutonium. These heavier-than-uranium, or "transuranic," elements do not produce nearly the amount of heat or penetrating radiation that fission products do, but they take much longer to decay. Transuranic wastes, sometimes called TRU, account for most of the radioactive hazard remaining in high-level waste after 1,000 years.
> 
> 
> Radioactive isotopes eventually decay, or disintegrate, to harmless materials. Some isotopes decay in hours or even minutes, but others decay very slowly. Strontium-90 and cesium-137 have half-lives of about 30 years (half the radioactivity will decay in 30 years). Plutonium-239 has a half-life of 24,000 years.
> 
> 
>
As said in the comments, the whole point of fission reactors is that you extract energy by turning uranium into something else and that "something else" is much more radioactive than uranium itself. The nuclear waste can be categorized according to their radioactivity:

* low-level waste (LLW)
* intermediate-level waste (ILW)
* high-level waste (HLW)

The vast majority of the waste (90% of total volume) is considered LLW which contributes to 1% of the total radioactivity. On the other hand, HLW accounts for 3% of the total volume of waste, but contains 95% of the total radioactivity.

[![enter image description here](https://i.stack.imgur.com/d9hXM.jpg)](https://i.stack.imgur.com/d9hXM.jpg)

The HLW is primarily both uranium fuel that has been used in a nuclear power reactor and is "spent" or no longer efficient in producing electricity (Spent fuel is thermally hot as well as highly radioactive and requires remote handling and shielding) and the fission products (why the fission product is radioactive is mentioned in Oscar's answer).

When the nucleus splits, most of the energy is released immediately and carried off by coolant to do useful work. This energy continues to be released for thousands of years after the atom splits. This delayed emissions mean that nuclear waste is still highly radioactive. Even the afterglow heat that is produced is what makes the waste hazardous.

[![enter image description here](https://i.stack.imgur.com/mUklC.png)](https://i.stack.imgur.com/mUklC.png)

The amount of HLW worldwide is currently increasing by about 12,000 tonnes every year. A 1000-megawatt nuclear power plant produces about 27 t of spent nuclear fuel (unreprocessed) every year.

**References**:

1. <https://en.wikipedia.org/wiki/Radioactive_waste>
2. <https://world-nuclear.org/nuclear-essentials/what-is-nuclear-waste-and-what-do-we-do-with-it.aspx>
3. <https://whatisnuclear.com/waste.html>
4. <https://world-nuclear.org/information-library/nuclear-fuel-cycle/nuclear-wastes/radioactive-wastes-myths-and-realities.aspx>