petroleum is the naturally forming liquid from the ground we extract petrol (gasoline for US) from, petroleum.
As the below commentor mentioned petroleum is the same as crude oil, just at a different stage in its refinement
Diesel is extracted from the crude oil at a different point in its production
the difference between petrol and diesel in vehicle use is that petrol is ignited by a spark, where as diesel is ignited by compression
Petrolium is a collective term for fossil hydrocarbons. But I assume here that you mean petrol or gasoline as it is sometimes marketed as just petrolium. Hydrocarbons can have different lengths of the carbon chains in them. When we pump it out from the ground it is a mixture of both short and long chains. But a refinery can separate these and also chemically alter one into the other. Shorter hydrocarbon chains are lighter, they flow easier and combust easier. Longer hydrocarbon chains are heavier, flow much slower, lubricates better and is harder to combust. The lightest hydrocarbons are gasses like methane, propane, butane, etc. The heaviest is tar which for all practical purposes is a solid.
Petrol are lighter then diesel. It is easier to ignite petrol, you can for example ignite it with a match which is hard to do with diesel. And diesel can be used as a light lubricant which is done in diesel engines to lubricate the fuel pump and injectors. These fuels are used for different types of engines. A petrol engine use the Otto cycle which mixes air and fuel well before making it detonate with a spark. This is hard to do with diesel, a lot of sources will say it is impossible. So a Diesel cycle engine compress the air much higher to generate enough pressure and heat for diesel to self ignite. And then inject diesel into the cylinder under these conditions. This creates a very different combustion, it takes longer, the fuel is not well mixed with the air, and the pressures involved are much higher. The advantage is that diesel is much more energy dense meaning that a liter of diesel have more energy then a liter of petrol.
When you distill Petroleum (or crude oil), you create fuel oil. There are several different types of fuel oil, but the difference between diesel and something like gasoline is the weight. So when fracking, lighter fluids like gasoline and kerosene are removed first. Heavier fluids, like diesel, are removed later since they are less volatile.
Bonus points: You can run waste oil (say from a deep fryer) as a fuel source into an older model diesel truck. Being mindful to change out the fuel filter once in a while, but as added fun, your exhaust smells like french fries or hot wings.
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