the soil ph can have 2 different impacts on the growth of plant roots. First and most important is how it affects the concentration of nutrients present in the soil itself. This variable varies depending on the ph tendency, in particular, nutrients like phosphorous, potassium ,sulphur, calcium and magnesium tend to drastically decrease in more acidic ph conditions (<6.0), on the other hand iron, manganese, boron, copper and zinc tend to lack in alkaline enviroments (>7.5) [figure 1].
The importance of these types of nutrients varies depending on the species of plant, in this case radish tends to show higher concentrations of Potassium, Calcium, Magnesium, Copper, Manganese, Phosphorous and Sodium. These all tend to grow in more alkali or neutral soil conditions rather than acidic as showed in figure
1. Another variable affected by a change in ph is the growth of microorganisms in the soil specifically affecting their respiration rate and the PLFA (phospholipid fatty acids) concentration [figure 3], which consist in the main component of the cell membrane of most microbes, including the soil ones. In this case too the graph [figure 2] seems to show a straight directly proportional relationship between respiration rate and PLFA concentration and PH growth, meaning that microorganisms optimum ph conditions tend to be either neutral or slightly alkali, particuralrly for respiration the best fit line on the graph displays a steeper line with a bigger gradient.
As a hypothesis i would say that the best results in terms of mass gaining and length should be matched by the radish seeds growing in an alkali or neutral enviroment, rather than in an acidic one To revisit my hypothesis and understand why the base solution should provide better nutrients for the radish rather than the other ones it is very important to understand the concept of CEC (Cation Exchange Capacity), this property of the soil is defined as: “The total number of cations a soil can hold–or its total negative charge–is the soil’s cation exchange capacity” the capacity of the soil to contain these ions is measured in millequivalents per 100 grams of soil (meq/100g) the increase of this soil property is usually associated with an increase of fertility, the reason being that the higher the CEC the higher the maximum amount of nutrients (ions) [figure 4]the soil is capable of holding and more likely it is to improve its productivity.
The increase in CEC is usually associated to a decrease in pH (showed in the graph [figure 5]). This has been prooved by Dr Lloyd A. Peterson who carried out an experiment of soil acidification through the use of N fertilizers  (the main chemical in these compounds is Ammonia, NH3, which as it’s released in the soil is converted into ammonium nitrate by the bacteria, during this biological process, 3 positive hydrogen ions are released in the soil per ammonia molecule converted. The increase in the H+ ions concentration makes the soil acidic.) which eventually lead to a much higher CEC rate, which should theoretically improve the soil fertility itself, but, as a side effect of the acidification is a drastic decline registered for the exchangeable base cations particularly in the ions Ca2+, which suffered of – 31% exchangeability and a – 36% for Mg2+ actually worsened the fertility of the soil since the ECEC (effective cation exchange capacity, calculated by adding the exchangeable base cations and the exchangeable acidity) actually turned out to be negative, meaning that the relation between pH and the ECEC itself is actually directly proportional.
To conclude, in the case of radish especially, the ions suffering from base cation exchangeability decrease, which are magnesium and calcium make up a big part of the nutrients absorbed by the seeds (see the background paragraph) this causes the radish growth to be damaged by an acidification of the soil. This explains why roots growing in acidic conditions should display the worst results, while the ones living in alkali and neutral soils should grow longer and heavier, because the ECEC along with the nutrients concentration in the soil varies depending on the change in pH, in this case with a linear directly proportional rate, meaning that as pH increases (towards alkali) so does the base cations nutrients concentration in the soil and the plant growth benefits from it.