Natural Sciences tutors

Cambridge offers biological sciences, physical sciences or a combination, followed by increasing specialisation. Bath provides another model. These examples are from the universities’ 2027-entry course pages; subject choices have prerequisites and restrictions, so check the course details before planning your study.

Here is a constructed example of the link between a physical idea and its mathematical model. An idealised radioactive sample starts with 800,000 undecayed parent nuclei and has a half-life of four minutes. The model predicts 400,000 remaining after four minutes. How many remain after eight minutes?

A tempting answer is zero: subtract another 400,000. But a half-life removes half of what remains, not the same number each time. Half of 400,000 is 200,000, so a better answer is: “After two half-lives, one quarter of the original parent nuclei remain.” These are modelled expected counts, rather than an exact prediction for every real sample.

For degree preparation, connect that reasoning to dN/dt = −λN. Here N is the number of undecayed parent nuclei, t is time, dN/dt is its rate of change, and λ is the positive decay constant. The minus sign says N decreases; it does not make the number of nuclei negative. With t in minutes, λ = ln(2)/4 ≈ 0.1733 per minute, where ln means the natural logarithm.

The solution is N(t) = N₀e^(−λt), where N₀ is the starting count and e is the base of the natural exponential. At t = 0 it gives N₀; at eight minutes it gives 800,000 × ¼ = 200,000. Because λ is measured per minute, λt has no units. Checking the starting value and units helps connect the symbols to the physical situation.

Try the next step: after twelve minutes, the model predicts 100,000 parent nuclei. If the starting count doubles to 1,600,000 while the half-life stays four minutes, the expected count after eight minutes doubles to 400,000. The fraction remaining is still one quarter.

Biological science also needs careful quantitative reasoning. Catalase speeds the breakdown of hydrogen peroxide into water and oxygen, so oxygen production can be used to follow the reaction. Consider these invented results for a data-reading exercise, using the early, approximately straight part of each oxygen-volume graph:

The 8 cm³ in condition A is an amount of product, not a rate. Estimate the initial rate from the gradient: change in volume divided by change in time. For A, 8 ÷ 20 = 0.4 cm³ per second; for B, 12 ÷ 20 = 0.6 cm³ per second. B’s estimated initial rate is 1.5 times A’s.

Suppose the two conditions represent different temperatures, with the same starting catalase and hydrogen peroxide concentrations, pH, reaction volume and collection method. The supported conclusion is that B had the higher estimated initial rate in this comparison. It does not establish the enzyme’s optimum temperature: two temperatures cannot locate the maximum. More temperatures, repeat measurements and consideration of measurement uncertainty would be needed.

A useful next task is to change the time interval and recalculate the gradients. Always carry the time unit into the answer, then state what the comparison does and does not show. This is practice in interpreting supplied data; practical work should follow your course’s supervision and instructions.

For Cambridge Natural Sciences 2027 entry, applicants must take the Engineering and Science Admissions Test (ESAT). The required modules are Mathematics 1 plus two chosen from Biology, Chemistry, Physics and Mathematics 2. If you are also applying to another course that uses ESAT, check its compulsory modules before making your choices.

ESAT tests the application of school science and mathematics. Preparation should therefore include using the knowledge in unfamiliar questions, alongside becoming familiar with the test’s format and question style. One useful approach is to attempt a provider practice question, explain why you chose your answer, and check whether an error came from the concept, algebra, units or reading the question. A tutor can then focus practice on that difficulty.

Start with the current specification and preparation material from UAT-UK. This Cambridge requirement should not be assumed for another Natural Sciences course: check the university’s instructions for your entry year, including any separate test registration.