Question 2 (4 points) a) It is estimated that 20% of a certain radioactive substance decays in 30 hours. What is the half life of this substance? b) If tomato juice with a pH of 4.2 is 75 times more acidic as milk what is the pH of milk? (Recall that lower pH's are more acidic!)

**a) Half-life Calculation**<br /><br />* **Understanding Decay:** A 20% decay means 80% (or 0.80) of the substance remains after 30 hours. The general formula for radioactive decay is:<br /><br /> N(t) = N₀ * (1/2)^(t/T)<br /><br /> Where:<br /> * N(t) is the amount remaining after time t<br /> * N₀ is the initial amount<br /> * t is the elapsed time<br /> * T is the half-life<br /><br />* **Setting up the equation:** We can set up the equation using the information given:<br /><br /> 0.80 = (1/2)^(30/T)<br /><br />* **Solving for T (half-life):**<br /><br /> Take the logarithm of both sides (base 1/2 or base 10 – it doesn't matter as long as you're consistent):<br /><br /> log(0.80) = (30/T) * log(1/2)<br /><br /> T = 30 * log(1/2) / log(0.80)<br /><br /> T ≈ 93.19 hours<br /><br /> Therefore, the half-life of the substance is approximately 93.19 hours.<br /><br /><br />**b) pH of Milk Calculation**<br /><br />* **Understanding pH:** The pH scale is logarithmic. A difference of 1 pH unit represents a tenfold difference in acidity. Since the tomato juice is 75 times more acidic than milk, we need to find the pH difference that corresponds to this factor.<br /><br />* **Calculating the pH difference:**<br /><br /> 75 is not a perfect power of 10, so we'll use logarithms:<br /><br /> pH difference = log₁₀(75) <br /> pH difference ≈ 1.88<br /><br />* **Finding the pH of milk:** Since tomato juice is *more* acidic (lower pH), we *add* the pH difference to the pH of tomato juice to find the pH of milk:<br /><br /> pH of milk = pH of tomato juice + pH difference<br /> pH of milk = 4.2 + 1.88<br /> pH of milk ≈ 6.08<br /><br /> Therefore, the pH of milk is approximately 6.08.<br />