Warm-Up Algebra 2 Honors 2/22/18 Simplify Ln e40x 2. Ln e(x+4) 3. ln e 2x + ln ex 4. Try solving: ey=ex+3 3 minutes to do 2 minutes to go over 5 total

The half-life of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of decay. Natural decay is modeled by the function below. 2 minutes 7 total

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g? Step 1 Find the decay constant for plutonium-239. N(t) = N0e–kt Use the natural decay function. 5 minutes 12 minutes 𝑁 0 2 = N0e–k(24,110) Substitute 24,110 for t and 𝑁 0 2 for N(t) because half of the initial quantity will remain.

ln 1 2 =ln 𝑒 −24110𝑘 ln 2 −1 = -24110k − ln 2 = -24110k Simplify and take ln of both sides ln 2 −1 = -24110k Write ½ as 2-1, and simplify the right side − ln 2 = -24110k ln2–1 = –1ln 2 = –ln 2 5 minutes 17 minutes k= 𝑙𝑛2 24110 Solve for k k≈0.000029

How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g? Step 2 Write the decay function and solve for t. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g? N(t) = N0e–0.000029t Substitute 0.000029 for k. Substitute 1 for N0 and 0.01 for N(t). 0.1 = 1e–0.000029t Take ln of both sides. ln 0.1 = ln e–0.000029t Simplify. ln 0.1 = –0.000029t 5 minutes 22 total 𝑡= 𝑙𝑛0.1 −0.000029 ≈80,000 It takes approximately 80,000 years to decay.

Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg. Step 1 Find the decay constant for Chromium-51. N(t) = N0e–kt Use the natural decay function. 3 minutes 25 minutes 650 2 = 650e–k(28) Substitute 24,110 for t and 𝑁 0 2 for N(t) because half of the initial quantity will remain.

ln 1 2 =ln 𝑒 −28𝑘 ln 2 −1 = -28k − ln 2 = -28k ln2–1 = –1ln 2 = –ln 2 Simplify and take ln of both sides ln 2 −1 = -28k Write ½ as 2-1, and simplify the right side − ln 2 = -28k ln2–1 = –1ln 2 = –ln 2 5 minutes 30 total k= 𝑙𝑛2 28 Solve for k k≈0.02475525

How long does it take for 650 mg of chromium-51 to decay to 200 mg. Step 2 Write the decay function and solve for t. How long does it take for 650 mg of chromium-51 to decay to 200 mg. N(t) = N0e–0.02475525t Substitute 0.000029 for k. Substitute 650 for N0 and 200 for N(t). 200= 650e–0.02475525t 5 minutes 35 minutes 200 650 = e–0.02475525t Divide by 650 on both sides

𝑙𝑛 200 650 = ln 𝑒 −0.02475525𝑡 Take ln of both sides. ln 200 650 = –0.02475525t Simplify. 5 minutes 40 minutes 𝑡= 𝑙𝑛 200 650 −0.02475525 ≈47 It takes approximately 47 days to decay.

Solutions with certain pH’s are classified by: less than 7 are acidic Chemists define the acidity or alkalinity of a substance according to the formula  "pH = –log[H+]"  where [H+] is the hydrogen ion concentration, measured in moles per liter. Solutions with certain pH’s are classified by: less than 7 are acidic greater than 7 are basic a pH of 7 (such as pure water) are neutral. 3 minutes 43 minutes

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This is less than 7 so it’s acidic Suppose that you test apple juice and find that the hydrogen ion concentration is [H+] = 0.0003. Find the pH value and determine whether the juice is basic or acidic. pH = –log[H+] pH = –log(.0003) = 3.52287874528... This is less than 7 so it’s acidic 3 minutes 48 minutes

This is greater than 7 so it’s basic You test some ammonia and determine the hydrogen ion concentration to be [H+] = 1.3 × 10–9. Find the pH value and determine whether the ammonia is basic or acidic. pH = –log[H+] pH = –log(1.3 × 10–9) = 8.88605664769… This is greater than 7 so it’s basic 2 minutes 50 minutes