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Chemistry applications of Separable ODE

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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 2
أستاذ المادة فؤاد حمزة عبد الشريفي       11/10/2018 16:21:27
Chemistry applications of Separable ODE
1. Radioactive decay
Example 1: The rate of decomposition of radium is proportional to the amount present at any time. The half-life of radioactive radium is 1600 years . If a sample initially contains 50 gm, how long will it be until it contains 45 gm ?
Solution: Let be the amount of radium present at time in years.





We have
Thus
The half-life of radium is 1600 years it s meaning that










2. Chemical Reaction
Example 2: During a chemical reaction, substance A is converted into B at a rate that is proportional to the square of the amount of substance A. When , 60 grams of A are present and after 1 hour , only 10 grams of A remain unconverted. How much of A is present after 2 hours?
Solution: Let be the amount of unconverted substance A at any time .From the given assumption about the conversion rate, we can write the differential equation as shown.



To solve for the constants and , use the initial conditions. That is,
We can determine that . Similarly, it follows that

So, the particular solution is


Using the model, you can determine that the amount of unconverted substance A after 2 hours is




Radioactive decay
(1) The rate of decomposition of radioactive einsteinium is proportional to the amount
present at any time. The half-life of radioactive einsteinium is 276 days . After 100 days, 0.5 gram remains. What was the initial amount?
(2) The rate of decomposition of radioactive einsteinium is proportional to the amount
present at any time. The half-life of radioactive radium is 1600 years. If a sample initially contains 30 grams, how much the amount will remain after 250 years?
Chemical Reaction
In exercises 3 and 4 use the chemical reaction model described in example 5 to find the amount (in grams) as a function of (in hours).




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