Finding coefficients for chemical reactions is often complicated, especially in the case of redox processes.
After understanding the exponential functionour next target is the natural logarithm. The natural log gives you the time needed to reach a certain level of growth.
Read more about e. Natural Log ln is the amount of time needed to reach a certain level of continuous growth Not too bad, right? E is About Growth The number e is about continuous growth. Natural Log is About Time The natural log is the inverse of e, a fancy term for opposite. Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln.
Now what does this inverse or opposite stuff mean? After 3 units of time, we end up with If we want growth of The natural log gives us the time needed to hit our desired growth. What is ln 1? Intuitively, the question is: How long do I wait to get 1x my current amount?
If we reverse it i. If we go backwards. In general, you can flip the fraction and take the negative: This means if we go back 1.
Ok, how about the natural log of a negative number? Logarithmic Multiplication is Mighty Fun How long does it take to grow 4x your current amount? Sure, we could just use ln 4. We can consider 4x growth as doubling taking ln 2 units of time and then doubling again taking another ln 2 units of time: Any growth number, like 20, can be considered 2x growth followed by 10x growth.
Or 4x growth followed by 5x growth. Or 3x growth followed by 6. This relationship makes sense when you think in terms of time to grow. If we want to grow 30x, we can wait ln 30 all at once, or simply wait ln 3to triple, then wait ln 10to grow 10x again. The net effect is the same, so the net time should be the same too and it is.
Well, growing 5 times is ln 5. Suppose we want 30x growth: We can consider the equation to be: If I double the rate of growth, I halve the time needed. The natural log can be used with any interest rate or time as long as their product is the same.
You can wiggle the variables all you want.
The Rule of 72 The Rule of 72 is a mental math shortcut to estimate the time needed to double your money. Now the question is easy: The Rule of 72 is useful for interest rates, population growth, bacteria cultures, and anything that grows exponentially.
Where to from here? I hope the natural log makes more sense — it tells you the time needed for any amount of exponential growth. The Natural Log of E Quick quiz: The math robot says:The HASPI Curriculum Resources are available free for use by educators.
All of the resources align with the Next Generation Science Standards (NGSS) and Common Core State Standards (CCSS). e IONIC EQUATIONS (for higher GCSE and AS students).
What is an 'ionic equation'? How do we construct and write ionic equations? In many reactions only certain ions change their 'chemical state' but other ions remain in exactly the same original physical and chemical state.
A good example is the reaction between hydrogen and fluorine in which hydrogen is being oxidized and fluorine is being reduced. H 2 + F 2 → 2 HF. We can write this overall reaction as two half-reactions. the oxidation reaction: H 2 → 2 H + + 2 e −. and the reduction reaction: F 2 + 2 e − → 2 F −.
Analyzing each half-reaction in isolation can often make the overall chemical. Nov 07, · Best Answer: In solving redox reactions (using ion-electron method), remember that we add H+ to balance the H and H2O to balance the O.
That's for those redox reactions that occur in acidic solution. For basic solutions, add OH- to balance H and H2O to balance O.
If none is specified in the problem, assume acidic initiativeblog.com: Resolved. Chemistry or Mathematics? Finding coefficients for chemical reactions is often complicated, especially in the case of redox processes.
The classical approach uses oxidation numbers and many chemists cannot imagine any way to find the solution other than balancing half-reactions for the process. Few areas of business attract as much attention as new ventures, and few aspects of new-venture creation attract as much attention as the business plan.