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Propagation of Error

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A 100 uL (microLiter) sample of a 7.0 millimolar protein is diluted to 500.0mL. If the error in measurement of the molarity(M) is ±0.02 mM, of the uL pipet is ±1 uL, and of the volume of the 500 mL flask is ±0.15 mL, determine how the molarity of the resulting solution should be reported. That is, what is the molarity of the resulting solution, and what error is there in this measurement? Recall that the molarity M of a solution is given by M= n/V where n is the number of moles of solute, and V is the volume of the solution in liters. MinitVinit=MfinalVfinal


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I would start from arranging convenient notations:

Initial Volume V_{init} = 100x10^{-6} (+/-) 1x10^{-6} liters
Initial Molarity M_{init} = 7x10^{-3} (+/-) 0.02x10^{-3} molars
Final Volume V_{final} = 500x10^{-3} (+/-) 0.15x10^{-3} liters
Final Molarity M_{final} = ?

If we disregarded the errors, we would calculate

M_{final} = V_{init} * M_{init} / V_{final} = 1.4x10^{-6} molars = 1.4 micromolars

The simplest way to estimate error propagation in multiplications and divisions is to deal with RELATIVE errors.

If we write the true value of V_{init} as

V_{init} = 100x10^{-6} * (1 + ...