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Calculate the equilibrium constant and estimate the error in it. In a keto-enol equilibrium measurement the fraction of enol at equilibrium is found to be 0.200 with a standard deviation of 0.004. Which can also be rewritten in a convenient form using relative errors: Where A, E and R are all constant for a reaction.Īpplying the propagation of error formula The temperature dependence of a rate constant is given by the Arrhenius equation:
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How is a rate constant affected by random errors in the temperature? Note that the 1% error in r has become a 3% error in V, in line with our rule of thumb. Calculate its volume and estimate the error.
#Arrhenius equation calculator series#
(It does to this first order approximation.) Remember that we have truncated the Taylor series after one term, and so this procedure will only work if the standard deviations are small enough that this truncation is a good approximation.Ī spherical water droplet has a radius measured to be 3.00 mm with an estimated standard deviation of 0.03 mm. So squaring and taking expectations (see previous tutorial),Īnd finally taking the square root we get a simple relationship between the standard deviation of x and the standard deviation of y:Īlthough we have proved this relationship for the true (underlying) standard deviations we assume that it also applies to the estimated standard deviations. The problem is that we do not know the actual error in x, but we can estimate its variance, which is the expectation of the squared error. The situation is particularly simple if the Taylor expansion can be truncated after the first correction term. This is an equation that tells us how the actual error in the measured x translates through the function to y. Now we take a Taylor expansion of the function f about the true value, Now X - m x = D X (the deviation of X from the theoretical mean), and similarly Y - m y = D Y.
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So we make an estimate of y, Y, from Y = f( X). The problem is that we do not know m x, we only know an experimental estimate of it, X. Of course this may not be true, a real statistician would want to know how the function transforms the underlying distribution to give a new distribution for Y, however if the distributions are narrow this is not a bad starting point. We assume that the true values obey the equation in statistical terms the true values are the means of the underlying distributions. A simple example might be the calculation of the volume of a cube after measuring one of its sides, or the calculation of the equilibrium constant for an isomerisation reaction from a measurement of the proportion of starting material isomerised at equilibrium. A measurement X of the quantity x is made, but the value of y is required. Suppose y and x are related through y = f( x). A related question is: how does uncertainty in the conditions affect a measurement (for example how do fluctuations in the temperature affect a rate constant measurement)? This section applies statistical methods to work out how errors in measured quantities affect the results of calculations. These are Taylor expansions, partial differentiation, and functions of several variables. Level 1 (gold) - this material needs some prerequisites that are covered in the first year mathematics for chemists course.