This is super confusing complex brain killing
A column packing for chromatography consists of a mixture of two types of particles. Assume that the particles in the batch being sampled are spherical with a radius of 0.5 mm. 36% of the particles appeared to be pink and are known to have a polymeric stationary phase attached. The average density of the mixture is 0.288 g cm-3. If 5.3126„b0.0003 g of the sample is weighed out, calculate,
(i) the number of particles in the sample
(ii) the % standard deviation in sampling the fraction of pink particles from this mass of the sample.
(iii) Which contributes the larger uncertainty to the procedure - weighing out the sample or the random error associated with sampling from the two-component mixture? Justify your answer with a calculation.
(iv) Assume that the % standard deviation in sampling is 8%. How many replicate samples of this packing material need to be taken so that there is 95% confidence that the mean is known to within „b6% of the true mean.
To solve this problem, we'll go step by step:
(i) To calculate the number of particles in the sample, we need to determine the mass of a single particle. Assuming the particles are spherical with a radius of 0.5 mm, we can calculate their volume using the formula V = (4/3) * π * r^3, where r is the radius.
V = (4/3) * π * (0.5 mm)^3 = 0.5236 mm^3
Next, we convert the volume to cm^3 since the density is given in g cm^-3:
V = 0.5236 mm^3 * (1 cm / 10 mm)^3 = 0.5236 cm^3
Now, we can calculate the mass of a single particle using the average density:
Mass = Density * Volume = 0.288 g cm^-3 * 0.5236 cm^3 = 0.1509 g
To find the number of particles in the sample, we divide the total mass of the sample by the mass of a single particle:
Number of particles = Mass of sample / Mass of a single particle
Number of particles = 5.3126 g / 0.1509 g ≈ 35.20
Therefore, the number of particles in the sample is approximately 35.20.
(ii) The % standard deviation in sampling the fraction of pink particles can be calculated using the formula:
% standard deviation = (Standard deviation / Mean) * 100
Given that the % standard deviation in sampling is 8%, we can set up the following equation:
8 = (Standard deviation / 36) * 100
Solving for the standard deviation:
Standard deviation = (8 * 36) / 100 = 2.88
Therefore, the standard deviation in sampling the fraction of pink particles is 2.88%.
(iii) To determine which contributes the larger uncertainty - weighing out the sample or the random error associated with sampling - we need to compare their respective standard deviations.
The standard deviation in weighing out the sample can be assumed to be very small, usually negligible. Let's assume it to be 0.001 g.
We can calculate the random error associated with sampling using the formula:
Random error = Standard deviation * √(Number of particles in the sample)
Random error = 2.88% * √(35.20) ≈ 0.3069%
Comparing the random error associated with sampling (0.3069%) to the standard deviation in weighing out the sample (0.001 g), it is clear that the random error associated with sampling is larger.
Therefore, the random error associated with sampling from the two-component mixture contributes the larger uncertainty to the procedure.
(iv) To calculate the number of replicate samples needed for 95% confidence that the mean is known to within 6% of the true mean, we can use the formula for sample size:
Sample size = (Z-value * Standard deviation / Desired precision)^2
Given that the % standard deviation in sampling is 8% and the desired precision is 6%, we can plug these values into the formula:
Sample size = (Z-value * 8 / 6)^2
To find the Z-value for 95% confidence, we can look it up in a standard normal distribution table. For 95% confidence, the Z-value is approximately 1.96.
Substituting the values:
Sample size = (1.96 * 8 / 6)^2 ≈ 10.272
Therefore, approximately 11 replicate samples of this packing material need to be taken to have 95% confidence that the mean is known to within 6% of the true mean.