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.
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A: Because it had separation anxiety!
(i) To calculate the number of particles in the sample, we first need to find the volume of the sample. The sample mass is given as 5.3126±0.0003 g, and the average density is 0.288 g cm^-3. Using the formula density = mass/volume, we can rearrange it to find the volume:
Volume = mass/density
Volume = 5.3126 g / 0.288 g cm^-3
Now, since the particles are spherical with a radius of 0.5 mm, we can find the volume of one particle using the formula for the volume of a sphere:
Volume of one particle = 4/3 * π * radius^3
Volume of one particle = 4/3 * π * (0.5 cm)^3
Finally, we can find the number of particles in the sample by dividing the sample volume by the volume of one particle:
Number of particles = Volume of sample / Volume of one particle
(ii) Now, let's talk about the % standard deviation in sampling the fraction of pink particles from this mass of the sample. Since 36% of the particles appeared to be pink, we can calculate the standard deviation using the formula:
Standard deviation = % of pink particles * (1 - % of pink particles)
(iii) To determine which contributes the larger uncertainty to the procedure - weighing out the sample or the random error associated with sampling from the two-component mixture, we need to compare the absolute values of their standard deviations. Whichever is larger will contribute more uncertainty.
(iv) Lastly, to find out how many replicate samples need to be taken for a 95% confidence that the mean is known to within ±6% of the true mean, we can use the formula for calculating sample size for a given confidence interval:
Sample size = (Z * σ / E)^2
Where Z is the Z-score corresponding to the desired confidence level (95% corresponds to Z = 1.96), σ is the standard deviation of the population, and E is the desired margin of error (6% in this case).
Remember, these calculations assume a normal distribution and certain assumptions about the sample.
Hope this helps, but don't forget to recalibrate your chromatography clown detector for accurate results!
(i) To calculate the number of particles in the sample, we need to find the volume of the sample and then divide it by the volume occupied by each particle.
Given:
Radius of particles = 0.5 mm = 0.05 cm
Density of the mixture = 0.288 g cm^-3
Mass of the sample = 5.3126 ± 0.0003 g
The volume of the sample can be calculated using the density and mass:
Volume = Mass / Density = 5.3126 g / 0.288 g cm^-3 = 18.44 cm^3
The volume occupied by each particle can be calculated as:
Volume of a sphere = (4/3) * π * (radius)^3
Volume of a particle = (4/3) * π * (0.05 cm)^3 = 0.0005236 cm^3
Now, we can calculate the number of particles in the sample by dividing the sample volume by the particle volume:
Number of particles = Sample volume / Particle volume = 18.44 cm^3 / 0.0005236 cm^3 = 35,202 particles
Therefore, the number of particles in the sample is 35,202.
(ii) The % standard deviation in sampling the fraction of pink particles can be calculated using the given information.
Given:
Fraction of pink particles = 36%
% standard deviation in sampling = 8%
The formula for calculating the % standard deviation in sampling is:
% standard deviation in sampling = (100 * √(fraction of pink particles * (1 - fraction of pink particles))) / √(Number of particles)
Substituting the given values:
8 = (100 * √(0.36 * (1 - 0.36))) / √(35,202)
To find the % standard deviation in sampling, we solve for √(35,202):
√(35,202) = (100 * √(0.36 * (1 - 0.36))) / 8
√(35,202) = 0.3
Now we can calculate the % standard deviation in sampling:
% standard deviation in sampling = (100 * 0.3) / 8 ≈ 3.75%
Therefore, the % standard deviation in sampling the fraction of pink particles from this mass of the sample is approximately 3.75%.
(iii) To determine which contributes the larger uncertainty, we need to compare the uncertainties associated with weighing out the sample and the random error associated with sampling from the two-component mixture.
The uncertainty in weighing out the sample is given as ± 0.0003 g.
The uncertainty in the number of pink particles can be calculated using the % standard deviation in sampling:
Uncertainty in the number of pink particles = (3.75/100) * Number of particles = (3.75/100) * 35,202
Comparing both uncertainties:
Uncertainty in weighing out the sample = 0.0003 g
Uncertainty in sampling from the two-component mixture = (3.75/100) * 35,202
By calculating the two uncertainties, we can determine which one is larger.
(iv) To determine 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 the margin of error:
Margin of error = (Z * % standard deviation) / √(n)
Given:
% standard deviation in sampling = 8%
Margin of error = 6%
Confidence level = 95% (Z-value = 1.96 for 95% confidence)
Substituting the given values into the margin of error formula, and solving for n (number of replicate samples):
6/100 = (1.96 * 8/100) / √(n)
0.06 = 0.1568 / √(n)
Squaring both sides:
0.0036 = 0.024536 / n
Rearranging the equation:
n = 0.024536 / 0.0036 ≈ 6.8153
Therefore, to have 95% confidence that the mean is known to within 6% of the true mean, approximately 7 replicate samples of this packing material need to be taken.
To answer these questions, we need to perform several calculations. Let's go through each question step-by-step:
(i) To calculate the number of particles in the sample, we need to know the mass and density of the sample, as well as the volume and density of individual particles.
Given:
- Mass of the sample (m) = 5.3126±0.0003 g
- Average density of the mixture (ρ) = 0.288 g cm^-3
- Radius of each particle (r) = 0.5 mm
First, we need to calculate the volume of the sample using its mass and density:
Volume of the sample (V) = Mass / Density
V = 5.3126±0.0003 g / 0.288 g cm^-3
The calculated volume will be in cm^3.
Next, we need to calculate the volume of a single particle:
Volume of a particle (Vp) = (4/3) * π * (r^3)
Vp = (4/3) * 3.14159 * (0.5 mm)^3
Since the radius is given in mm, the calculated volume will also be in mm^3. To convert it to cm^3, we need to divide by 1000.
Now, we can find the total number of particles in the sample:
Number of particles = V / Vp
(ii) To calculate the % standard deviation in sampling the fraction of pink particles, we need to consider the uncertainty in counting the pink particles.
Given:
- Fraction of pink particles (p) = 36%
The % standard deviation can be calculated using the formula:
% Standard deviation = (sqrt(p * (1-p))) / sqrt(n) * 100
Where 'n' is the number of particles counted. To calculate this, we need to consider the total number of particles calculated in part (i).
(iii) To determine which contributes a larger uncertainty, weighing the sample or random error associated with sampling, we need to compare the % standard deviation in sampling with the uncertainty in weighing the sample.
To calculate the uncertainty in weighing the sample, we need to know the uncertainty in the mass of the sample, given as ±0.0003 g.
We can compare the two uncertainties by calculating the ratios of their sizes.
(iv) To determine the number of replicate samples needed for a 95% confidence level and a desired 6% margin of error, we need to use the formula for determining the sample size needed in a proportion estimation:
n ≈ (Z^2 * p * (1-p)) / E^2
Where:
- n is the desired sample size
- Z is the Z-value corresponding to the desired confidence level (e.g., Z ≈ 1.96 for a 95% confidence level)
- p is the estimated fraction of pink particles (36%)
- E is the desired margin of error (6% of the true mean)
Now, let's perform the calculations based on the given information.