Convert Units: q,C from gr/lb, gr/gal to mg/g, mg/dm^3

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In summary: So many errors are caught this way.In summary, the conversation discusses the use of the Langmuir equation to calculate contaminant removal in waste water. The values for the constants b and k are given in grains per lb and grains per gal, and the task is to determine the values for b' and k' in mg per g and mg per dm^3. Through dimensional analysis, it is determined that b and k are not dimensionless and have a relation to the units of q and C. The conversation ends with the acknowledgement that checking dimensions can help catch errors.
  • #1
foxtrotalpha
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Hey guys I am having a big problem with this question :frown:

Homework Statement



The adsorbtion isotherm for the removal of a contaminant from waste water is givern by the langmuir equation: q=(bkC)/(1+bC) where q is the loading of contaminant of the adsorbent and C is the concentration of the contaminant in solution. Literature data gives values for the constants b and k of 1.16 and 130 respectively for the case where q is grains per lb and C is grains per gal . Determine the values for b' and k' for the Langmuir equation between loading q' in mg per g and C' in mg per dm^3 .

Data : 1 lb=7000 gr(grains)

Homework Equations





The Attempt at a Solution


I am sort of stuck as the 2 variables will have changes in their unis simultaneously, all I have done is to convert 1mg=0.01544 gr and 1 dm^3=0.220 gal .
 
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  • #2
Dimensional analysis will probably work here:
With equation: $$q=\frac{bkC}{1+bC}$$ ... are k and b dimensionless?
Did the values you looked up have units at all? There's a clue right there.

What are the units of b and k in terms of the units of q and C?

You can work it out - either by reading the tables or dimensional analysis:
i.e. notice that, in the denominator "1+bC" has to make sense in terms of units?
so (square brackets reads "units of"): [1+bC]=[1]+[bC] means that must have some relation to [C].Note: 1 cubic decimeter = 1 liter.
 
  • #3
Simon Bridge said:
Dimensional analysis will probably work here:
With equation: $$q=\frac{bkC}{1+bC}$$ ... are k and b dimensionless?
Did the values you looked up have units at all? There's a clue right there.

What are the units of b and k in terms of the units of q and C?

You can work it out - either by reading the tables or dimensional analysis:
i.e. notice that, in the denominator "1+bC" has to make sense in terms of units?
so (square brackets reads "units of"): [1+bC]=[1]+[bC] means that must have some relation to [C].


Note: 1 cubic decimeter = 1 liter.


THANKS ! , you made me realized that b and k are not dimensionless at all ,before this I had always assumed them to be dimensionless.
 
  • #4
Well done.

Checking the dimensions is very powerful.
 
  • #5


Hello,

Thank you for reaching out for help with this problem. It seems that you are trying to convert the units for the Langmuir equation from grains per pound and grains per gallon to milligrams per gram and milligrams per cubic decimeter. This type of conversion is common in scientific research, and it is important to understand how to do it accurately.

First, let's review the given information. The Langmuir equation is q=(bkC)/(1+bC), where q is the loading of contaminant on the adsorbent and C is the concentration of the contaminant in solution. The constants b and k have values of 1.16 and 130, respectively, when q is in grains per pound and C is in grains per gallon.

To convert these units to milligrams per gram and milligrams per cubic decimeter, we need to use conversion factors. As you mentioned, 1 pound is equal to 7000 grains, and 1 gallon is equal to 7000 grains. We also know that 1 milligram is equal to 0.01544 grains, and 1 decimeter cubed (dm^3) is equal to 1000 milliliters (mL) or 1000 cubic centimeters (cm^3).

Using these conversion factors, we can convert the units for q and C as follows:

q (grains per pound) = q' (milligrams per gram) * (1 pound/7000 grains)
C (grains per gallon) = C' (milligrams per cubic decimeter) * (1 gallon/7000 grains)

Substituting these values into the original Langmuir equation, we get:

q' = (1.16 * 130 * C')/(1+1.16 * 130 * C')

Simplifying this equation, we get:

q' = (150.8 * C')/(1+150.8 * C')

This is the Langmuir equation for the loading q' in milligrams per gram and the concentration C' in milligrams per cubic decimeter. The constants b' and k' are now equal to 150.8 and 1.16*130, respectively.

I hope this helps! Remember, when converting units, it is important to keep track of the conversion factors and make sure the units are consistent throughout the equation. Let me know if you have any other questions.

Best,
 

1. How do I convert grams per pound (gr/lb) to milligrams per gram (mg/g)?

To convert from gr/lb to mg/g, you can use the conversion factor of 1000 mg/g for every 1 gr/lb. This means that you need to multiply the value in gr/lb by 1000 to get the equivalent value in mg/g.

2. What is the conversion factor for grams per gallon (gr/gal) to milligrams per decimeter cubed (mg/dm^3)?

The conversion factor for gr/gal to mg/dm^3 is 1000 mg/dm^3 for every 1 gr/gal. This means that you need to multiply the value in gr/gal by 1000 to get the equivalent value in mg/dm^3.

3. Can I use the same conversion factor for both gr/lb to mg/g and gr/gal to mg/dm^3?

Yes, you can use the same conversion factor of 1000 for both conversions. This is because both conversions involve converting from a unit of mass (grams) to a unit of volume (liters or decimeters cubed).

4. How do I convert from milligrams per gram (mg/g) to milligrams per decimeter cubed (mg/dm^3)?

To convert from mg/g to mg/dm^3, you need to divide the value in mg/g by 1000. This is because 1 dm^3 is equal to 1000 cm^3, and since we are converting from grams to milligrams, we need to account for the difference in units.

5. Is there an easy way to remember the conversion factor for these units?

One way to remember the conversion factor is to think of it as moving the decimal point 3 places to the right. For example, 1 gr/lb is equivalent to 1000 mg/g (move decimal point 3 places to the right) and 1 gr/gal is equivalent to 1000 mg/dm^3 (move decimal point 3 places to the right).

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