Rates and concentration

Rates with two different units, concentration as a rate (g/L, mg/mL, % w/v and % v/v), diluting a solution, and drip rates for intravenous fluids.

A rate tells you how much of one thing there is for each unit of another: kilometres per hour, dollars per kilogram, milligrams per millilitre. Concentration is just a rate that measures how much of a substance is packed into each unit of liquid. Pharmacists, nurses, farmers and pool owners all work with it.

Revise: unit rates and converting km/h to m/s are in Year 11 Units, rates and conversions.

Converting rates

A rate has two units, so a conversion may need two steps: one for the top unit and one for the bottom. Change one unit at a time and ask whether the number should get bigger or smaller.

Example 1●

A tap leaks $45$ mL per minute. Write this as litres per day.

$45 \times 1440 = 64{,}800$ mL per day

Change the time unit first: there are 60 × 24 = 1440 minutes in a day, so the amount per day is 1440 times bigger.

$64{,}800 \div 1000 = 64.8$ L per day

Then change mL to L by dividing by 1000.

Concentration

$\text{concentration} = \dfrac{\text{amount of substance}}{\text{volume of solution}}$
$\%\ \text{w/v}$ = grams per $100$ mL. $\%\ \text{v/v}$ = mL per $100$ mL.
So $\text{amount} = \text{concentration} \times \text{volume}$.

"w/v" means weight over volume: a $0.9\%$ w/v saline drip has $0.9$ g of salt in every $100$ mL. "v/v" means volume over volume: a drink labelled $5\%$ v/v has $5$ mL of alcohol in every $100$ mL.

Example 2●

How much salt is in a $250$ mL bag of $0.9\%$ w/v saline?

$0.9 \times 2.5 = 2.25$ g

0.9% w/v means 0.9 g in every 100 mL, and 250 mL is 2.5 lots of 100 mL.

Example 3●●

A pool chemical is sold as $12.5$ g/L. How many grams are in a $20$ L drum, and what is the strength in mg/mL?

$12.5 \times 20 = 250$ g

Amount = concentration × volume.

$12.5$ g/L $= 12.5$ mg/mL

1 g/L means 1000 mg in 1000 mL, which is 1 mg/mL, so the number stays the same.

Diluting

When you add water, the amount of substance does not change. It is just spread through more liquid, so the concentration falls. That single idea solves every dilution problem: work out the amount, then divide by the new volume.

Dilution

amount before = amount after, so $C_1 V_1 = C_2 V_2$

Example 4●●

A cleaner mixes $40$ mL of $5\%$ w/v disinfectant with $360$ mL of water. What is the new concentration?

$5 \times 40 \div 100 = 2$ g

Find the amount of disinfectant in the stock: 5 g per 100 mL.

$40 + 360 = 400$ mL

The new volume is stock plus water.

$2 \div 400 \times 100 = 0.5\%$ w/v

Concentration = amount ÷ volume, written per 100 mL for a percentage.

InteractiveDilution beakers

Try this

Check Example 4 on Disinfectant. Then predict the strength if you add $760$ mL of water instead of $360$, and test it. Can you make exactly $1\%$ w/v from $5\%$ stock? Next switch to Medicine and find how much water turns $10$ mL of $50$ mg/mL stock into $5$ mg/mL.

Drip rates

In hospitals, fluid is given through a drip. A pump is set in mL per hour, while a gravity drip is counted in drops per minute. The tubing's drop factor says how many drops make $1$ mL.

Drip rate

$\text{drops/min} = \dfrac{\text{volume} \times \text{drop factor}}{\text{time}}$
with volume in mL, drop factor in drops/mL and time in minutes.

Example 5●●●

A patient is to receive $1200$ mL over $10$ hours through tubing with a drop factor of $20$ drops/mL. Find the rate in mL/h and in drops per minute.

$1200 \div 10 = 120$ mL/h

mL per hour is volume ÷ hours.

$1200 \times 20 = 24{,}000$ drops

Total drops = volume × drop factor.

$24{,}000 \div 600 = 40$ drops per minute

Divide by the time in minutes: 10 h = 600 min.

Your turn

How much water must be added to $50$ mL of $20\%$ w/v fertiliser to make a $2\%$ w/v solution?

Show the working

Amount: $20 \times 50 \div 100 = 10$ g. For $2\%$, $10$ g must be in $10 \div 2 \times 100 = 500$ mL. Water to add: $500 - 50 = 450$ mL.

Flashcards

% w/v

Grams per 100 mL

Dilution rule

The amount stays the same: C₁V₁ = C₂V₂

Drip rate

drops/min = mL × drop factor ÷ minutes

Practice

A tap leaks $45$ mL per minute. How many litres is that per day?
How many grams of salt are in $250$ mL of $0.9\%$ w/v saline?
A $375$ mL can is $5\%$ v/v alcohol. How many mL of alcohol does it contain?
$3$ g/L is the same as:
$40$ mL of $5\%$ w/v disinfectant is mixed with $360$ mL of water. What is the new concentration, as a % w/v?
When water is added to a solution, which stays the same?
How many mL of water must be added to $50$ mL of $20\%$ w/v fertiliser to make $2\%$ w/v?
$10$ mL of $50$ mg/mL stock is diluted to $5$ mg/mL. How many mL of water are added?
Exam style: $1200$ mL is to be given over $10$ hours with a drop factor of $20$ drops/mL. Find the drip rate in drops per minute.
Exam style: For the same drip, what is the rate in mL per hour?