Pharmaceutical Calculations (1)

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Pharmaceutical Calculations (1)
The absolute basics - Mass and concentrations
Phil Rowe Liverpool School of Pharmacy
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Pharmaceutical Calculations
Access to PowerPoints
All the PowerPoints concerning pharmaceutical
calculations are available from http//www.staff.
livjm.ac.uk/phaprowe (Select Pharmaceutical
calculations)
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Measures of mass (1)grams
kg 103 g 1 mg 10-3 microgram 10-6 ng 10-9 pg
10-12
Most likely to be met as a blood concentration of
drug, not as a dose.
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Measures of mass (1)grams
Must be able to inter-convert these quickly and
accurately
12.5grams _____ mg 0.25mg _____
micrograms 40mg ______ grams 0.06g ______
micrograms 7,500 pg _____ micrograms
Work these out before turning to next page!
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Measures of mass (1)grams
Answers
12.5grams 12,500 mg 0.25mg 250
micrograms 40mg 0.04 grams 0.06g 60mg
60,000 micrograms 7,500 pg 7.5 ng 0.0075
micrograms
Down one unit, so x1,000
Down one unit, so x1,000
Up one unit, so 1,000
Down two units, so x 1,000,000
Up two units,so 1,000,000
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Measures of mass (2)Moles
The RMM of the substance expressed in grams. RMM
of aspirin is 180, so 1 Mole 180g 1 mMole
180 mg 1 microMole 180 microgram etc
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Measures of mass (2)Moles
Example questions. For aspirin 1) 0.18g
____ mMole2) 0.09mg ____ microMole3) 7.2
micrograms ____ nMole
Work these out before turning to next pages!
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Measures of mass (2)Moles
Answer 1 0.18g 180mg 1mMole
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Measures of mass (2)Moles
Answer 2 180mg 1mMole1mg 1/180 mMole0.09mg
0.09/180mMole 0.0005mMole
0.5microMole
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Measures of mass (2)Moles
Answer 3 180 microgram 1 microMole1 microgram
1/180 microMole7.2 microgram 7.2/180
microMole
0.04microMole 40
nanoMole
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Measures of concentration

• Mass/volume e.g. mg/100ml
• Molarity
• (w/v)
• (v/v)
• (w/w)
• ppm
• Ratio e.g. 1 in 50,000

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Using the standard formula for concentration in
any of its re-arranged forms
C M/Ve.g. What concentration will arise if we
dissolve mass (M) to make volume (V) of
solution? M C.Ve.g. What mass do we need to
make up to volume (V) in order to produce
concentration (C)? OrWhat mass is contained in
volume (V) of a solution of concentration (C)? V
M/Ce.g. To what volume should we dilute mass
(M) in order to produce concentration (C)?
OrWhat volume of a solution of concentration (C)
contains mass (M)?
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Measures of concentration (1)mass/volume (e.g.
mg/100ml)

• 500mg dissolved in water to give a final volume
  of 100ml 500mg/100ml
• Can be re-expressed as
• ____ mg/ml
• ____ mg per 5ml
• ____ mg/L
• ____ g/L

Oral doses often quoted per 5ml
Work these out before turning to next page!
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Measures of concentration (1)mass/volume

• Answers
• 500mg/100ml can be re-expressed as
• 5mg/ml
• 25mg/5ml
• 5,000mg/L
• 5g/L

Re-expressing simple mass/volume concentrations
is tackled fully in the next presentation
Pharmaceutical Calculations (2)
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Measures of concentration (1)mass/volume
Example questions 1) How much sucrose is
contained in 5ml of a solution of concentration
20g/L? 2) What will be the concentration of
sucrose (in mg/100ml) if 1g is dissolved in 5L 3)
To what volume should we make up a solution of
40g of sucrose if it is to produce a
concentration of 0.1g per 5ml?
Work these out before turning to next page!
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Measures of concentration (1)mass/volume
Answer 1 How much sucrose is contained in 5ml of
a solution of concentration 20g/L? M C.V
20g/L x 5ml (Must resolve units)
20g/1,000ml x 5ml 0.1g 100mg
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Measures of concentration (1)mass/volume
Answer 2 What will be the concentration of
sucrose (in mg/100ml) if 1g is dissolved in 5L C
M/V 1g/5L (Answer is to be in mg
ml) 1,000mg/5,000ml 200mg/1,000ml
20mg/100ml
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Measures of concentration (1)mass/volume
Answer 3 To what volume should we make up a
solution of 40g of sucrose if it is to produce a
concentration of 0.1g per 5ml? Get rid of the
Conc per 5ml they are always awkward0.1g per
5ml 0.2g/10ml 20g/L V M/C 40g / 20g/L
2L
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Measures of concentration (2)molarity
A molar solution contains 1 mole per litre

• RMM aspirin 180.
• 180g/L 1M
• 180mg/L 1mM
• 180microgram/L 1microMolar
• etc

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Measures of concentration (2)molarity
Example questions 1) 0.09g of aspirin is
dissolved in water to produce 250ml solution.
What is the resultant concentration in mM
units? 2) What volume of a 50mM solution will
contain 0.9g of aspirin? 3) What mass of aspirin
will be contained in 2.5L of a 0.1M solution?
Work these out before turning to next pages!
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Measures of concentration (2)molarity
Answer 1 0.09g of aspirin is dissolved in water
to produce 250ml solution. What is the resultant
concentration in mM units? Mixed units. Final
answer to be mMolar, so convert grams to
moles180g 1 Mole0.09g 0.09/180 Mole
0.0005 Mole 0.5 mMole C M/V
0.5mMole/250ml 2mMole/L 2 mMolar
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Measures of concentration (2)molarity
Answer 2 What volume of a 50mM solution will
contain 0.9 gram of aspirin? Mixed units
(molaritiy and mass). Final answer to be in
mass, so convert mM to g/L 1 mM 180mg/L50 mM
9,000mg/L 9g/L V M/C 0.9g
/ 9g/L 0.1L 100ml
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Measures of concentration (2)molarity
Answer 3 What mass of aspirin will be contained
in 2.5L of a 0.1M solution? Mixed units. Final
answer to be a mass, so convert Molar to g/L 1 M
180g/L0.1M 18g/L M V.C 2.5L x 18g/L
45g
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Measures of concentration (3)(w/v)
Grams per 100ml 5g/L 0.5g/100ml
0.5(w/v) 35mg/L 0.035g/L 0.0035g/100ml
0.0035(w/v)
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Measures of concentration (3)(w/v)
Example questions 1) What volume of a 0.015
(w/v) solution contains 900mg of sodium
chloride? 2) What concentration expressed as
(w/v) will arise if 10g of sodium chloride is
dissolved to make 2.5 litres of solution? 3)
What mass of sodium chloride should be used to
create 375ml of a 0.04 (w/v) solution? Answer
in units of mg.
Work these out before turning to next pages!
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Measures of concentration (3)(w/v)
Answer 1 What volume of a 0.015 (w/v) solution
contains 900mg of sodium chloride? (w/v) is
awkward, so get everything into grams and
litres0.015(w/v) 0.015g/100ml 0.15g/L900mg
0.9g V M/C 0.9g / 0.15g/L 6L
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Measures of concentration (3)(w/v)
Answer 2 What concentration expressed as
(w/v) will arise if 10g of sodium chloride is
dissolved to make 2.5L of solution?C M/V
10g/2.5L 4g/L 0.4g/100ml
0.4(w/v)
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Measures of concentration (3)(w/v)
Answer 3 What mass of sodium chloride should be
used to create 375ml of a 0.04 (w/v) solution?
Answer in units of mg. Get everything into grams
and litres0.04(w/v) 0.04g/100ml
0.4g/L375ml 0.375LM C.V 0.4g/L
x 0.375L 0.15g 150mg
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Measures of concentration (4)(v/v)
ml per 100ml
Used for mixtures of liquids. e.g. 50ml of
ethanol diluted to 1L with water 50ml ethanol/L
5ml/100ml 5 (v/v)
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(v/v) different equations needed
In this case the equation C M/V (and its
re-arranged forms) makes no sense, as we are not
starting with a known mass of material. Instead,
we start with a volume (V1) and dilute it to a
final volume (V2). The relevant equations are
then C V1/V2 V1 C.V2 V2 V1/C

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(v/v) Ensuring that units match
When using concentrations expressed as (v/v)
there are an awful lot of volumes involved and
you need to ensure that units match. There are a
number of approaches that would work, but here is
a suggestion Express V1 (Vol of active
ingredient) in units of ml Express V2 (Vol of
final solution) in units of Litres Concentration
must then be in units of ml/L
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Measures of concentration (4)(v/v)
Example questions 1) What concentration will
arise Expressed as (v/v) if 750 microlitres of
ethanol is diluted to a final volume of 150ml? 2)
To what volume in litres would we need to
dilute 8ml of ethanol to create a 0.32(v/v)
solution? 3) What volume of ethanol should we
dilute to 5L to produce a 2.5(v/v) solution?
Work these out before turning to next pages!
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Measures of concentration (4)(v/v)
Answer 1 What concentration will arise
Expressed as (v/v) if 750 microlitres of
ethanol is diluted to a final volume of
150ml? Get V1 in units of ml and V2 in litres.
Conc will be in ml/L750microlitres
0.75ml150ml 0.15LC V1/V2 0.75ml /
1.5L 5ml/LRe-express as (v/v)
0.5ml/100ml 0.5(v/v)
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Measures of concentration (4)(v/v)
Answer 2 To what volume in litres would we
need to dilute 8ml of ethanol to create a
0.32(v/v) solution? V1 already in ml. Get conc
as ml/L. V2 will then be in litres 0.32
0.32ml/100ml 3.2ml/L V2 V1/C
8ml / 3.2ml/L 2.5L
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Measures of concentration (4)(v/v)
Answer 3 What volume of ethanol should we dilute
to 5L to produce a 2.5(v/v) solution? V2 already
in litres. Get conc as ml/L. V1 will then be in
ml.2.5 2.5ml/100ml 25ml/L V1 C x
V2 25ml/L x 5L 125ml
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Measures of concentration (5)(w/w)
g per 100g
Used (for example) in ointments e.g. 0.5g active
ingredient in 25g ointment 0.5g/25g 2g/100g
2 (w/w)
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(w/w) also needs different approach
C M/V (and its re-arranged forms) again make no
sense, as we are relating concentration to a mass
of final product rather than a volume. We have a
mass (M1) of active ingredient present in a final
mass (M2) of product. We will use C M1 /
M2 M1 C x M2 M2 M1 / C
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(w/w) Ensuring that units match
When using concentrations expressed as (w/w) we
now have a plethora of masses and confusion over
units can arise. Again, there are a number of
approaches that would work, but here is a
suggestion Express M1 (Mass of active
ingredient) in units of g Express M2 (Mass of
final product) in units of kg Concentration must
then be in units of g/kg
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Measures of concentration (5)(w/w)
Example questions 1) Calculate the concentration
as (w/w) if 300mg of hydrocortisone is
contained in 30g of cream? 2) What quantity in
grams of a clobetasol propionate cream
0.05(w/w) would contain 12.5mg of the active
ingredient? 3) How much betamethasone should be
used to make 5kg of cream 0.1(w/w)?
Work these out before turning to next pages!
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Measures of concentration (5)(w/w)
Answer 1 Calculate the concentration as (w/w)
if 300mg of hydrocortisone is contained in 30g of
cream? Get M1 into grams and M2 into kg. Conc
will be in g/kg300mg 0.3g30g 0.03kg C
0.3g / 0.03kg 10g / kgRe-express conc as
(w/w) 1g / 100g 1(w/w)
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Measures of concentration (5)(w/w)
Answer 2 What quantity in grams of a
clobetasol propionate cream 0.05(w/w) would
contain 12.5mg of the active ingredient? Get M1
in g and conc in g/kg. M2 will be in kg.12.5mg
0.0125g0.05(w/w) 0.05g/100g
0.5g/kg M2 M1 / C 0.0125g /
0.5g/kg 0.025kg 25g
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Measures of concentration (5)(w/w)
Answer 3 How much betamethasone should be used
to make 5kg of cream 0.1(w/w)? M2 already in
kg. Get conc in g/kg. M1 will be in
g.0.1(w/w) 0.1g/100g
1g/kg M1 C x M2 1g/kg x 5kg 5g
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Measures of concentration (6)ppm
mg per litre
1ppm 1gram in 1,000,000ml
1mg/1,000ml 1mg/L
e.g. 400mg of fluoride (as stannous fluoride)
dissolved in 500L of water 400mg/500L
0.8mg/L 0.8ppm
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Measures of concentration (6)ppm
Example questions 1) What mass of fluoride is
present in 0.2ml of drinking water with a
fluoride content of 0.9ppm? 2) What
concentration expressed as ppm would be
achieved if 1gram of fluoride were added to every
2,000 litres of water? (Assume water is
initially completely free of fluoride.) 3) If
drinking water contains 0.8ppm of fluoride, how
much would you need to drink to ingest 10mg of
fluoride?
Work these out before turning to next pages!
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Measures of concentration (6)ppm
Answer 1 What mass of fluoride is present in
0.2ml of drinking water with a fluoride content
of 0.9ppm? Easiest in micrograms and ml0.9ppm
0.9mg/L 900microgram/L 0.9microgram/ml M C
x V 900microgram/ml x 0.2ml 0.18
microgram 180ng
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Measures of concentration (6)ppm
Answer 2 What concentration expressed as ppm
would be achieved if 1gram of fluoride were added
to every 2,000 litres of water? (Assume water is
initially completely free of fluoride.) 1g/2,000L
1,000mg/2,000L 0.5mg/L
0.5ppm
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Measures of concentration (6)ppm
Answer 3 If drinking water contains 0.8ppm of
fluoride, how much would you need to drink to
ingest 10mg of fluoride? Convert ppm to
mg/L0.8ppm 0.8mg/L V M/C 10mg /
0.8mg/L 12.5 L
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Measures of concentration (7)ratio (e.g. 1 in
50,000)
gram per (number) ml
1 in 50,000 1 gram in 50,000ml
e.g. Xylocaine (local anaesthetic injection) may
contain adrenaline to reduce local blood flow and
delay the removal of the anaesthetic from the
site of application. Concentration of adrenaline
is quoted as 1 in 200,000.
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Measures of concentration (7)ratio (e.g. 1 in
50,000)
Example questions 1) What concentration
expressed as ratio will arise if 50mg of
adrenaline is dissolved in a total volume of ten
litres? 2) What volume of adrenaline solution (1
in 200,000) will contain 25microgram of
adrenaline? 3) What mass of adrenaline should be
made up to 2L to create a 1 in 200,000 solution?
Work these out before turning to next pages!
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Measures of concentration (7)ratio (e.g. 1 in
50,000)
Answer 1 What concentration expressed as a
ratio will arise if 50mg of adrenaline is
dissolved in a total volume of ten litres? We
need to re-arrange until it becomes 1g in ml C
M/V 50mg/10L 1,000mg/200L
1g/200,000ml 1 in 200,000
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Measures of concentration (7)ratio (e.g. 1 in
50,000)
Answer 2 What volume of adrenaline solution (1
in 200,000) will contain 25microgram of
adrenaline? Get everything into mg and litre
units1 in 200,000 1g/200,000ml
1mg/200ml
5mg/L25microgram 0.025mg V M/C 0.025mg
/ 5mg/L 0.005L 5ml
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Measures of concentration (7)ratio (e.g. 1 in
50,000)
Answer 3 What mass of adrenaline should be made
up to 2L to create a 1 in 200,000 solution?
Re-express the ratio in mg and litre units1 in
200,000 1g/200,000ml
1mg/200ml 5mg/L M V x C
2L x 5mg/L 10mg
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A special type of calculation, only easily
possible where concentration is expressed as
(w/w)
A concentration of 1g/100ml can be achieved by
dissolving 1g of solid and making the volume up
to 100ml. Note that we are forced to talk in
terms of the volume finally achieved. Life
becomes much more complicated if we give
instructions such as Take 1g and add 100ml of
water. We do not immediately know the final
volume and so it becomes much more difficult to
calculate what the concentration would be. The
same objection applies to concentrations
expressed as (w/v) or molarity etc. There are
similar problems with (v/v), because if you mix
10ml of ethanol and 90ml of water you will not
obtain 100ml of mixture (There is
shrinkage). These problems can be overcome by
using Displacement values but these will not be
tackled in this presentation.
Continued on next slide
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A special type of calculation, only easily
possible where concentration is expressed as
(w/w)
However, (w/w) is special, because this is the
one case where we might say Prepare a cream by
mixing 5g of active ingredient with 95g of
excipient and we would be able to calculate the
final weight of cream (100g) and so its
concentration can be calculated to be
5(w/w). This opens up a type of calculation not
easily achievable when concentration is expressed
in other ways. A couple of example questions
follow
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Mixing masses and expressing result as (w/w)
Example questions 1) How much excipient should
we add to 500g of zinc oxide to produce a
24(w/w) cream? 2) If we mix 500g of
hydrocortisone with 49.5kg of excipient, what
concentration (Expressed as w/w) will arise?
Work these out before turning to next pages!
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Mixing masses and expressing result as (w/w)
Answer 1 How much excipient should we add to 500g
of zinc oxide to produce a 24(w/w) cream? M1
already in g. Get conc in g/kg. M2 will be in
kg24(w/w) 24g/100g
240g/kg M2 M1 / C 500g / 240g/kg
2.083kg BUT BE VERY CAREFUL!!!What we have
just calculated is the final mass of product.
What we were asked to calculate was the mass of
excipient to add.Excipient Final mass of
product mass of active ingredient
2.083kg 0.5kg 1.583kg
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Mixing masses and expressing result as (w/w)
Answer 2 If we mix 500g of hydrocortisone with
49.5kg of excipient, what concentration
(Expressed as w/w) will arise? AGAIN BE
CAREFUL!!! M2 is not 49.5kg it should be the
total mass i.e. 49.5 0.5 50kgM1 is already
in g and M2 in kg. Conc will be in g/kg C M1 /
M2 500g / 50kg 10g/kgRe-express as
(w/w) 1g/100g 1(w/w)
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Mixing masses and expressing result as (w/w)
Calculating mass of active ingredient (M1) to add
to a stated amount of excipient
In the previous 2 cases we successfully
calculated either C or M2. However, calculating
M1 (the mass of active ingredient that should
added to a stated quantity of excipient to
produce a stated concentration) is not so easy!
Cant use the obvious M1 C x M2 because M2
cant be calculated without knowing M1!
Solution on next slide
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Mixing masses and expressing result as (w/w)
Solution If M1 and M2 represent the masses of
active ingredient and final product and C
concentration expressed as a proportion (e.g. 50
0.5), then M1 C x M2 M2 M1 mass of
excipient (Ex), so M1 C x (M1 Ex) M1 C.M1
C.Ex M1 C.M1 C.ExM1(1 C) C.ExM1
C.Ex 1 C
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Mixing masses and expressing result as (w/w)
Example - Calculating mass of active ingredient
(M1) to add to a stated amount of excipient
What mass of zinc oxide should we add to 3kg of
excipient to produce a 20(w/w) ointment?
M1 C.Ex 1 C
0.2 x 3kg 1 0.2
0.6kg 0.8
0.75kg
Check M2 0.75kg 3kg 3.75kg C M1/M2
750g / 3.75kg 200g/kg 20g /100g
20(w/w)
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What you should be able to do

• Use all units within the range of pico (10-12)
  to kilo (103). e.g. pg, ng, microgram, mg, g
  kg for mass.
• Understand all of the 7 measures of
  concentration set out in this presentation.
• Calculate the concentration if the mass that is
  dissolved in a solution and its volume are
  stated. (C M/V)
• Calculate the mass that needs to be dissolved to
  yield a stated concentration and volume of
  solution. (M C x V)
• Calculate the volume of solution that needs to
  be prepared if it is to contain a stated mass and
  be of a stated concentration. (V M/C)
• In the special case of (w/w), calculate
  concentration from mass of active ingredient and
  mass of other constituent(s).