Mass Concentration (m/v or w/v) Chemistry Tutorial

Key Concepts

⚛ Mass concentration⁽¹⁾ tells us what mass of solute is present in a given volume of solution.

⚛ Mass concentration is usually abbreviated as m/v (or as w/v). ⁽²⁾

· You may also see mass concentration given the symbol γ

⚛ Common units for mass concentration are g L^-1 (g/L) in which

· mass of solute is measured in grams (g)

· volume of solution is measured in litres (L)

⚛ Note⁽³⁾: 1 L = 1 dm³ therefore mass concentration may also be expressed as g dm^-3 (g/dm³)

· mass of solute is measured in grams (g)

· volume of solution is measured in cubic decimetres (dm³)

⚛ Mass concentration (m/v) can be calculated from the mass of solute and the volume of solution:

mass concentration = γ = m/v =

mass of solute
volume of solution

⚛ The mass of solute present in a given volume of solution for which the mass concentration (m/v or w/v) is known can be calculated:

mass of solute = mass concentration × volume of solution

⚛ The volume of solution that contains a given mass of solute can be calculated if the mass concentration (m/v or w/v) is known:

volume of solution =

mass of solute
mass concentration

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Calculating Mass Concentration (m/v or w/v)

A solution is a mixture in which a solute is dissolved in a solvent. The most common solvent is water. Dissolving a solute in water produces an aqueous solution.

The mass concentration of a solution tells us what mass of solute present is present in a given volume of a solution. Common units for mass concentration are grams per litre, g L^-1 or g/L.

For example, if an aqueous solution of sodium chloride, NaCl(aq), has a mass concentration of 3.5 g L^-1 then 3.5 grams of solute (NaCl) are present in 1 litre of solution (NaCl(aq)).

Other possible units for mass concentration include mg L^-1, μg L^-1, mg mL^-1, μg mL^-1, etc.

For example, if an aqueous solution of potassium nitrate, KNO₃(aq), has a mass concentration of 6.25 mg mL^-1 then 6.25 milligrams of solute (KNO₃) are present in 1 millilitre of solution (KNO₃(aq)).

The units of mass concentration are always the mass of solute units divided by volume of solution units, this means that we calculate mass concentration by dividing mass of solute by the volume of solution:

mass concentration

mass of solute
volume of solution

For this reason, mass concentration is usually abbreviated as m/v (or as w/v), that is, mass/volume (or weight/volume).

mass concentration = m/v = w/v

Always check that your units are consistent, if m/v has units of g L^-1 then solute mass has units of g and solution volume has units of L. There are times when you may need to convert units to make them consistent, examples of this are given in the worked examples in each of the following sections.

To calculate the mass concentration (m/v or w/v) of a solution:

Step 1. Identify the solute and solution
Step 2. Write the equation: mass concentration = mass of solute ÷ volume of solution
Step 3. Identify the mass of the solute (include units)
Step 4. Identify the volume of the solution (include units)
Step 5. Substitute the values for mass and volume into the equation and solve (include units)
Step 6. Write the answer: m/v = (include units)

Worked Examples: Calculating the Mass Concentration of a Solution

Question 1. Determine the mass concentration of an aqueous solution of silver nitrate, AgNO₃(aq), if 6.28 grams of solid silver nitrate, AgNO₃(s), is dissolved in enough distilled water to make 2.00 litres of solution.

Solution:

Step 1. Identify the solute and solution

solute is silver nitrate, AgNO₃(s)

solution is an aqueous solution of silver nitrate, AgNO₃(aq) (because water is the solvent)

Step 2. Write the equation: mass concentration = mass of solute ÷ volume of solution

m/v = mass concentration

mass of AgNO₃(s)
volume of AgNO₃(aq)

Step 3. Identify the mass of the solute (include units)

mass of solute = m(AgNO₃(s)) = 6.28 g

Step 4. Identify the volume of the solution (include units)

volume of solution = v(AgNO₃(aq)) = 2.00 L

Step 5. Substitute the values for mass and volume into the equation and solve (include units)

m/v = mass concentration	=	m(AgNO₃(s)) v(AgNO₃(aq))
	=	6.28 g 2.00 L
	=	3.14 g L^-1

Note that 3 significant figures are justified.

Step 6. Write the answer: m/v = (include units)

m/v = 3.14 g L^-1

Question 2. 0.125 grams of copper(II) sulfate was dissolved in enough distilled water to make 250.0 mL of solution.
Calculate the mass concentration of this solution in g L^-1

Solution:

Step 1. Identify the solute and solution

solute is copper(II) sulfate, CuSO₄(s)

solution is an aqueous solution of copper(II) sulfate, CuSO₄(aq)

Step 2. Write the equation: mass concentration = mass of solute ÷ volume of solution

m/v = mass concentration

mass of CuSO₄(s)
volume of CuSO₄(aq)

Step 3. Identify the mass of the solute (include units)

mass of solute = m(CuSO₄(s)) = 0.125 g

Step 4. Identify the volume of the solution (include units)

volume of solution = v(CuSO₄(aq)) = 250.0 mL

Convert the volume units from mL to L because we have been asked to find m/v in units of g L^-1 not g mL^-1

v(CuSO₄(aq)) = 250.0 mL = 250.0 mL/1000 mL/L = 0.2500 L

Step 5. Substitute the values for mass and volume into the equation and solve (include units)

m/v = mass concentration	=	m(CuSO₄(s)) v(CuSO₄(aq))
	=	0.125 g 0.2500 L
	=	0.500 g L^-1

Note that 3 significant figures are justified.

Step 6. Write the answer: m/v = (include units)

m/v = 0.500 g L^-1

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Calculating Mass of Solute from m/v

We can rearrange the mathematical equation for mass concentration to calculate the mass of solute we need to dissolve in a given volume in order to achieve a given mass concentration.

First we write the equation:

m/v = mass concentration

mass of solute
volume of solutiion

Then we multiply both sides of the equation by the volume of solution:

m/v × volume of solution	=	mass of solute ~~volume of solution~~	× ~~volume of solution~~
m/v × volume of solution	=	mass of solute

And we can use this equation, mass(solute) = m/v × volume(solution), to calculate the mass of solute required using the following steps:

Step 1. Identify the solute and solution
Step 2. Write the equation: mass of solute = mass concentration × volume of solution
Step 3. Identify the mass concentration of the solution (include units)
Step 4. Identify the volume of the solution (include units)
Step 5. Substitute the values for mass concentration and volume into the equation and solve (include units)
Step 6. Write the answer: mass(solute) = (include units)

Worked Examples: Calculating the Mass of a Solute from m/v (or w/v)

Question 1. Determine the mass of solid sodium hydroxide, NaOH(s), needed to make 2.50 L of an aqueous solution of sodium hydroxide, NaOH(aq), with a mass concentration of 10.00 g L^-1.

Solution:

Step 1. Identify the solute and solution

solute is sodium hydroxide, NaOH(s)

solution is aqueous solution of sodium hydroxide, NaOH(aq)

Step 2. Write the equation: mass of solute = mass concentration × volume of solution

m(NaOH(s)) = m/v × v(NaOH(aq))

Step 3. Identify the mass concentration of the solution (include units)

m/v = 10.00 g L^-1

Step 4. Identify the volume of the solution (include units)

v(solution) = 2.50 L

Step 5. Substitute the values for mass concentration and volume into the equation and solve (include units)

m(solute) = 10.00 g ~~L^-1~~ × 2.50 L = 25.0 g

Note that 3 significant figures are justified.

Step 6. Write the answer: mass(solute) = (include units)

m(NaOH(s)) = 25.0 g

Question 2. What mass of solid potassium chloride, KCl(s), must be dissolved in enough distilled water to make 100.0 mL of solution with a mass concentration of 3.79 g L^-1?

Solution:

Step 1. Identify the solute and solution

solute is potassium chloride, KCl(s)

solution is an aqueous solution of potassium chloride, KCl(aq) (because water is the solvent)

Step 2. Write the equation: mass of solute = mass concentration × volume of solution

m(KCl(s)) = m/v × v(KCl(aq))

Step 3. Identify the mass concentration of the solution (include units)

m/v = 3.79 g L^-1

Step 4. Identify the volume of the solution (include units)

v(KCl(aq)) = 100.0 mL

Convert mL to L to make the units consistent with the mass concentration given in units of L^-1

v(KCl(aq)) = 100.0 mL ÷ 1000 mL/L = 0.1000 L

Step 5. Substitute the values for mass concentration and volume into the equation and solve (include units)

m(solute) = 3.79 g ~~L^-1~~ × 0.1000 L = 0.379 g

Note that 3 significant figures are justified.

Step 6. Write the answer: mass(solute) = (include units)

m(KCl(s)) = 0.379 g

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Calculating Volume of Solution from m/v

In the section above we found that mass(solute) = m/v × volume(solution). We can rearrange this mathematical equation to find the volume of solution given the mass of solute and the mass concentration of the solution.

First we write the equation:

mass of solute

m/v × volume of solution

Then we divide both sides of the equation by m/v (the mass concentration of the solution):

mass of solute m/v	=	~~m/v~~ × volume of solution ~~m/v~~
mass of solute m/v	=	volume of solution

To calculate the volume of solution we divide the mass of the solute by the mass concentration of the solution.

Follow these steps to calculate the volume of the solution:

Step 1. Identify the solute and solution
Step 2. Write the equation: volume of solution = mass of solute ÷ mass concentration
Step 3. Identify the mass of the solute (include units)
Step 4. Identify the mass concentration of the solution (include units)
Step 5. Substitute the values for mass and mass concentration into the equation and solve (include units)
Step 6. Write the answer: v = (include units)

Worked Examples: Calculating the Volume of Solution from m/v (or w/v)

Question 1. Determine the volume of KBr(aq) that contains 1.75 g of solute if the mass concentration of the solution is 22.5 g L^-1.

Solution:

Step 1. Identify the solute and solution

solute is KBr(s)

solution is KBr(aq)

Step 2. Write the equation: volume of solution = mass of solute ÷ mass concentration

v(KBr(aq)) = m(KBr(s)) ÷ m/v

Step 3. Identify the mass of the solute (include units)

m(KBr(s)) = 1.75 g

Step 4. Identify the mass concentration of the solution (include units)

m/v = 22.5 g L^-1

Step 5. Substitute the values for mass and mass concentration into the equation and solve (include units)

v(KBr(aq)) = 1.75 g ÷ 22.5 g L^-1 = 0.0778 L

Note that 3 significant figures are justified.

Step 6. Write the answer: v = (include units)

v(KBr(aq)) = 0.0778 L

Question 2. An aqueous solution of sodium hydrogencarbonate, NaHCO₃(aq), has a mass concentration of 5.472 g L^-1. Calculate the volume of solution that contains 6.523 × 10^-3 kg of sodium hydrogencarbonate.

Solution:

Step 1. Identify the solute and solution

solute is sodium hydrogencarbonate, NaHCO₃(s)

solution is an aqueous solution of sodium hydrogencarbonate, NaHCO₃(aq)

Step 2. Write the equation: volume of solution = mass of solute ÷ mass concentration

v(NaHCO₃(aq)) = m(NaHCO₃(s)) ÷ m/v

Step 3. Identify the mass of the solute (include units)

m(NaHCO₃(s)) = 6.523 × 10^-3 kg

Convert this mass from kg to g because the concentration is given in units of g L^-1

m(NaHCO₃(s)) = 6.523 × 10^-3 kg × 1000 g/kg = 6.523 g

Step 4. Identify the mass concentration of the solution (include units)

m/v = 5.472 g L^-1

Step 5. Substitute the values for mass and mass concentration into the equation and solve (include units)

v(NaHCO₃(aq)) = 6.523 g ÷ 5.472 g L^-1 = 1.192 L

Note that 4 significant figures are justified.

Step 6. Write the answer: v = (include units)

v(NaHCO₃(aq)) = 1.192 L

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Problem Solving: Mass Concentration

The Problem: In an experiment to precipitate all the I^-(aq) in 20.00 mL of KI(aq) as PbI₂(s), a student adds 18.36 mL of a stock solution of Pb(NO₃)₂(aq) with a mass concentration of 14.98 g L^-1 to the KI(aq).

Determine the mass of Pb(NO₃)₂ added to KI(aq).

Solving the Problem using the StoPGoPS model for problem solving:

STOP!	State the question.	What is the question asking you to do? Determine the mass of Pb(NO₃)₂ m(Pb(NO₃)₂) = ?
PAUSE!	Pause to Plan.	What information (data) have you been given? m/v(Pb(NO₃)₂(aq)) = 14.98 g L^-1 v(Pb(NO₃)₂(aq)) = 18.36 mL volume of KI(aq) is not relevant What is your plan for solving this problem? (i) Write the mathematical equation to calculate mass of Pb(NO₃)₂: m(Pb(NO₃)₂) = m/v(Pb(NO₃)₂(aq)) × v(Pb(NO₃)₂(aq)) (ii) Convert v(Pb(NO₃)₂(aq)) from mL to L (mass concentration is given in g L^-1 not g mL^-1) v(Pb(NO₃)₂(aq)) L = v(Pb(NO₃)₂(aq)) mL/1000 mL/L (iii) Substitute the values for m/v in g L^-1 and v in L into the equation and solve for mass of Pb(NO₃)₂
GO!	Go with the Plan.	(i) Write the mathematical equation to calculate mass of Pb(NO₃)₂: m(Pb(NO₃)₂) = m/v(Pb(NO₃)₂(aq)) × v(Pb(NO₃)₂(aq)) (ii) Convert v(Pb(NO₃)₂(aq)) from mL to L (mass concentration is given in g L^-1 not g mL^-1) v(Pb(NO₃)₂(aq)) L = v(Pb(NO₃)₂(aq)) mL/1000 mL/L v(Pb(NO₃)₂(aq)) L = 18.36 mL/1000 mL/L = 0.01836 L (iii) Substitute the values for m/v in g L^-1 and v in L into the equation and solve for mass of Pb(NO₃)₂ m(Pb(NO₃)₂) = 14.98 g ~~L^-1~~ × 0.01836 L = 0.2750 g (4 significant figures are justified)
PAUSE!	Ponder Plausability.	Have you answered the question that was asked? Yes, we have calculated the mass of Pb(NO₃)₂ added. Is your solution to the question reasonable? Let's work backwards by using our calculated mass and volume to determine the mass concentration of the solution: m = 0.2750 g v = 0.01836 L m/v = 0.2750 g/0.01836 L = 14.98 g L^-1 Since this m/v value is the same as the mass cconcentration given in the question, we are reasonably confident that our answer is correct.
STOP!	State the solution.	What is the mass of Pb(NO₃)₂? m(Pb(NO₃)₂) = 0.2750 g

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Sample Question: Mass Concentration

A student has been given a stock solution of aqueous potassium hydroxide with a mass concentration of 28.054 g L^-1.

The student needs to add 7.014 g of potassium hydroxide to an acid in order to neutralise it.

What volume, in litres, of KOH(aq) must the student add to the acid?

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Footnotes:

(1) Mass concentration is also known as mass density and given the symbol ρ. We are avoiding this term because of the confusion it can cause with density.

(2) Do not confuse mass concentration (m/v or w/v) with mass/volume percentage concentration, %(m/v) (weight/volume percentage concentration, %(w/v)).

(3) The SI unit for length is the metre, so the SI unit for volume is the cubic metre, m³.
The SI unit for mass is the kilogram (kg), so the SI units for mass concentration are kg m^-3 (kg/m³) which is a riduculously large unit for a chemistry lab.
Hence, chemists usually use grams per cubic decimetre, g dm^-3 (g/dm³).
Note that 1 dm³ = 1 L (1 litre), so an acceptable non-SI unit for mass concentration is g L^-1 (g/L)
Australian schools generally use litres rather than cubic decimetres so all the following calculations use litres rather than decimetres.

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