Question #36be0

1 Answer
May 13, 2017

Answer:

#"Mass %" = b/(b + 1000/M) × 100 %#

Explanation:

The formula for molality #b# is

#color(blue)(bar(ul(|color(white)(a/a)b = "moles of solute"/"kilograms of solvent" color(white)(a/a)|)))" "#

It will be more convenient during the derivation to express the mass of solvent in grams rather than kilograms.

Then,

#b = n/(10^"-3" m_text(s)) = (1000n)/(m_text(s))#

where

#n color(white)(ll)=# moles of solute
#m_text(s) =# grams of solvent

Since

#n = "mass"/"molar mass" = m/M#

we can write

(1) #color(white)(mmmm)b = (1000m)/(Mm_text(s)#

where

#m =# mass of solute
#M =# molar mass of solute

The formula for mass percent is

#color(blue)(bar(ul(|color(white)(a/a)"Mass %" = "mass"/"total mass" × 100 %color(white)(a/a)|)))" "#

or

(2) #color(white)(mmm)"Mass %" = (m)/(m + m_text(s)) × 100 %#

where

#m_text(s) =# mass of solvent

From (1),

(3) #color(white)(mmmm)m = (bMm_text(s))/1000#

Substitute (3) into (2)

#"Mass %" = ((bMcolor(red)(cancel(color(black)(m_text(s)))))/1000)/((bMcolor(red)(cancel(color(black)(m_text(s)))))/1000 + stackrelcolor(blue)(1)(color(red)(cancel(color(black)(m_text(s)))))) × 100 % = ((bM)/1000)/((bM)/1000 + 1) × 100 %#

Multiply numerator and denominator by #1000/M#. Then

#color(blue)(bar(ul(|color(white)(a/a)"Mass %" = b/(b + 1000/M) × 100 %color(white)(a/a)|)))" "#

Example

Calculate the mass percent of a 1.00 mol/kg solution of #"NaCl"#.

#"Mass %" = b/(b + 1000/M) × 100 % = 1.00 /(1.00 + 1000/58.44) × 100 %#

#= 1.00/(1.00 + 17.11) × 100 % = 1.00/18.11 × 100 % = 5.52 %#