# Question #6d84d

##### 1 Answer

#### Explanation:

As you know, **percent concentration by mass** is defined as the *mass of the solute*, which in your case is sugar, divided by the **total mass of the solution**, and multiplied by

#color(blue)("% w/w" = "mass of solute"/"mass of solution" xx 100)#

The problem provides you with the mass of the solution, which is said to be equal to

#m_"sol" = "5.2752 g"#

You will also need to use the **density** of water, which at room temperature is equal to

#rho_w = "0.9982 g mL"^(-1)#

http://antoine.frostburg.edu/chem/senese/javascript/water-density.html

and the density of solid sugar, or sucrose, which is listed as

#rho_s = "1.587 g mL"^(-1)#

Now, your goal here will be to find the mass of solute,

As you know, density is defined as mass per unit of volume, which means that you can express volume using density and mass

#color(blue)(rho - m/V implies V = m/(rho))#

Now, the **total volume** of the solution will be equal to the *volume of sugar*, *volume of water*,

#V_"sol" = V_s + V_w#

This is equivalent to

#V_"sol" = m_s/(rho_s) + m_w/(rho_w)" " " "color(red)("(*)")#

Here

You also know that the **total mass of solution**,

#m_"sol" = m_s + m_w#

This means that the mass of water can be written as

#m_w = m_"sol" - m_s#

Plug this into equation

#V_"sol" = m_s/(rho_s) + m_"sol"/(rho_w) - m_s/(rho_w)#

Rearrange to isolate

#m_s * (1/rho_s - 1/rho_w) = V_"sol" - m_"sol"/rho_w#

#m_s * ((rho_w - rho_s)/(color(red)(cancel(color(black)(rho_w))) * rho_s)) = (V_"sol" * rho_w - m_"sol")/color(red)(cancel(color(black)(rho_w)))#

Finally, you will get

#m_s = (V_"sol" * rho_w - m_"sol") * rho_s/(rho_w - rho_s)#

Plug in your values to get the value of

#m_s = (5.00 color(red)(cancel(color(black)("mL"))) * "0.9982 g" color(red)(cancel(color(black)("mL"^(-1)))) - "5.2752 g") * (0.1.587 color(red)(cancel(color(black)("g mL"^(-1)))))/(0.9982color(red)(cancel(color(black)("g mL"^(-1)))) - 1.587color(red)(cancel(color(black)("g mL"^(-1)))))#

#m_s = "0.766 g"#

This means that the solution's percent concentration by mass is

#"% w/w" = (0.766 color(red)(cancel(color(black)("g"))))/(5.2752color(red)(cancel(color(black)("g")))) xx 100 = color(green)("14.5%") -># rounded to threesig figs

To test the result, calculate the density of the sugar solution

#rho_"sol" = "5.2752 g"/"5.00 mL" = "1.055 g mL"^(-1)#

This density is characteristic of a sugar solution that is approximately