# Question 03a15

Mar 16, 2016

The concentration of the sap should be 0.059 mol/L.

#### Explanation:

The sap will rise until the hydrostatic pressure due to the height of the sap column just balances the osmotic pressure.

The formula for osmotic pressure is

color(blue)(|bar(ul( Π = cRT)|), where

• $c$ = molarity of solute
• $R$ = universal gas constant ("8.314 kPa·L·K"^"-1""mol"^"-1")
• $T$ = temperature ($\text{288.15 K}$)

The formula for hydrostatic pressure is

color(blue)(|bar(ul( P = hρg)|)

where

• $h$ is the height of the column
• ρ is the density of the sap ($\text{1100 kg·m"^"-3}$)
• $g$ is the acceleration due to gravity ($\text{9.81 m·s"^"-2}$)

Our first task is to determine the hydrostatic pressure of a 13 m high column of sap.

P = hρg = "13 m" × "1100 kg·m"^"-3" × "9.81 m·s"^"-2" = 1.40 × 10^5color(white)(l) "kg·m"^"-1""s"^"-2"
= 1.40 × 10^5color(white)(l) "Pa" = "140 kPa"

Now, we calculate the concentration of sugar sap needed to generate this osmotic pressure.

Π = cRT

c = Π/(RT) = (140 color(red)(cancel(color(black)("kPa"))))/(8.314 color(red)(cancel(color(black)("kPa")))·"L"·color(red)(cancel(color(black)("K"^"-1")))"mol"^"-1" × 288.15 color(red)(cancel(color(black)("K")))) = "0.059 mol/L"#

The concentration of sugar sap is 0.059 mol/L.