Question #03a15

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 (#"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 (#"1100 kg·m"^"-3"#)
• #g# is the acceleration due to gravity (#"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.