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.