Question

If the fluid inside a tree is about \(0.1 M\) more concentrated in solute than the groundwater that bathes the roots, how highwill a column of fluid rise in the tree at \(25^{\circ} \mathrm{C}\) ? Assume that the density of the fluid is \(1.0 \mathrm{~g} / \mathrm{cm}^{3}\). (The density of mercury is \(\left.13.6 \mathrm{~g} / \mathrm{cm}^{3} .\right)\)

Short answer

Given the fluid inside a tree is $0.1\,\text{M}$ more concentrated in solute than the groundwater, we can calculate the height of the fluid column in the tree at $25^{\circ} \mathrm{C}$ using osmotic pressure. The osmotic pressure (π) is given by the Van 't Hoff equation: \[\pi = i M R T.\] After finding the osmotic pressure, convert it to Pascals and use the formula \(P = h\rho g\) to find the height of the fluid column, where \(P\) is the osmotic pressure, \(h\) is the height, \(\rho\) is the density, and \(g\) is the gravitational acceleration. Finally, make sure to convert the height to the desired unit, such as meters, centimeters, or millimeters.

Step-by-step solution

Write the formula for osmotic pressure

The formula for osmotic pressure (π) is given by Van 't Hoff equation: \[\pi = i M R T,\] where \(i\) is the van 't Hoff factor, \(M\) is the molarity of the solute, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Calculate osmotic pressure

Given that the fluid inside the tree is \(0.1\,\text{M}\) more concentrated in solute than the groundwater. The Van 't Hoff factor (\(i\)) is assumed to be \(1\), as no specific solute is mentioned. We have the temperature given as \(25^\circ\text{C}\), which converts to \(298\,\text{K}\) in Kelvin. The ideal gas constant (\(R\)) is \(0.0821\,\dfrac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}}\). Using the formula for osmotic pressure, we can find the osmotic pressure (π): \[\pi = (1)(0.1\,\text{M})(0.0821\,\dfrac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})(298\,\text{K}).\]

Convert osmotic pressure to pascals

Since we have the osmotic pressure in atm, we should convert it to pascals (Pa) to find the height. \(1\,\text{atm} = 101325\,\text{Pa}\). So, \[\pi\,\text{(Pa)} = \pi\,\text{(atm)}\times 101325\,\dfrac{\text{Pa}}{\text{atm}}.\]

Calculate the height of the fluid column

We know that pressure is related to height, density, and gravitational acceleration as: \[P = h\rho g,\] where \(P\) is the osmotic pressure, \(h\) is the height of the fluid column, \(\rho\) is the density of the fluid, and \(g\) is the gravitational acceleration. We are given the density of the fluid, \(\rho = 1.0\,\dfrac{\text{g}}{\text{cm}^3}\), which can be converted to SI units: \[\rho = 1000\,\dfrac{\text{kg}}{\text{m}^3}.\] The gravitational acceleration is approximately \(g = 9.81\,\dfrac{\text{m}}{\text{s}^2}\). We can now solve for the height of the fluid column, \(h\): \[h = \dfrac{P}{\rho g}.\]

Convert height to meters

After calculating the height, the answer will be in meters. Make sure to convert it to a suitable unit if necessary (e.g., centimeters, millimeters, etc.), depending on the magnitude of the value.

Key concepts

Van 't Hoff equation

The Van 't Hoff equation is essential for understanding osmotic pressure in solutions. It is given by the equation: \[ \pi = i M R T \] Here, \(\pi\) represents the osmotic pressure, which is a measure of the pressure required to stop the osmotic flow of water. The Van 't Hoff factor \(i\) accounts for ionization effects in solutes, which means how many particles a solute breaks into in solution. For many simple organic molecules in dilute solution, such as glucose, \(i\) is approximately 1.

• \(M\) is the molarity, the concentration of solute in the solution.
• \(R\) is the ideal gas constant, with a value of \(0.0821\,\dfrac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\).
• \(T\) is the absolute temperature measured in Kelvin (K). Note that to convert degrees Celsius (°C) to Kelvin, you add 273.15.

This equation describes how changes in solute concentration, temperature, and properties of the solute affect the osmotic pressure. It helps predict how much osmotic pressure will be exerted by a given solution, which directly influences many biological and chemical processes.

molarity

Molarity is a key concept when studying solutions and is defined as the number of moles of solute per liter of solution. It is expressed by the unit molar (M), where 1 M is equivalent to 1 mole of solute in 1 liter of solution. This concentration measure is crucial in calculating various parameters like osmotic pressure using the Van 't Hoff equation.
To calculate molarity, you need two pieces of information:

• The number of moles of solute, which can be found using the formula: \(\text{moles} = \dfrac{\text{mass of solute}}{\text{molar mass of solute}}\).
• The volume of solution in liters.

The equation for molarity is:\[ M = \dfrac{\text{moles of solute}}{\text{volume of solution in liters}} \] Understanding molarity allows us to comprehend concentration-related aspects of a solution's behavior, especially in processes like osmosis and chemical reactions.

osmotic potential

Osmotic potential, also known as solute potential, refers to the potential energy of water molecules due to the presence of solute particles in a solution. It quantifies the tendency of water to move across a semi-permeable membrane during osmosis.
In biological contexts, osmotic potential plays a critical role in the movement of water into and out of cells and other compartments. Water will always flow towards the region with a lower (more negative) water potential, which is often the area with higher solute concentration.

• Osmotic potential is inversely related to the solute concentration. More solute particles result in a lower (or more negative) osmotic potential.
• It is one component of the overall water potential, which also includes pressure potential. Together, these determine the direction water will flow.

By understanding osmotic potential, we gain insights into processes such as plant water absorption from the soil, where the concentration gradient drives water movement from root to tip.

fluid dynamics

Fluid dynamics encompasses the study of fluids (liquids and gases) in motion. It is a significant field in physics and engineering, providing insights into how fluids behave under various forces.
In the context of osmotic pressure, fluid dynamics helps explain how liquids like water move through and within organisms, such as rise within trees due to osmosis. Here are some basic concepts:

• Pressure and gravity are key forces influencing fluid movement. Pressure differences can be caused by temperature changes, concentration gradients, or external forces.
• The relationship between pressure, height, density, and gravity in a fluid column is given by the formula: \( P = h \rho g \), where \( P \) is pressure, \( h \) is height, \( \rho \) is fluid density, and \( g \) is gravitational acceleration.
• In biological and environmental systems, these principles explain phenomena such as blood circulation in animals or water flow through plant xylem.

Applying fluid dynamics to osmotic processes enables us to predict how solutions will behave in natural systems, contributing to our understanding of ecology and physiology.