A very long pipe is capped at one end with a semipermeable membrane. How deep (in meters) must t

step 1 Okay, in this question, we have a very, very, very long tube. Okay. And at the end of this tube,

A very long pipe is capped at one end with a semipermeable...

Key Concepts

Van’t Hoff Equation

The van’t Hoff equation relates the osmotic pressure of a dilute solution to the concentration of the solute, the temperature, and a constant that accounts for the dissociation of the solute into multiple particles. This equation, in the form ? = iMRT, where ? is the osmotic pressure, i is the van’t Hoff factor, M is the molarity, R is the gas constant, and T is the absolute temperature, provides a fundamental link between thermodynamic properties and osmotic effects.

Osmotic Pressure

Osmotic pressure is the pressure required to stop the net flow of water across a semipermeable membrane due to osmosis. It is quantitatively determined using the concentration of solute particles in a solution, reflecting how solute presence reduces the chemical potential of the solvent. This pressure is a key factor in balancing osmotic flows in biological and engineering systems.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium at any given point within the fluid, due to the gravitational force acting on the fluid column. It increases linearly with depth and is given by the product of the fluid's density, the acceleration due to gravity, and the height (or depth) of the fluid column. This concept is important in understanding how pressure differences can counteract osmotic pressures.

Osmosis

Osmosis is the movement of a solvent, typically water, across a semipermeable membrane from a region of lower solute concentration to a region of higher solute concentration. This process occurs until a balance in chemical potential is reached, and it explains many natural phenomena, such as water absorption by plant roots and the behavior of cells in different environments.

Semipermeable Membrane

A semipermeable membrane is a barrier that allows certain molecules or ions to pass through by diffusion and sometimes by facilitated transport, while blocking others. In the context of osmosis, these membranes let water molecules pass but restrict the movement of solute particles, creating conditions for establishing osmotic gradients.

Transcript

00:01 Okay, in this question, we have a very, very, very long tube.

00:07 Okay.

00:08 And at the end of this tube, we have a membrane.

00:12 Let me draw here, a little membrane right here.

00:16 So only water can get through this membrane.

00:20 And we're going to submerge this tube, this pipe, into seawater.

00:28 Okay? so what's going to happen is the first you might think well water will get into the tube right but there is a pressure that it's keeping the water outside and that's the osmotic pressure right because if any water gets through the membrane it will be pure water in here so the pure water won't be happy here you will want to go back to the more concentrated environment which is the seawater.

01:02 Okay, so in order to overcome this, the pressure of the water column.

01:09 So here we have a pressure due to the column of water.

01:14 This pressure has to be greater than the osmotic pressure.

01:18 Okay, so our task in this problem is first to find the osmotic pressure of the seawater and then compute how deep this pipe, has to go in order to the two pressures to be equal and then the pure water be allowed to enter the pipe.

01:42 Okay.

01:44 So first of all, let's compute the osmotic pressure.

01:48 We know that as a formula, crti.

01:53 Okay.

01:55 Let's plug in the numbers.

01:57 So c is the concentration, malls per liter.

02:00 We have that.

02:01 The problem told us that 0 .7 moles per liter are i'm going to use 0 a2 atm liter kelvin mole and then the temperature is 20 celsius that would be 293 kelvin and i remember for an acl you should have memorized by now that this is equal to so if you compute the pressure, you should get something around 33 .7 .8m.

02:42 So it's a very, it's a huge pressure.

02:46 Okay.

02:46 So you would think to overcome this huge pressure, you would need to go really, really, really deep.

02:55 So next step.

02:57 Now, what's the formula for the pressure of a water column? it's the density times the height of that column of water, which in our case it means the depth and the gravity.

03:16 We have dg and we have the pressure, we just computed, so we can find age by taking p and divided by d and g.

03:27 So let's try to do this.

03:31 So age will be equal 33 .7 atm.

03:38 Okay? and then you divide this by 1 .03 grams per centimeters to the minus 3.

03:47 And then you put the gravity meters per second and minus 2.

03:52 And then you should see a problem here.

03:56 The units are all awkward.

04:00 So when you have a situation like this, the most safe thing to do is to fall back to si units.

04:07 So this is si units, the gravity is an si units.

04:12 The density and the pressure is not.

04:14 So let's convert them.

04:17 First, 33 .7 atm...