The concentration of a solution is a measure of the amount of solute that is dissolved in the solvent. While concentrations can be used to describe any mixture, we will typically deal with binary mixtures (two components) that are liquid solutions (a liquid solvent with something disolved in it). There are a number of ways to define concentration but all of them have the same essential idea of expressing the amount of material (solute) that is dissolved in the mixture compared to the amount of liquid (solvent).
By far the most common concentration term for a chemist is molarity. It is the ratio of moles of solute to the overall volume of the solution in liters.
\[{\rm molarity} = {{\rm mol\;solute}\over{\rm L\;solution}}\]
And although IUPAC recommends that the units for molarity be explicitly shown (mol/L), it is still common practice to use the abbreviation of a capital "M" for molarity. So an example of this would be a 0.25 M HCl solution or even say a solution of 1.5 × 10-5 M NO3− ions.
Another common unit is that of molality. For this description of concentration the solute is measured in moles (just like molarity), but the denominator is the mass of the solvent (not solution) in kg.
\[{\rm molality} = {{\rm mol\;solute}\over{\rm kg\;solvent}}\]
Molality is a more robust unit of concentration than molarity since the number of moles and mass don't change with conditions (temperature and pressure). However, this unit is rarely used. Also, because the mass of 1 L of water is 1 kg, molarity and molality as essentially the same for aqueous solutions (at room temperature).
The last common chemical unit of concentration is that of the mole fraction. It is just the ratio of the moles of measured component (be it solute or solvent - it doesn't matter) to ALL the moles (total) in the mixture.
Consider the mole fraction of A in a mixture of A, B, and C.
Let \(x_{\rm A}\) represent the mole fraction of A.
\[ x_{\rm A} = {n_{\rm A}\over {n_{\rm B} + n_{\rm C} + n_{\rm A}}}\]
Where \(n\) is the number of moles in each component. Note that mole fraction is a fraction and is a unitless number between 0 and 1.
There are other concentration measures that you will encounter as well. These are typically practical laboratory units (ones that are easy to measure). For most chemistry problems you would typically need to convert from these units to molarity. Examples of these units are percent by weight (or percent by volume). Mass percent is simply
\[\%{\rm conc} = {{\rm mass\;of\;solute}\over{\rm mass\;of\;solution}}\times 100\% \]
Below is a video lecture on concentrations in two parts
Part 1 of video on concentrationsColligative properties are properties of solutions that depend on the particular solvent and on the concentration, but they do not depend on the nature of the solute. To a first degree, all colligative properties are related to ideal solutions. These are solutions in which the solute and solvent have identical intermolecular forces. If this is the case, then the enthalpy of solution, \(\Delta H_{solution} = 0\) since making the solution has no effect on the potential energy of the molecules (ions). However, mixing the solution will raise the entropy of the solution, \(\Delta S_{solution} > 0\). Thus, colligative properties are what we would call an entropic effect (they are a result of entropy).
There are four properties that we will look at. All are simply related to the concentration of the solution. However, for historical reasons the units for the concentration are different for the different phenomena
They are
The lowering of Vapor Pressure (Raoult’s Law). The vapor pressure of a solution will be lower than the pure substance. This is the basis of understanding distillation. It is also the same idea as boiling point elevation.
Boiling Point Elevation. The boiling point of a solution will be higher than the pure substance. This is the same as the lowering of the vapor pressure. We have a handy formula for the boiling point.
Freezing Point Depression. The freezing point of a solution will be lower than the pure substance. This is why you salt the sidewalk to melt ice. And why you use ice and salt to make ice cream.
Osmosis. Explains the movement of solvent between solutions of different concentrations separated by a membrane. A critical concept for cell biology. An important means to purify water.
Answer: IT IS ALL IN THE ENTROPY!
Remember we have assumed the solutions are ideal. Thus the difference is the entropy, not the enthalpy.
Because what matters is the total amount of solute, it is important to realize that ionic solids break up into multiple ions in solution. Thus 1 M of NaCl solution has an effective concentration of 2M. This is because it has a 1 M concentration of Na+ ions and a 1 M concentration of Cl- ions. It is very important to account for all the dissolved particles in soltion.
The number of ions that form up dissolving a particular ionic solid is called the van't Hoff factor and is often denoted by a lower case i. For molecular solids i=1 since they don't dissociate in water. For ionic compounds, i is equal to the number of ions in the compound.
So from an ideal solution perspective the freezing point of a 1 M solution of sugar (i=1) will be identical to a 0.333 M solution of RbCl2 (i=3).
Note: In chemistry we generally assume that ions are dissociated when they are in solution. In fact, at high concentrations positive and negative ions can form pairs that stay in solution. This is different from reaching the solubility limit and having solids precipitated out of solution. Ion pairing reduces the vant' Hoff factor (i) from whole numbers. Measuring a colligative effect is a means to measure "i" to gauge the effect of ion pairing. For example you might predict that a 0.1 m solution of sodium chloride would be effectively 0.2 m in concentration (i=2). However, in practice you might note that the change in properties for the solution is for a concentration of 0.187 m. This difference is the result of ion pairing. We will always be using the "ideal" vant' Hoff factor that assumes total dissociation of ions unless it is explicitly noted.
The vapor pressure of solution is lower than that of the pure liquid. More correctly, the vapor pressure of solution containing a non-volatile (zero vapor pressure) solute is lower than that of the pure solvent. So if you dissolve sugar into water, the vapor pressure of the solution is lower than that of pure water. There are a number of ways to think about this phenomena but it important to realize that the main effect is a result of entropy.
The higher the entropy of a substance the lower its free energy. If we compare the entropy of a solution to that of a pure solvent, the entropy of the solution will be higher. Therefore, the free energy of the solution will be lower than that of the pure solvent. Lower free energy is more stable. From a thermodynamic standpoint, the solution is more stable than a pure solvent. Because the solution is more stable, fewer molecules are leaving to the gas phase. This lower evaporation rate leads to a lower vapor pressure.
From an ideal solution perspective (ΔHsolution=0) this effect is straight forward to calculate. The vapor pressure of the solution simply depends on the mole fraction of solvent. This idea is know as Raoult's Law. It states
\[P_{\rm solution} = \chi _{solvent} \; P^{\circ}\]
Where \(P^{\circ}\) is the vapor pressure of the pure solvent. What matters is not what the solute is, but how many moles of it there are. The more moles of solute, the lower the mole fraction of solvent, the lower the vapor pressure. Since we often write properties of solution based on the concentration of solute (rather than solvent) this formula can be re-written as a change in vapor pressure as
\[\Delta P = -\chi _{solute} \; P^{\circ}\]
It is important to note that this effect is relatively small. That is because even in very very concentrated solutions, the majority of the mixture is still solvent. Thus the solvent mole fraction is always very close to 100%. The only way to dramatically lower the mole fraction is to examine mixture of two liquids (rather than solids dissolved in liquids).
For vapor pressure lowering the effect also depends on the total concentration of all solutes. However, because the concentration is written as a mole fraction it is difficult to include the vant' Hoff factor i in the formula. This is because the mole fraction is a ratio. In this case the ratio of the moles of solute : the total number of moles. The total number of moles depends on the moles of solute and the moles of solvent. So, for an ionic solute remember to take into account the number ions formed in solution when calculating the moles of solute.
Vapor PressureBoiling point elevation is an identical concept to vapor pressure reduction. Since the vapor pressure of the solution is lower than that of the pure solvent, you need to raise the temperature to an even higher point compared to the pure solvent to get the solution to boil. Conceptually the effect is the same as vapor pressure lowering. The solution has a lower free energy than the pure solvent. This means that it is more stable over a larger temperature range. On a phase diagram this results in an expansion of the "liquid" region which results in an increase in the boiling point (and a decrease in the freezing point). This can be seen on the diagram below:
Quantitatively, the change in temperature can be calculated from a number of factors that include the enthalpy of vaporization, the pure boiling point, and the concentration of the solution. These are typically all wrapped up in a single constant resulting in a simple formula.
\[ \Delta T = iK_b \; m\]
Kb is a constant that depends on the solvent and m is the total solute concentration in molality. Kb is called the boiling point elevation constant or the ebullioscopic constant. The little "i" in formula is the van't Hoff factor for how many ions an electrolyte (ionic solute) breaks up into.
This effect is generally very small. Value of Kb depend on the solvent but they are typically in the range of 0.5 - 6 °C molal-1. For example, the Kb for water is only 0.5 °C molal-1. So even a highly concentrated 1 M NaCl solution has a boiling point increase of only 1 °C. (i = 2 For water 1 M = 1 m. ΔT = (2)(0.5 °C m-1)(1 m) = 1 °C)
Boiling Point ElevationThe freezing point for a solution goes down compared to the pure solvent for the same reason the boiling point goes up: the solution is more stable (it has a lower free energy) compared to the pure solvent. Since it is relatively more stable it exists over a wider temperature range. This can be easily seen again on a phase diagram where the solution is depicted with the "larger" liquid region.
Quantitatively, the change in freezing point can be calculated again in an approximate formula based on the molality of the solution.
Freezing Point depression
\[ \Delta T = -iK_f \; m\]
Again \(K_f\) is a constant that depends on the solvent and m is the total solute concentration in molality. Kf is called the freezing point depression constant or cryoscopic constant.
Sometimes this formula doesn't have the negative sign and you simply need to remember that freezing point goes down. i is the vant' Hoff factor and for ionic solutes accounts for the total number of ions the solute breaks up into.The effect of freezing point depression is generally larger than that of boiling point elevation as values of Kf are typically larger than those of Kb. They range from around 1.5 to as much as 40 °C molal-1. For water, Kb = 1.8 °C molal-1. So a 1 M NaCl solution should freeze at -3.6 °C. (i=2 since there are two ions) sFor water 1 M = 1m. ΔT = -(2)(1.8 °C m-1)(1 m) = -3.6 °C)
Here is video about Freezing Point Depression using a molecular solute thus making the van't Hoff factor equal to one (i = 1).
Here is video about Freezing Point Depression using an ionic solute where the van't Hoff factor is equal to three (i = 3).
Osmosis is a critical phenomena related to the movement of solvent across semi-permeable membranes. A semi-permeable membrane is one through which solvent molecules can pass, but solute molecules cannot. This may seem like a very odd material (and in some ways it is), but biological cells are composed of such membranes. Thus the importance of osmosis in biology, as it has a strong effect on how solvent is shuttled in and out of cells.
First, what is osmosis? Osmosis is the spontaneous movement of solvent through a semi-permeable membrane. It occurs when two solutions of different concentrations are separated by such a membrane. Remember, the free energy of a solution depends on its concentration. Higher concentration solutions have lower free energy. That is higher concentration solution are thermodynamically more stable. When two solutions of different concentrations are placed on opposite sides of a membrane, there is a free energy difference between the two sides. As a result, the solvent will move in such a way as to lower its free energy. This means it will pass through the membrane from the low concentration side to the higher concentration side. This movement of solvent is called osomsis. The solvent is said to "osmose". The osmosis will continue until both side of the membrane have the same concentration (same free energy). This is the equilibrium state. Take the example shown below.
A tube has a semi-permeable membrane at the center with two solutions on either side. The solution on the left has a higher concentration than the one on the right. Thus the solvent will move from the righthand side to the lefthand side to get to the solution of lower free energy. This will lead to a dilution of the solution that was initially more concentrated and a concentration of the solution that was initially more dilute. The flow will continue until the two sides are equal in free energy. The movement of solvent from one side to the other, will lead to a height difference between the two sides. This height can be converted into a pressure. The pressure is given as
\[ \Pi = \rho g h\]
Where Π is the pressure and is related to the density of the solution, \(\rho\), the acceleration due to gravity, \(g\), and the height of the solution, \(h\). It is important to take care with units with this formula. In general, it is best to use densities in kg m-3, g = 9.8 m s-2, and heights in meters. This will yield pressures in Pascals. This pressure is also related to the concentration difference between the two solutions. It is given as
\[ \Pi = iMRT\]
Where again Π is the osmostic pressure, M is the difference in concentration of the solutions, R is the ideal gas constant, and T is the temperature is in Kelvin. Again, units are critical. If the concentration is in molarity (moles per liter), then using R in L-atm will give pressures in atm. Remember the concentration that matters is the total concentration of all solutes. Any ions that dissociate you need to account for the van't Hoff factor, i. If you are calculating the "osmotic pressure" of a solution. It is assumed this is a comparison to a pure solvent. Thus the difference in concentration is simply the concentration of the solution.
Unlike boiling point elevation and freezing point depression, the effects of osmosis can be large. For a 1 M NaCl solution the osmotic pressure is 48.9 atm (i = 2, Π = iMRT = (2)(1 mol L-1)(0.08206 L-atm K-1 mol-1)(298 K) = 48.9 atm. This is a substantial pressure.
Another way to think about the osmotic pressure is that it is the pressure that needs to be applied to stop the osmotic flow. This leads to another important concept: reverse osmosis. This the non-spontaneous transport of the solvent through the semipermeable membrane from concentrated solutions to dilute. This is a process that can be used to purify water. How can the solvent flow be reversed? By greatly increasing the free energy of the concentrated solution by adding pressure. Applying a pressure equal to the osmotic pressure will stop the osmosis. Applying a pressure greater than the osmotic pressure will reverse the osmosis. As we apply pressure to the concentrated side, the solvent will flow the other direction through the membrane resulting in pure solvent. One challenge with reverse osmosis is that as you produce pure solvent, the solution gets more and more concentrated. The means the osmotic pressure is increasing and a ever higher pressure is required to continue the process. A major problem with reverse osmosis for water purification is that the membranes can burst at such high applied pressures.
A couple of other thoughts. You might wonder how it is possible to construct a semi-permeable membrane that can pass the solvent but not solute. In general these are simply barriers with very small holes. If the solvent is small and the solute is big, the semi-permeable membrane acts essentially as a filter. But how does the membrane work when the solvent is water and the solute is an ion like Na+? Isn't Na+ much smaller than a water molecule? Yes and no. A Na+ in the gas phase is much smaller than a water molecule. But we are looking at a Na+ in solution. More correctly we are interested in Na+(aq). In aqueous solution, the Na+ is strongly interacting with a number of water molecules. Thus it is not smaller than a water molecule, but instead it is the size of 6-8 water molecules. Now we can construct a barrier that will let the water pass and hold back the ions. In addition, the ions are charged and the water is neutral. Thus we have additional means of "filtering" out the solute.
Finally, osmosis is very important for biological cells. If we place cells into solutions in which the concentration of the solution outside the cell is much lower than inside the cell, water will spontaneously move through the cell membrane into the higher concentration inside. If the concentration difference is sufficiently high this process will continue until the cell wall burst. Conversely, if we place the cells into a solution in which the concentration outside the cell is higher than inside, the water in the cells will spontaneous move out of the cells into the outer solution. This will effectively remove water from the cells. But if we ensure the concentration inside and outside is identical, then there will be no free energy difference between the inside and the outside and the rate or water leaving the cell will be identical to the rate of water entering the cell. These three conditions are depicted below. When the outer solution has a higher concentration than inside the cell, called hypertonic, the cells shrivel. When the concentration outside is lower than inside, called hypotonic, the water moves in and the cells burst. When the concentrations are the same, isotonic, the cells free exchange water with the solution without resulting in a concentration change.
So you want to calculate a change for a solution compared to a pure solvent? You need the formula for the colligative property. There are three things to remember about such calculations.
1. They assume what matters is the total concentration of "stuff" in the solution (not what that stuff is). This is what we mean by colligative property. What matters is the concentration and the solvent.
2. The above statement is an approximation, so don't expect these answers will be exact in the real world.
3. The formulas all have different historical derivations. As such, they tend to use different concentration units. The concept is always the same. In the details of the calculations, be sure to pay attention to the units.
Boiling Point elevation
\[ \Delta T = iK_b \; m\]
\(K_b\) is a constant that depends on the solvent and m is the total solute concentration in molality. The little "i" is the van't Hoff factor for how many ions an electrolyte breaks up into. If you have a mixture of many solutes. You need to remember the concentration that matters is the total concentration of solute "particles". So in this case you have to add up i times the molality for each solute. This also assumes that the solute is non-volatile (that is the vapor pressure of the solute itself is approximately zero). Therefore this will work well for solutions like salt in water since NaCl is non-volatile. It won't work well for solutions like ethyl alcohol in water since the alcohol itself also has a vapor pressure.
Freezing Point depression
\[ \Delta T = -iK_f \; m\]
Again \(K_f\) is a constant that depends on the solvent and m is the total solute concentration in molality. Sometimes this formula doesn't have the negative sign and you simply need to remember that freezing point goes down.
Vapor Pressure
There are two formulas for vapor pressure of a solution. They are really exactly the same. One is written in term of the concentration of the solute and the other in the concentration of the solvent. The solvent formula is the following
\[P_{\rm solution} = \chi _{solvent} \; P^{\circ}\]
Where \(P^{\circ}\) is the vapor pressure of the pure solvent. This relationship is known as Raout's Law. The formula can be re-written as a change in vapor pressure as
\[\Delta P = -\chi _{solute} \; P^{\circ}\]
Because this formula is written in terms of mole fraction, it is not easy to include the i for the van't Hoff factor in this formula since uses mole fraction to measure the concentration (and the moles of solute will appear in both the numerator and denominator of the fraction). You still need to include the fact that you have multiple ions for an ionic solute. For example if you have a 1m solution of NaCl in water. This is 1 mole of NaCl in 1 kg of water. 1 mole of NaCl will lead to 2 moles of solute (1 mole of Na+ and 1 mole of Cl-). 1kg of water is 55.55 moles of water. Therefore the mole fraction of solute is 0.035 [(2)/(2+55.55)]. Note: this is very close to what you get if you just look at the ratio of moles of solute to moles of solvent 0.036 (2/55.55).
Osmotic Pressure
The osmotic pressure for a solution is found from the concentration in molarity and is the same regardless of the solvent. It is
\[ \Pi = iMRT\]
where the osmotic pressure is Π, i is the van't Hoff factor, M is the molarity of the solute, R is the ideal gas constant (typically in units of L-atm K-1 mol-1, and T is the temperature in the Kelvin. Note the units of the gas constant will determine the pressure units.
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