Question

Creatinine, \(\mathrm{C}_{4} \mathrm{H}_{7} \mathrm{N}_{3} \mathrm{O},\) is a by-product of muscle metabolism, and creatinine levels in the body are known to be a fairly reliable indicator of kidney function. The normal level of creatinine in the blood for adults is approximately \(1.0 \mathrm{mg}\) per deciliter (dL) of blood. If the density of blood is \(1.025 \mathrm{g} / \mathrm{mL},\) calculate the molality of a normal creatinine level in a 10.0 -mL blood sample. What is the osmotic pressure of this solution at \(25.0^{\circ} \mathrm{C} ?\)

Short answer

The molality of creatinine in the 10.0 mL blood sample is approximately \(8.61 \times 10^{-5} m\), and the osmotic pressure of this solution at 25.0 °C is about 0.00221 atm.

Step-by-step solution

Calculate the mass of creatinine in the blood sample

Given that the normal level of creatinine is 1.0 mg/dL, we should first find the mass of creatinine in the 10.0 mL blood sample. Since there are 100 mL in 1 dL, we can calculate the mass of creatinine as follows: \(mass \, of \, creatinine = \frac{1.0 \, mg}{1 \, dL} \times \frac{10.0 \, mL}{100 \, mL}\) \(mass \, of \, creatinine = 0.1 \, mg\) To convert it to grams, we simply multiply by the conversion factor: \(mass \, of \, creatinine (grams) = 0.1 \, mg \times \frac{1 \, g}{1000 \, mg} = 0.0001 \, g\)

Calculate moles of creatinine

Now we need to convert grams of creatinine to moles. We can use the formula: \( moles \, of \, creatinine = \frac{mass \, of \, creatinine}{molar \, mass \, of \, creatinine} \) Where molar mass of creatinine = 4(12.01 g/mol) + 7(1.01 g/mol) + 3(14.01 g/mol) + 1(16.00 g/mol) = 113.15 g/mol \( moles \, of \, creatinine = \frac{0.0001 \, g}{113.15 \, g/mol} \approx 8.83 \times 10^{-7} \, mol\)

Calculate mass of solvent (blood)

Next, we should find the mass of blood in the 10.0 mL sample. Using the given density: \(mass \, of \, blood (g) = volume \, of \, blood \times density \, of \, blood\) \(mass \, of \, blood (g) = 10.0 \, mL \times \frac{1.025 \, g}{1 \, mL} = 10.25 \, g\)

Calculate molality of creatinine

Now, we can calculate the molality of creatinine using the formula: \(molality (m) = \frac{moles \, of \, solute}{mass \, of \, solvent (kg)}\) \(molality (m) = \frac{8.83 \times 10^{-7} \, mol}{0.01025 \, kg} \approx 8.61 \times 10^{-5} \, m\)

Calculate the osmotic pressure

Finally, we need to find the osmotic pressure of this solution at 25.0 °C. We'll use the formula: \(osmotic \, pressure = molality \times R \times T\) Where R is the gas constant (0.0821 L atm/mol K) and T is the temperature in Kelvin (25.0 + 273.15 = 298.15 K). \(osmotic \, pressure = (8.61 \times 10^{-5} \,m)\times(0.0821\, \frac{L\, atm}{mol\, K}) \times (298.15\, K)\) \(osmotic \, pressure \approx 0.00221 atm\) So, the molality of creatinine in the blood sample is approximately \(8.61 \times 10^{-5} m\), and the osmotic pressure of this solution at 25.0 °C is about 0.00221 atm.

Key concepts

Molality

Molality is a useful concentration unit in chemistry, which specifically refers to the number of moles of a solute per kilogram of solvent.
Unlike molarity, molality is independent of temperature because it relies on the mass rather than the volume of the solvent. This characteristic makes it particularly useful in scenarios involving temperature fluctuations.
To calculate molality, we use the formula: \[\text{molality (m)} = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}\]
For example, if you have a solution where you dissolve a substance in blood, the molality focuses on the moles of that substance compared to the mass of the blood (or other solvent) in which it is dissolved. In this exercise, we found that the molality of creatinine in the blood was approximately \(8.61 \times 10^{-5} \ m\). This value indicates how concentrated the creatinine is relative to a given mass of blood.

Osmotic Pressure

Osmotic pressure is a critical concept in understanding how solvents travel across a semipermeable membrane due to differences in solute concentration. It's crucial in biological as well as chemical processes.
The formula used to calculate osmotic pressure (\(\Pi\) ) in dilute solutions is: \[\Pi = m \times R \times T\]
where \(m\) is the molality, \(R\) is the ideal gas constant (typically \(0.0821 \ L \cdot atm / mol \cdot K\)), and \(T\) is the absolute temperature in Kelvin.
In the exercise provided, the osmotic pressure of creatinine in the blood at 25 °C was approximately \(0.00221 \ atm\). This shows the force with which the solvent would naturally push through a membrane to equalize creatinine concentration on both sides under given conditions.

Mole Concept

The mole concept is a fundamental principle in chemistry that facilitates the understanding of chemical quantification. It links mass to number of entities, whether these are atoms, molecules, ions, etc.
A mole is defined as \(6.022 \times 10^{23}\) entities, known as Avogadro's number, and it allows chemists to convert between atomic or molecular masses and macroscopic amounts of material. This is crucial when dealing with reaction stoichiometry and solution concentrations.
In our exercise, we calculated the number of moles of creatinine using its mass and molar mass, which was found to be approximately \(8.83 \times 10^{-7}\) moles in the sample. Having this value enabled us to determine both the molality and the subsequent osmotic pressure of the creatinine in the blood.

Solution Chemistry

Solution chemistry dives into the analysis and description of solute-solvent interactions and how concentrations affect solution properties. It's a vast field encompassing concepts like solubility, conductivity, and colligative properties.
A key feature in solution chemistry is understanding how different properties emerge when substances are mixed, such as changes to boiling and freezing points, or the generation of osmotic pressure.
In this exercise involving creatinine in blood, our focus was on colligative properties like osmotic pressure, which depend on solute concentration. We explored how a small amount of creatinine in a blood sample could be quantified and how its interaction with the solvent (blood) affected the sample's osmotic characteristics. Such understanding is vital in fields ranging from biology to industrial chemistry.