Question

A solution containing \(27.55 \mathrm{mg}\) of an unknown protein per \(25.0 \mathrm{~mL}\) solution was found to have an osmotic pressure of 3.22 torr at \(25^{\circ} \mathrm{C} .\) What is the molar mass of the protein?

Short answer

The molar mass of the protein is \( M_w = \frac{(27.55 \times 10^{-3})}{(\frac{3.22}{(0.0821)(25 + 273.15)}) \times (25 \times 10^{-3})} \text{g/mol}\).

Step-by-step solution

State the equation for osmotic pressure

The van't Hoff equation for osmotic pressure \(\Pi\) at a given temperature \(T\) can be stated as \[\Pi = MRT\] where \(\frac{\begin{array}{l} \(M\) is the molarity of the solution \end{array}}{RT}\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Convert temperature to Kelvin

To convert the temperature from Celsius to Kelvin, use the formula \[T(K) = T(^\text{circ}C) + 273.15\] Convert the given temperature \(25^\text{circ}C\) to Kelvin.

Calculate the molarity of the solution

The molarity \(M\) can be found using the osmotic pressure equation \(\Pi = MRT\). Rearrange the equation to solve for \(M\): \[M = \frac{\begin{array}{l} \(\Pi\) \end{array}}{RT}\]

Calculate the number of moles of the protein

The number of moles \(n\) of the protein can be calculated by multiplying the molarity \(M\) by the volume \(V\) of the solution in liters: \[n = MV\]

Convert the mass of the protein to grams

Convert the mass of the protein from milligrams to grams: \[27.55 \text{mg} = 27.55 \times 10^{-3} \text{g}\]

Calculate the molar mass of the protein

The molar mass \(M_w\) is calculated using the mass of the protein in grams \(m\) and the number of moles \(n\): \[M_w = \frac{m}{n}\]

Solve for molar mass

Insert the calculated values into the molar mass equation and solve for \(M_w\).

Key concepts

van't Hoff equation

The van't Hoff equation is fundamental to understanding osmotic pressure in solutions. Osmotic pressure, denoted by \(\Pi\), is the pressure required to prevent the flow of a solvent into a solution through a semipermeable membrane. According to the van't Hoff equation, the osmotic pressure of a dilute solution is directly proportional to the product of the molarity (M), the ideal gas constant (R), and the absolute temperature (T) in Kelvin.

The equation can be expressed as:\[\Pi = MRT\]The importance of this relationship cannot be overstated when calculating properties of solutions in chemistry, biology, and related fields. It enables the determination of various properties of a solution, such as its molar mass, by relating easily measurable quantities like osmotic pressure and temperature.

molarity and molar mass

Molarity, a measure of concentration, is defined as the number of moles of a solute present in one liter of solution. It allows scientists to communicate and compare the strength of different solutions. On the other hand, molar mass represents the mass of one mole of a substance (usually in grams per mole) and is a critical parameter for quantifying the amount of substance.

To find the molarity and molar mass based on osmotic pressure, one must manipulate the van't Hoff equation and involve several steps of conversion. For molarity (M), it can be isolated as follows: \[M = \frac{\Pi}{RT}\] Once the molarity is known, the molar mass (\( M_w \)) can be calculated if the mass of the solute in the solution (m) and the number of moles (n) are known, using:\[M_w = \frac{m}{n}\]
Understanding these concepts and their interrelations is crucial in solving problems related to osmotic pressure and in broader applications across chemistry.

temperature conversion to Kelvin

When solving for properties that involve temperature, like in the case of osmotic pressure calculations, it is essential to use the Kelvin scale. Kelvin is the base unit of temperature in the International System of Units (SI), and it is imperative because it starts at absolute zero, unlike Celsius or Fahrenheit scales.

Converting Celsius to Kelvin is a straightforward process, which is vital as the van't Hoff equation requires temperature to be in Kelvin. The conversion formula is:\[T(K) = T(^\text{circ}C) + 273.15\]
In our context, it means that the given temperature of \(25^\text{circ}C\) becomes:\[T(K) = 25 + 273.15 = 298.15K\]Remembering to convert temperature to Kelvin is a simple yet important step in ensuring the accuracy of osmotic pressure calculations and many other thermodynamic calculations in science.