Question

A solution containing 27.55 mg of an unknown protein per 25 mL solution was found to have an osmotic pressure of 3.22 torr at 25 C. What is the molar mass of the protein?

Answer 1

The molar mass of the protein is calculated using the osmotic pressure equation and the given concentration, osmotic pressure, and temperature. After conversions and calculations, the molar mass is found to be 6376 g/mol.

Step-by-step explanation:

To calculate the molar mass of the protein, we will use the osmotic pressure equation, which is derived from the ideal gas law: \(\Pi = nRT/V\), where \(\Pi\) is the osmotic pressure, \(n\) is the number of moles of solute, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(V\) is the volume of the solution.

First, we convert the osmotic pressure from torr to atm using the conversion factor 1 atm = 760 torr:
Osmotic pressure = 3.22 torr x (1 atm / 760 torr) = 0.00424 atm

Next, convert the temperature from Celsius to Kelvin:
Temperature (K) = 25 °C + 273.15 = 298.15 K

Now we can rearrange the osmotic pressure equation to solve for \(n\), the number of moles:

\(n = \Pi V / (RT)\)

Substitute the values into the equation and use \(R = 0.0821 (L \cdot atm) / (K \cdot mol)\) which is the ideal gas constant value:

\(n = (0.00424 atm \cdot 0.025 L) / (0.0821 (L \cdot atm) / (K \cdot mol) \cdot 298.15 K)\) = 4.32 x 10^{-6} moles

The mass of the protein is given as 27.55 mg, which we convert to grams:
Mass of Protein = 27.55 mg x (1 g / 1000 mg) = 0.02755 g

Finally, calculate the molar mass (M) by dividing the mass by the number of moles:
Molar mass (M) = Mass of protein / n

Molar mass (M) = 0.02755 g / 4.32 x 10^{-6} moles

Molar mass (M) = 6376 g/mol

The molar mass of the protein is therefore 6376 g/mol.

Answer 2

The molar mass of the unknown protein is approximately \(0.00664\) g/mol.

To find the molar mass of the unknown protein, we can use the formula for osmotic pressure:

`\[ \Pi = (n)/(V)RT \]`

Where:

-
`\(\Pi\)` is the osmotic pressure (in atm),

-
`\(n\)` is the number of moles of solute (protein),

-
`\(V\)`is the volume of the solution (in liters),

-
`\(R\)` is the ideal gas constant, and

-
`\(T\)` is the temperature (in Kelvin).

The formula can be rearranged to solve for the molar mass (\(M\)):

`\[ M = \frac{\text{mass of solute}}{n} = \frac{\text{mass of solute} * V}{\Pi * RT} \]`

Given:

- Mass of protein = 27.55 mg = 0.02755 g (since 1 mg = 0.001 g)

- Volume of solution = 25 mL = 0.025 L (since 1 mL = 0.001 L)

- Osmotic pressure = 3.22 torr = 3.22/760 atm (since 1 atm = 760 torr)

- Temperature = 25°C = 298 K (since Kelvin = Celsius + 273)

The ideal gas constant \(R\) is 0.0821 L·atm/(K·mol).

Let's calculate the molar mass.

The molar mass of the unknown protein is approximately \(0.00664\) g/mol.