A 50.00 mL sample of an aqueous solution contains 1.08 g of human serum albumin in the blood. What is the molar mass of plasma protein? The solution has an osmotic pressure of 5.85 mmHg at 298 K. What is the mass of the albumin
The Correct Answer and Explanation is:
To determine the molar mass of human serum albumin (a plasma protein), we can use the formula for osmotic pressure:Π=MRT\Pi = MRTΠ=MRT
Where:
- Π\PiΠ is the osmotic pressure (in atm),
- MMM is the molarity (mol/L),
- RRR is the ideal gas constant (0.0821 L·atm/mol·K),
- TTT is the temperature in Kelvin.
Step 1: Convert osmotic pressure to atm
Given:Π=5.85 mmHg\Pi = 5.85\ \text{mmHg}Π=5.85 mmHg
Convert to atm:Π=5.85 mmHg760 mmHg/atm=0.007697 atm\Pi = \frac{5.85\ \text{mmHg}}{760\ \text{mmHg/atm}} = 0.007697\ \text{atm}Π=760 mmHg/atm5.85 mmHg=0.007697 atm
Step 2: Rearranged osmotic pressure formula to solve for molarity
M=ΠRT=0.007697(0.0821)(298)=0.00769724.5158≈3.14×10−4 mol/LM = \frac{\Pi}{RT} = \frac{0.007697}{(0.0821)(298)} = \frac{0.007697}{24.5158} \approx 3.14 \times 10^{-4}\ \text{mol/L}M=RTΠ=(0.0821)(298)0.007697=24.51580.007697≈3.14×10−4 mol/L
Step 3: Use volume to calculate moles
Volume = 50.00 mL = 0.05000 Lmoles=M×V=(3.14×10−4)(0.05000)≈1.57×10−5 mol\text{moles} = M \times V = (3.14 \times 10^{-4})(0.05000) \approx 1.57 \times 10^{-5}\ \text{mol}moles=M×V=(3.14×10−4)(0.05000)≈1.57×10−5 mol
Step 4: Calculate molar mass
Given mass of albumin = 1.08 gMolar mass=massmoles=1.081.57×10−5≈68,790 g/mol\text{Molar mass} = \frac{\text{mass}}{\text{moles}} = \frac{1.08}{1.57 \times 10^{-5}} \approx 68,790\ \text{g/mol}Molar mass=molesmass=1.57×10−51.08≈68,790 g/mol
Final Answer:
- Molar mass of albumin = 68,790 g/mol
Explanation:
This problem involves colligative properties, specifically osmotic pressure, which is useful for determining the molar mass of large biomolecules like proteins. Osmotic pressure depends on the number of particles (moles of solute) rather than their size or nature. The key relationship is Π=MRT\Pi = MRTΠ=MRT, where Π\PiΠ is osmotic pressure, MMM is molarity, RRR is the gas constant, and TTT is temperature in Kelvin.
First, osmotic pressure is given in millimeters of mercury (mmHg), which we must convert to atmospheres since the gas constant R=0.0821R = 0.0821R=0.0821 uses atm. Dividing the pressure in mmHg by 760 gives the value in atm. The temperature is already in Kelvin, so we can proceed to calculate molarity.
Next, the rearranged formula M=Π/RTM = \Pi / RTM=Π/RT allows us to solve for molarity using the known pressure, gas constant, and temperature. We then multiply this molarity by the volume of the solution in liters to find the number of moles of protein present.
Knowing the mass of the protein (1.08 grams) and the number of moles it represents, we divide the mass by the moles to find the molar mass. This results in a very high molar mass, approximately 68,790 g/mol, which is expected for a large protein like albumin.
This method is often used in biochemistry to estimate molecular weights of macromolecules that cannot be measured easily through simpler methods due to their large size.
