Question

The blood sugar (glucose) level of a diabetic patient is approximately \(0.140 \mathrm{~g}\) of glucose \(/ 100 \mathrm{~mL}\) of blood. Every time the patient ingests \(40 \mathrm{~g}\) of glucose, her blood glucose level rises to approximately \(0.240 \mathrm{~g} / 100 \mathrm{~mL}\) of blood. Calculate the number of moles of glucose per milliliter of blood and the total number of moles and grams of glucose in the blood before and after consumption of glucose. (Assume that the total volume of blood in her body is \(5.0 \mathrm{~L}\).

Short answer

Before ingestion: 0.03885 mol (7.00 g); After ingestion: 0.06665 mol (12.00 g) of glucose.

Step-by-step solution

Calculate Moles of Glucose per Milliliter of Blood (Before Ingestion)

To find the moles of glucose per milliliter of blood before ingestion, we need to determine the moles of glucose corresponding to the initial concentration of glucose which is 0.140 g/100 mL.\[\text{Moles of glucose} = \frac{\text{mass of glucose (g)}}{\text{molar mass of glucose (g/mol)}}\]The molar mass of glucose (C6H12O6) is 180.16 g/mol. Thus, the moles of glucose per 100 mL is:\[\text{Moles in 100 mL} = \frac{0.140 \text{ g}}{180.16 \text{ g/mol}} = 0.000777 \text{ mol}\]Then, moles per milliliter is:\[\text{Moles per mL} = \frac{0.000777 \text{ mol}}{100 \text{ mL}} = 7.77 \times 10^{-6} \text{ mol/mL}\]

Calculate Total Moles and Grams of Glucose in Blood (Before Ingestion)

The total volume of blood is given as 5.0 L, which is 5000 mL. We can find the total moles of glucose in 5.0 L by multiplying the moles per mL by the total blood volume:\[\text{Total moles in 5.0 L} = 7.77 \times 10^{-6} \text{ mol/mL} \times 5000 \text{ mL} = 0.03885 \text{ mol}\]Convert moles back to grams using the molar mass of glucose:\[\text{Total grams in blood} = 0.03885 \text{ mol} \times 180.16 \text{ g/mol} = 7.00 \text{ g}\]

Calculate Moles of Glucose per Milliliter of Blood (After Ingestion)

With a blood glucose level at 0.240 g/100 mL after ingestion, calculate the moles of glucose per 100 mL:\[\text{Moles in 100 mL after ingestion} = \frac{0.240 \text{ g}}{180.16 \text{ g/mol}} = 0.001333 \text{ mol}\]Then, moles per mL:\[\text{Moles per mL} = \frac{0.001333 \text{ mol}}{100 \text{ mL}} = 1.333 \times 10^{-5} \text{ mol/mL}\]

Calculate Total Moles and Grams of Glucose in Blood (After Ingestion)

Calculate the total moles of glucose in the blood after the ingestion:\[\text{Total moles in 5.0 L} = 1.333 \times 10^{-5} \text{ mol/mL} \times 5000 \text{ mL} = 0.06665 \text{ mol}\]Convert moles back to grams:\[\text{Total grams in blood} = 0.06665 \text{ mol} \times 180.16 \text{ g/mol} = 12.00 \text{ g}\]

Conclusion

Before ingestion, the glucose concentration is 7.77 × 10^{-6} mol/mL and the total amount in blood is 0.03885 mol or 7.00 grams. After ingestion, it increases to 1.333 × 10^{-5} mol/mL with 0.06665 mol or 12.00 grams of glucose.

Key concepts

Blood Glucose Level

Blood glucose level refers to the concentration of glucose present in the blood. For diabetic patients, maintaining an optimal blood glucose level is crucial for health management. The blood glucose level is typically measured in terms of mass of glucose per specific volume of blood, usually grams per 100 milliliters (g/100 mL).
This exercise provided that before consuming glucose, the patient's blood glucose level was 0.140 g/100 mL. After consumption, it increased to 0.240 g/100 mL. Changes in blood glucose levels are common and need constant monitoring in diabetic patients to avoid adverse health effects.

• Before ingestion: 0.140 g/100 mL
• After ingestion: 0.240 g/100 mL

Monitoring of these levels can help manage energy use, detect potential medical issues early, and tailor dietary and medical treatment to patient needs.

Molar Mass of Glucose

The molar mass of a compound is the mass in grams of one mole of its particles. In this context, glucose has the chemical formula C₆H₁₂O₆. The molar mass is calculated by summing the atomic masses of all atoms in a glucose molecule.
The calculation for glucose (C₆H₁₂O₆) is as follows:

• Carbon (C): 6 atoms × 12.01 g/mol = 72.06 g/mol
• Hydrogen (H): 12 atoms × 1.01 g/mol = 12.12 g/mol
• Oxygen (O): 6 atoms × 16.00 g/mol = 96.00 g/mol

Summing these gives the total molar mass of glucose:
\[ 72.06 + 12.12 + 96.00 = 180.16 ext{ g/mol} \]This molar mass is crucial for converting between grams and moles, a key part of calculating concentrations and reactions involving glucose.

Glucose Concentration

Glucose concentration in the blood is an important measure of metabolic health. It is often expressed as the amount of glucose present in a unit volume of blood, typically grams per deciliter or per 100 mL.
In the exercise, glucose concentration changes before and after consumption of additional glucose:

• Initially, 0.140 g per 100 mL of blood
• Later, increased to 0.240 g per 100 mL of blood

To work with chemical calculations, converting this concentration to moles is helpful because reactions typically deal with moles.
Moles per volume is determined using the formula:
\[ ext{Moles of glucose} = rac{ ext{mass of glucose}}{ ext{molar mass of glucose}} \]Having found the moles per unit volume, it's possible to calculate total glucose in terms of moles in the entire blood volume, leveraging metrics familiar in chemistry.

Conversion of Moles to Grams

Converting moles to grams involves using the molar mass of the substance. This process is essential in translating the stoichiometric calculations of chemistry into real-world quantities.
In the steps provided in the exercise, after calculating moles of glucose from blood, these moles were converted back to grams to find how much glucose is present in the blood in practical terms.
The conversion formula is:\[ ext{grams} = ext{moles} imes ext{molar mass} \]
For instance, before ingestion, 0.03885 moles of glucose is present:
\[ 0.03885 ext{ mol} imes 180.16 ext{ g/mol} = 7.00 ext{ g} \]After ingestion, the moles increase to 0.06665, translating to:\[0.06665 ext{ mol} imes 180.16 ext{ g/mol} = 12.00 ext{ g} \]This quantitative understanding aids in assessing how dietary intake impacts blood glucose levels from a chemical perspective.