Question

The general test for Type 2 diabetes is that the blood sugar (glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) ) level should be below \(120 \mathrm{mg}\) per deciliter. Convert this concentration to molarity.

Short answer

The molarity is approximately \(0.00667\, \mathrm{mol/L}\).

Step-by-step solution

Calculate the Molar Mass

First, calculate the molar mass of glucose (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)). Add up the atomic masses: Carbon (C) is roughly \(12\, \mathrm{g/mol}\), Hydrogen (H) is \(1\, \mathrm{g/mol}\), and Oxygen (O) is \(16\, \mathrm{g/mol}\). Thus, for glucose, the molar mass is \(6(12) + 12(1) + 6(16) = 180\, \mathrm{g/mol}\).

Convert Concentration to Grams Per Liter

Convert the given concentration of \(120\, \mathrm{mg/dL}\) to \(\mathrm{g/L}\. \) Since \(1\, \mathrm{dL} = 0.1\, \mathrm{L}\), this converts as follows: \(120\, \mathrm{mg/dL}\) is \(1200\, \mathrm{mg/L}\), and \(1200\, \mathrm{mg/L} = 1.2\, \mathrm{g/L}\).

Calculate Molarity

Now, use the molar mass to convert from \(\mathrm{g/L}\) to molarity (moles per liter). Use the formula:\[\text{Molarity} = \frac{\text{mass in grams}}{\text{molar mass}}\].Plug in the values: \(\frac{1.2\, \mathrm{g/L}}{180\, \mathrm{g/mol}} = 0.00667\, \mathrm{mol/L}\).

Conclusion

The molarity of glucose in the blood should be approximately \(0.00667\, \mathrm{mol/L}\).

Key concepts

Molar Mass

To understand molarity, it's vital first to grasp the concept of molar mass. Molar mass is the mass of one mole of a substance. For molecular compounds like glucose, it's derived from summing the atomic masses of all atoms in the molecule. Glucose has the formula \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\). You calculate its molar mass by adding together the atomic masses of each element it contains:

• Carbon (C): Approximately \(12\, \mathrm{g/mol}\)
• Hydrogen (H): \(1\, \mathrm{g/mol}\)
• Oxygen (O): \(16\, \mathrm{g/mol}\)

Multiply these atomic masses by their respective quantities in glucose:

• Carbon: \(6 \times 12 = 72\)
• Hydrogen: \(12 \times 1 = 12\)
• Oxygen: \(6 \times 16 = 96\)

Finally, add these values: \(72 + 12 + 96 = 180\, \mathrm{g/mol}\). Therefore, glucose has a molar mass of \(180\, \mathrm{g/mol}\). This information is crucial for converting mass into moles, which is a step in finding molarity.

Concentration Conversion

Converting concentrations from one unit to another is often necessary in chemistry, especially when dealing with solutions. Here, we need to convert a concentration from \(\mathrm{mg/dL}\) to \(\mathrm{g/L}\). The given concentration was \(120\, \mathrm{mg/dL}\). It's useful to know that \(1\, \mathrm{dL}\) (deciliter) equals \(0.1\, \mathrm{L}\) (liter). Converting \(120\, \mathrm{mg/dL}\) to \(\mathrm{g/L}\) involves a two-step process:

• First, convert milligrams to grams: \(120\, \mathrm{mg} = 0.12\, \mathrm{g}\)
• Then, convert \(\mathrm{g/dL}\) to \(\mathrm{g/L}\): Since \(1\, \mathrm{dL} = 0.1\, \mathrm{L}\), multiply \(0.12\) by \(10\) to align with \(\mathrm{L}\): \(0.12 \times 10 = 1.2\, \mathrm{g/L}\)

This conversion allows us to move on to determining molarity, which requires the concentration to be expressed in terms of \(\mathrm{g/L}\).

Glucose

Glucose is a simple sugar with the molecular formula \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\). It's a key energy source for the human body, and its concentration in the blood can be an indicator of health conditions, such as diabetes.When measuring glucose levels, doctors often use the jednotka\(\mathrm{mg/dL}\). For scientific calculations, it is easier to use molarity, which indicates the number of moles of glucose per liter of blood.In the exercise at hand, we first calculated the molar mass of glucose to be \(180\, \mathrm{g/mol}\). This helps us convert a given mass into moles. By converting glucose concentration from \(120\, \mathrm{mg/dL}\) to \(\mathrm{g/L}\) and then using the molar mass, we found the molarity to be approximately \(0.00667\, \mathrm{mol/L}\). This molarity tells us how many moles of glucose are dissolved per liter of blood.Understanding glucose concentration in molarity helps health professionals better analyze blood sugar levels and determine if someone is at risk for diabetes or other conditions.