Question

A small kidney stone (Chemistry in Action on p. 176) might contain \(0.50 \mathrm{~g}\) of uric acid \(\left(\mathrm{C}_{5} \mathrm{H}_{4} \mathrm{~N}_{4} \mathrm{O}_{3}\right)\). How many micromoles of uric acid are contained in this stone?

Short answer

The stone contains approximately 2975 micromoles of uric acid.

Step-by-step solution

Calculate Molar Mass of Uric Acid

First, we need to determine the molar mass of uric acid by adding up the atomic masses based on its molecular formula, \(\mathrm{C}_{5} \mathrm{H}_{4} \mathrm{N}_{4}\mathrm{O}_{3}\). Using the periodic table: Carbon (C) has an atomic mass of 12.01 g/mol, Hydrogen (H) is 1.01 g/mol, Nitrogen (N) is 14.01 g/mol, and Oxygen (O) is 16.00 g/mol. The molar mass calculation is as follows: \[ 5(12.01) + 4(1.01) + 4(14.01) + 3(16.00) = 168.11 \text{ g/mol} \]

Convert Grams to Moles

Next, we convert the mass of uric acid from grams to moles. Given that the stone contains 0.50 g of uric acid, we use the molar mass calculated in Step 1: \[ \text{Moles of uric acid} = \frac{0.50 \text{ g}}{168.11 \text{ g/mol}} \approx 0.002975 \text{ moles} \]

Convert Moles to Micromoles

Finally, we convert the number of moles to micromoles. There are 1,000,000 micromoles in a mole, so: \[ \text{Micromoles of uric acid} = 0.002975 \text{ moles} \times 1,000,000 \text{ µmol/mole} \approx 2975 \text{ µmol} \]

Key concepts

Molar Mass Calculation

The molar mass of a compound is a critical concept in chemistry, allowing us to connect the mass of a substance to the number of particles or moles present. To calculate the molar mass of uric acid, we need to sum up the atomic masses of all the elements in its molecular formula, which is C\(_5\)H\(_4\)N\(_4\)O\(_3\).

Here's how it's done:

• Carbon (C) has an atomic mass of 12.01 g/mol. Multiply this by the 5 atoms of carbon: 5 x 12.01 = 60.05 g/mol.
• Hydrogen (H) has an atomic mass of 1.01 g/mol. Multiply this by the 4 atoms of hydrogen: 4 x 1.01 = 4.04 g/mol.
• Nitrogen (N) has an atomic mass of 14.01 g/mol. Multiply this by the 4 atoms of nitrogen: 4 x 14.01 = 56.04 g/mol.
• Oxygen (O) has an atomic mass of 16.00 g/mol. Multiply this by the 3 atoms of oxygen: 3 x 16.00 = 48.00 g/mol.

Add all these values together to find the molar mass of uric acid:
\[ 60.05 + 4.04 + 56.04 + 48.00 = 168.11 \text{ g/mol} \]
The molar mass is a link between the molecular scale and the scale of everyday measurements in grams.

Converting Grams to Moles

Once we know the molar mass, converting grams of a substance to moles becomes straightforward and crucial for stoichiometry. This step allows us to bridge between the mass you have and the amount, in moles, of a substance.

For the 0.50 g of uric acid in the kidney stone:

• Divide the mass of the substance by its molar mass to obtain the number of moles.
• The formula is: \( \text{moles} = \frac{\text{grams}}{\text{molar mass}} \)
• Substituting in the values: \( \text{moles of uric acid} = \frac{0.50 \text{ g}}{168.11 \text{ g/mol}} \approx 0.002975 \text{ moles} \)

This calculation allows us to understand precisely how much of the uric acid, in terms of quantity of molecules, is present in the stone.

Converting Moles to Micromoles

After determining the number of moles, the next common conversion is to micromoles, particularly useful in fields requiring precise measurements, like biochemistry.

Micromoles provide a finer scale, beneficial when dealing with small quantities:

• There are 1,000,000 micromoles in a single mole. This conversion factor helps scale your moles to a more workable number.
• Simply multiply the number of moles by 1,000,000 to convert to micromoles.
• In this case: \( 0.002975 \text{ moles} \times 1,000,000 \text{ µmol/mole} = 2975 \text{ µmol} \)

This conversion illustrates how even a small mass can correspond to a significant number of molecules, providing deeper insights for analysis and practical applications.