Question

A patient needs \(100 . \mathrm{g}\) of glucose in the next \(12 \mathrm{~h}\). How many liters of a \(5 \%(\mathrm{~m} / \mathrm{v})\) glucose solution must be given?

Short answer

2 L

Step-by-step solution

- Understand the percentage concentration

The glucose solution has a concentration of 5%, which means there are 5 grams of glucose in every 100 milliliters of solution.

- Set up the equation

We need to find the volume of the 5% solution that provides 100 grams of glucose. Set up the equation: \[ \text{Volume (L)} \times \frac{5 \text{ grams}}{100 \text{ mL}} = 100 \text{ grams} \]

- Convert volume into liters

Since the volume is in liters for the final answer, convert 100 mL to liters. Remember that 1000 mL = 1 L. So, \[ \frac{5 \text{ grams}}{0.1 \text{ L}} \text{ per L of solution} \]

- Solve for the volume

To find the volume, rearrange the equation from Step 2: \[ \text{Volume (L)} = \frac{100 \text{ grams}}{5 \text{ grams/0.1 L}} = \frac{100 \text{ grams}}{50 \text{ grams/L}} = 2 \text{ L} \]

Key concepts

percentage concentration

Percentage concentration in a solution tells us how much solute (in this case, glucose) is present in a fixed amount of solution. For example, a 5% (m/v) glucose solution means there are 5 grams of glucose in every 100 milliliters of the solution. This is written as 5% (m/v), which stands for mass/volume. Understanding this concept helps us determine how much glucose is present in any given volume of this solution. To calculate the amount needed for a different volume, we can use the ratio given.

volume calculation

Volume calculation involves finding the amount of space a liquid occupies. In this problem, we start with the known need of 100 grams of glucose and use the percentage concentration to find the required volume. We know the concentration is 5 grams of glucose per 100 milliliters of solution. Using an equation, we can solve for the volume in liters. The equation used is: \[ \text{Volume (L)} \times \frac{5 \text{ grams}}{100 \text{ mL}} = 100 \text{ grams} \] By solving this, we can find the volume of the solution that will provide exactly the desired amount of glucose.

mass/volume solution

A mass/volume (m/v) solution is one where the amount of solute (mass) is measured against the volume of the solution. To calculate how much of a solution is needed to obtain a specific mass of solute, we set up a proportion based on the known concentration. For instance, a 5% (m/v) glucose solution has 5 grams of glucose in every 100 mL. To meet the patient's needs of 100 grams of glucose, we need to figure out the corresponding volume, which we calculated to be 2 liters (L). This is essential in medical dosages and chemical preparations to ensure accurate concentrations.

unit conversion

Unit conversion is crucial in scientific problems for consistency and accuracy. In this context, we convert between milliliters (mL) and liters (L). Since 1 liter (L) equals 1000 milliliters (mL), knowing this conversion helps us solve for volumes more efficiently. In the problem, we converted the 5 grams per 100 mL concentration to 5 grams per 0.1 L to match our final volume needed in liters. Using this conversion ensures that all parts of our equation use the same units, making it easier to solve and understand the solution accurately.