Question

In a role-playing video game (RPG) your character is a Human Paladin that can carry \(115 \mathrm{lb}\) of gear. Your character is carrying 92 lb of gear and a vial of strength potion (which allows you to carry an additional \(50.0\) lb of gear). If you find a cave filled with mass potions (used for resisting strong winds), after using the strength potion, how many vials can you collect if the vials each contain \(50.0 \mathrm{~mL}\) of mass potion with a density of \(193 \mathrm{~g} / \mathrm{mL}\) ? The vials have negligible mass.

Short answer

You can collect 3 vials.

Step-by-step solution

Calculate Additional Capacity

First, we need to determine how much more gear your character can carry after using the strength potion. The strength potion allows carrying an additional 50 lb. Thus, the new carrying capacity will be:\[115 \text{ lb} + 50 \text{ lb} = 165 \text{ lb}.\]

Determine Remaining Capacity

Next, subtract the current amount of gear being carried (92 lb) from this new total capacity to find the remaining capacity:\[165 \text{ lb} - 92 \text{ lb} = 73 \text{ lb}.\]

Convert Remaining Capacity to Grams

Convert the remaining capacity from pounds to grams. Given that 1 lb = 453.592 grams, the remaining capacity in grams is:\[73 \text{ lb} \times 453.592 \text{ g/lb} = 33122.216 \text{ g}.\]

Calculate the Mass per Vial

Calculate the mass of the potion in one vial. Given the potion's density is 193 g/mL and each vial contains 50.0 mL:\[193 \text{ g/mL} \times 50.0 \text{ mL} = 9650 \text{ g}.\]

Determine Number of Vials

Divide the total remaining capacity in grams by the mass of each vial to find the number of vials that can be carried:\[\frac{33122.216 \text{ g}}{9650 \text{ g}} \approx 3.43\]Since you can't carry a fraction of a vial, you can carry 3 vials.

Key concepts

Density Calculation

Understanding density is crucial in solving many chemistry problems involving mass and volume.
Density is defined as the mass of an object divided by its volume. The formula for density is: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] In our exercise, the density of the mass potion is given as 193 g/mL. This tells us that for every milliliter (mL) of potion, there is 193 grams (g) of mass.
To find out the mass of the potion in a single vial, we multiply the potion's density by the volume of the vial: \[ 193 \text{ g/mL} \times 50.0 \text{ mL} = 9650 \text{ g} \] In this instance, since each vial contains 50.0 mL of potion, one vial has a mass of 9650 g. This information is essential for knowing how much weight your character can carry when you find vials of mass potion in the cave.

Unit Conversion

Unit conversion is a fundamental skill in chemistry for ensuring that all measurements are in the appropriate units.
In this problem, we convert between pounds (lb) and grams (g), as the game's carrying capacities are in pounds, but the potion's mass is given in grams. The conversion factor between pounds and grams is: \[ 1 \text{ lb} = 453.592 \text{ g} \] To convert the remaining capacity of 73 lb to grams: \[ 73 \text{ lb} \times 453.592 \text{ g/lb} = 33122.216 \text{ g} \] By converting 73 lb to grams, we see that the remaining carrying capacity is 33122.216 grams. This conversion ensures that we can accurately compare the weight of our vials (in grams) with the character's remaining carrying capacity.

Capacity Calculation

In chemistry, capacity calculations help us determine how much of a substance can fit within a given space or weight limit.
For this RPG scenario, we start by adding the extra carrying capacity provided by the strength potion to your character's existing capacity: \[ 115 \text{ lb} + 50 \text{ lb} = 165 \text{ lb} \] Then, we subtract the weight of the already carried gear to find the remaining capacity: \[ 165 \text{ lb} - 92 \text{ lb} = 73 \text{ lb} \] Once we convert this remaining 73 lb capacity to grams, we calculate how many vials of the potion can be carried: \[ \frac{33122.216 \text{ g}}{9650 \text{ g/vial}} = 3.43 \text{ vials} \] Since partial vials can't be carried, your character can carry 3 full vials of the mass potion. Understanding these steps in capacity calculation allows you to achieve precise and practical results for this game scenario.