Math / Data & Statistics

QuestionOne cup (250. mL) of coffee contains 125 mg of caffeine. One cup of tea has 0.100 g of caffeine. A can of Diet Coke ( 355 mL ) contains $50 . \mathrm{mg}$ of caffeine, and a can of Surge $(355 \mathrm{~mL})$ contains about 65 mg of caffeine. Caffeine is $\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}$ . a. Which drink has the most caffeine per milliliter? Show how you arrived at your conclusion. (3 points) b. Calculate the molarity of caffeine in the drink you chose in part a. (5 points)

Studdy Solution

STEP 1

What is this asking? Which drink has the highest caffeine concentration and what is that concentration in molarity? Watch out! The caffeine amounts are given in different units (mg and g), so we need to be careful with unit conversions!
Also, remember molarity is moles per liter, not moles per milliliter.

STEP 2

1. Calculate Caffeine Concentration per Milliliter
2. Determine the Drink with the Highest Concentration
3. Calculate the Molar Mass of Caffeine
4. Calculate the Molarity

STEP 3

Let's calculate the caffeine concentration for each drink in milligrams of caffeine per milliliter of drink.
This will help us compare them directly.
For coffee, we have $\frac{125 \text{ mg}}{250 \text{ mL}}$ .

STEP 4

Dividing gives us $0.500 \frac{\text{mg}}{\text{mL}}$ for coffee.
See how we kept the units with the number?
That's super important so we don't get confused later!

STEP 5

For tea, we have $0.100$ g of caffeine.
Since there are $1000$ mg in $1$ g, we multiply $0.100 \text{ g} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} = 100 \text{ mg}$ of caffeine in one cup of tea.

STEP 6

Now, we can divide this by the volume, $250$ mL, to get $\frac{100 \text{ mg}}{250 \text{ mL}} = 0.400 \frac{\text{mg}}{\text{mL}}$ .

STEP 7

Diet Coke has $\frac{50 \text{ mg}}{355 \text{ mL}} \approx 0.141 \frac{\text{mg}}{\text{mL}}$ .
We're rounding to three significant figures here to match the given values.

STEP 8

Finally, Surge has $\frac{65 \text{ mg}}{355 \text{ mL}} \approx 0.183 \frac{\text{mg}}{\text{mL}}$ .

STEP 9

Comparing the values we just calculated, we see that coffee has the highest concentration of caffeine at $0.500 \frac{\text{mg}}{\text{mL}}$ .

STEP 10

The formula for caffeine is $\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}$ .
To calculate the molar mass, we'll add the atomic masses of all the atoms in one molecule of caffeine.

STEP 11

We have $8$ carbon atoms, $10$ hydrogen atoms, $4$ nitrogen atoms, and $2$ oxygen atoms.
Using the periodic table, we find the molar mass: $8 \cdot 12.01 \frac{\text{g}}{\text{mol}} + 10 \cdot 1.01 \frac{\text{g}}{\text{mol}} + 4 \cdot 14.01 \frac{\text{g}}{\text{mol}} + 2 \cdot 16.00 \frac{\text{g}}{\text{mol}}$ .

STEP 12

This sums to $96.08 + 10.10 + 56.04 + 32.00 = 194.22 \frac{\text{g}}{\text{mol}}$ .
So, the molar mass of caffeine is $194.22 \frac{\text{g}}{\text{mol}}$ .

STEP 13

We found that coffee has $0.500 \frac{\text{mg}}{\text{mL}}$ of caffeine.
To calculate the molarity, we need the concentration in moles per liter.

STEP 14

First, let's convert milligrams to grams: $0.500 \frac{\text{mg}}{\text{mL}} \cdot \frac{1 \text{ g}}{1000 \text{ mg}} = 0.000500 \frac{\text{g}}{\text{mL}}$ .

STEP 15

Next, convert grams to moles using the molar mass: $0.000500 \frac{\text{g}}{\text{mL}} \cdot \frac{1 \text{ mol}}{194.22 \text{ g}} \approx 0.00000257 \frac{\text{mol}}{\text{mL}}$ .

STEP 16

Finally, convert milliliters to liters: $0.00000257 \frac{\text{mol}}{\text{mL}} \cdot \frac{1000 \text{ mL}}{1 \text{ L}} \approx 0.00257 \frac{\text{mol}}{\text{L}}$ .

STEP 17

Coffee has the highest caffeine concentration at $0.500 \frac{\text{mg}}{\text{mL}}$ .
The molarity of caffeine in coffee is approximately $0.00257$ M.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Caffeine Concentration per Milliliter
2. Determine the Drink with the Highest Concentration
3. Calculate the Molar Mass of Caffeine
4. Calculate the Molarity

STEP 3

Let's calculate the caffeine concentration for each drink in milligrams of caffeine per milliliter of drink.
This will help us compare them directly.
For coffee, we have $\frac{125 \text{ mg}}{250 \text{ mL}}$ .

STEP 4

Dividing gives us $0.500 \frac{\text{mg}}{\text{mL}}$ for coffee.
See how we kept the units with the number?
That's super important so we don't get confused later!

STEP 5

For tea, we have $0.100$ g of caffeine.
Since there are $1000$ mg in $1$ g, we multiply $0.100 \text{ g} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} = 100 \text{ mg}$ of caffeine in one cup of tea.

STEP 6

Now, we can divide this by the volume, $250$ mL, to get $\frac{100 \text{ mg}}{250 \text{ mL}} = 0.400 \frac{\text{mg}}{\text{mL}}$ .

STEP 7

Diet Coke has $\frac{50 \text{ mg}}{355 \text{ mL}} \approx 0.141 \frac{\text{mg}}{\text{mL}}$ .
We're rounding to three significant figures here to match the given values.

STEP 8

Finally, Surge has $\frac{65 \text{ mg}}{355 \text{ mL}} \approx 0.183 \frac{\text{mg}}{\text{mL}}$ .

STEP 9

Comparing the values we just calculated, we see that coffee has the highest concentration of caffeine at $0.500 \frac{\text{mg}}{\text{mL}}$ .

STEP 10

The formula for caffeine is $\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}$ .
To calculate the molar mass, we'll add the atomic masses of all the atoms in one molecule of caffeine.

STEP 11

STEP 12

This sums to $96.08 + 10.10 + 56.04 + 32.00 = 194.22 \frac{\text{g}}{\text{mol}}$ .
So, the molar mass of caffeine is $194.22 \frac{\text{g}}{\text{mol}}$ .

STEP 13

We found that coffee has $0.500 \frac{\text{mg}}{\text{mL}}$ of caffeine.
To calculate the molarity, we need the concentration in moles per liter.

STEP 14

First, let's convert milligrams to grams: $0.500 \frac{\text{mg}}{\text{mL}} \cdot \frac{1 \text{ g}}{1000 \text{ mg}} = 0.000500 \frac{\text{g}}{\text{mL}}$ .

STEP 15

Next, convert grams to moles using the molar mass: $0.000500 \frac{\text{g}}{\text{mL}} \cdot \frac{1 \text{ mol}}{194.22 \text{ g}} \approx 0.00000257 \frac{\text{mol}}{\text{mL}}$ .

STEP 16

Finally, convert milliliters to liters: $0.00000257 \frac{\text{mol}}{\text{mL}} \cdot \frac{1000 \text{ mL}}{1 \text{ L}} \approx 0.00257 \frac{\text{mol}}{\text{L}}$ .

STEP 17

Coffee has the highest caffeine concentration at $0.500 \frac{\text{mg}}{\text{mL}}$ .
The molarity of caffeine in coffee is approximately $0.00257$ M.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Caffeine Concentration per Milliliter2. Determine the Drink with the Highest Concentration3. Calculate the Molar Mass of Caffeine4. Calculate the Molarity

STEP 3

Let's **calculate** the caffeine concentration for each drink in milligrams of caffeine per milliliter of drink.This will help us compare them directly.For coffee, we have 125 mg250 mL\frac{125 \text{ mg}}{250 \text{ mL}}250 mL125 mg​.

STEP 4

Dividing gives us 0.500mgmL0.500 \frac{\text{mg}}{\text{mL}}0.500mLmg​ for coffee.See how we kept the units with the number?That's super important so we don't get confused later!

STEP 5

For tea, we have 0.1000.1000.100 g of caffeine.Since there are 100010001000 mg in 111 g, we multiply 0.100 g⋅1000 mg1 g=100 mg0.100 \text{ g} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} = 100 \text{ mg}0.100 g⋅1 g1000 mg​=100 mg of caffeine in one cup of tea.

STEP 6

Now, we can **divide** this by the volume, 250250250 mL, to get 100 mg250 mL=0.400mgmL\frac{100 \text{ mg}}{250 \text{ mL}} = 0.400 \frac{\text{mg}}{\text{mL}}250 mL100 mg​=0.400mLmg​.

STEP 7

Diet Coke has 50 mg355 mL≈0.141mgmL\frac{50 \text{ mg}}{355 \text{ mL}} \approx 0.141 \frac{\text{mg}}{\text{mL}}355 mL50 mg​≈0.141mLmg​.We're rounding to three significant figures here to match the given values.

STEP 8

Finally, Surge has 65 mg355 mL≈0.183mgmL\frac{65 \text{ mg}}{355 \text{ mL}} \approx 0.183 \frac{\text{mg}}{\text{mL}}355 mL65 mg​≈0.183mLmg​.

STEP 9

Comparing the values we just calculated, we see that coffee has the **highest** concentration of caffeine at 0.500mgmL0.500 \frac{\text{mg}}{\text{mL}}0.500mLmg​.

STEP 10

The formula for caffeine is C8H10 N4O2\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}C8​H10​ N4​O2​.To **calculate** the molar mass, we'll **add** the atomic masses of all the atoms in one molecule of caffeine.

STEP 11

STEP 12

This **sums** to 96.08+10.10+56.04+32.00=194.22gmol96.08 + 10.10 + 56.04 + 32.00 = 194.22 \frac{\text{g}}{\text{mol}}96.08+10.10+56.04+32.00=194.22molg​.So, the molar mass of caffeine is 194.22gmol194.22 \frac{\text{g}}{\text{mol}}194.22molg​.

STEP 13

We found that coffee has 0.500mgmL0.500 \frac{\text{mg}}{\text{mL}}0.500mLmg​ of caffeine.To **calculate** the molarity, we need the concentration in moles per liter.

STEP 14

First, let's convert milligrams to grams: 0.500mgmL⋅1 g1000 mg=0.000500gmL0.500 \frac{\text{mg}}{\text{mL}} \cdot \frac{1 \text{ g}}{1000 \text{ mg}} = 0.000500 \frac{\text{g}}{\text{mL}}0.500mLmg​⋅1000 mg1 g​=0.000500mLg​.

STEP 15

Next, convert grams to moles using the molar mass: 0.000500gmL⋅1 mol194.22 g≈0.00000257molmL0.000500 \frac{\text{g}}{\text{mL}} \cdot \frac{1 \text{ mol}}{194.22 \text{ g}} \approx 0.00000257 \frac{\text{mol}}{\text{mL}}0.000500mLg​⋅194.22 g1 mol​≈0.00000257mLmol​.

STEP 16

Finally, convert milliliters to liters: 0.00000257molmL⋅1000 mL1 L≈0.00257molL0.00000257 \frac{\text{mol}}{\text{mL}} \cdot \frac{1000 \text{ mL}}{1 \text{ L}} \approx 0.00257 \frac{\text{mol}}{\text{L}}0.00000257mLmol​⋅1 L1000 mL​≈0.00257Lmol​.

STEP 17

Coffee has the highest caffeine concentration at 0.500mgmL0.500 \frac{\text{mg}}{\text{mL}}0.500mLmg​.The molarity of caffeine in coffee is approximately 0.002570.002570.00257 M.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate Caffeine Concentration per Milliliter
2. Determine the Drink with the Highest Concentration
3. Calculate the Molar Mass of Caffeine
4. Calculate the Molarity

Let's calculate the caffeine concentration for each drink in milligrams of caffeine per milliliter of drink.
This will help us compare them directly.
For coffee, we have $\frac{125 \text{ mg}}{250 \text{ mL}}$ .

Dividing gives us $0.500 \frac{\text{mg}}{\text{mL}}$ for coffee.
See how we kept the units with the number?
That's super important so we don't get confused later!

For tea, we have $0.100$ g of caffeine.
Since there are $1000$ mg in $1$ g, we multiply $0.100 \text{ g} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} = 100 \text{ mg}$ of caffeine in one cup of tea.

Now, we can divide this by the volume, $250$ mL, to get $\frac{100 \text{ mg}}{250 \text{ mL}} = 0.400 \frac{\text{mg}}{\text{mL}}$ .

Diet Coke has $\frac{50 \text{ mg}}{355 \text{ mL}} \approx 0.141 \frac{\text{mg}}{\text{mL}}$ .
We're rounding to three significant figures here to match the given values.

Finally, Surge has $\frac{65 \text{ mg}}{355 \text{ mL}} \approx 0.183 \frac{\text{mg}}{\text{mL}}$ .

Comparing the values we just calculated, we see that coffee has the highest concentration of caffeine at $0.500 \frac{\text{mg}}{\text{mL}}$ .

The formula for caffeine is $\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}$ .
To calculate the molar mass, we'll add the atomic masses of all the atoms in one molecule of caffeine.

This sums to $96.08 + 10.10 + 56.04 + 32.00 = 194.22 \frac{\text{g}}{\text{mol}}$ .
So, the molar mass of caffeine is $194.22 \frac{\text{g}}{\text{mol}}$ .

We found that coffee has $0.500 \frac{\text{mg}}{\text{mL}}$ of caffeine.
To calculate the molarity, we need the concentration in moles per liter.

First, let's convert milligrams to grams: $0.500 \frac{\text{mg}}{\text{mL}} \cdot \frac{1 \text{ g}}{1000 \text{ mg}} = 0.000500 \frac{\text{g}}{\text{mL}}$ .

Next, convert grams to moles using the molar mass: $0.000500 \frac{\text{g}}{\text{mL}} \cdot \frac{1 \text{ mol}}{194.22 \text{ g}} \approx 0.00000257 \frac{\text{mol}}{\text{mL}}$ .

Finally, convert milliliters to liters: $0.00000257 \frac{\text{mol}}{\text{mL}} \cdot \frac{1000 \text{ mL}}{1 \text{ L}} \approx 0.00257 \frac{\text{mol}}{\text{L}}$ .

Coffee has the highest caffeine concentration at $0.500 \frac{\text{mg}}{\text{mL}}$ .
The molarity of caffeine in coffee is approximately $0.00257$ M.