Math / Algebra

QuestionCalculate the energy required to heat 122.0 mg of ethanol from $-3.1^{\circ} \mathrm{C}$ to $17.3{ }^{\circ} \mathrm{C}$ . Assume the specific heat capacity of ethanol under these conditions is $2.44 \mathrm{~J} \cdot \mathrm{~g}^{-1} \cdot \mathrm{~K}^{-1}$ . Round your answer to 3 significant digits.

Studdy Solution

STEP 1

What is this asking? How much energy do we need to warm up this little bit of ethanol? Watch out! Don't forget to convert milligrams to grams and Celsius to Kelvin!
Also, keep track of your units throughout the calculation.

STEP 2

1. Convert units
2. Calculate the change in temperature
3. Calculate the energy required

STEP 3

Alright, let's convert the mass of ethanol from milligrams to grams because the specific heat capacity is given in joules per gram Kelvin.
We're given $122.0$ mg of ethanol.
Since there are $1000$ mg in $1$ g, we divide $122.0$ by $1000$ to get $0.1220$ g.
See? Easy peasy!

STEP 4

Now, let's convert those Celsius temperatures to Kelvin.
Remember, to go from Celsius to Kelvin, we just add $273.15$ to the Celsius temperature.
So, our initial temperature of $-3.1^\circ \mathrm{C}$ becomes $-3.1 + 273.15 = 270.05 \mathrm{K}$ .
And our final temperature of $17.3^\circ \mathrm{C}$ becomes $17.3 + 273.15 = 290.45 \mathrm{K}$ .

STEP 5

The change in temperature, represented by $\Delta T$ , is simply the final temperature minus the initial temperature.
So, $\Delta T = 290.45 \mathrm{K} - 270.05 \mathrm{K} = 20.40 \mathrm{K}$ .
That's how much our ethanol is warming up!

STEP 6

Now for the grand finale!
We can calculate the energy required (represented by $q$ ) using the formula: $q = mc\Delta T$ , where $m$ is the mass, $c$ is the specific heat capacity, and $\Delta T$ is the change in temperature.

STEP 7

Let's plug in our values: $q = (0.1220 \mathrm{~g}) \cdot (2.44 \mathrm{~J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}) \cdot (20.40 \mathrm{K})$ .
Notice how the units of grams and Kelvin divide to one, leaving us with just joules, which is the unit of energy!

STEP 8

Multiplying these values, we get $q = 6.05832 \mathrm{~J}$ .

STEP 9

Finally, we need to round our answer to 3 significant digits.
So, our final answer is $6.06 \mathrm{~J}$ .
Boom!

STEP 10

The energy required to heat the ethanol is $6.06 \mathrm{~J}$ .

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Studdy Solution

STEP 1

What is this asking? How much energy do we need to warm up this little bit of ethanol? Watch out! Don't forget to convert milligrams to grams and Celsius to Kelvin!
Also, keep track of your units throughout the calculation.

STEP 2

1. Convert units
2. Calculate the change in temperature
3. Calculate the energy required

STEP 3

Alright, let's convert the mass of ethanol from milligrams to grams because the specific heat capacity is given in joules per gram Kelvin.
We're given $122.0$ mg of ethanol.
Since there are $1000$ mg in $1$ g, we divide $122.0$ by $1000$ to get $0.1220$ g.
See? Easy peasy!

STEP 4

STEP 5

The change in temperature, represented by $\Delta T$ , is simply the final temperature minus the initial temperature.
So, $\Delta T = 290.45 \mathrm{K} - 270.05 \mathrm{K} = 20.40 \mathrm{K}$ .
That's how much our ethanol is warming up!

STEP 6

Now for the grand finale!
We can calculate the energy required (represented by $q$ ) using the formula: $q = mc\Delta T$ , where $m$ is the mass, $c$ is the specific heat capacity, and $\Delta T$ is the change in temperature.

STEP 7

Let's plug in our values: $q = (0.1220 \mathrm{~g}) \cdot (2.44 \mathrm{~J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}) \cdot (20.40 \mathrm{K})$ .
Notice how the units of grams and Kelvin divide to one, leaving us with just joules, which is the unit of energy!

STEP 8

Multiplying these values, we get $q = 6.05832 \mathrm{~J}$ .

STEP 9

Finally, we need to round our answer to 3 significant digits.
So, our final answer is $6.06 \mathrm{~J}$ .
Boom!

STEP 10

The energy required to heat the ethanol is $6.06 \mathrm{~J}$ .

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Studdy Solution

STEP 1

What is this asking? How much energy do we need to warm up this little bit of ethanol? Watch out! Don't forget to convert milligrams to grams and Celsius to Kelvin!Also, keep track of your units throughout the calculation.

STEP 2

1. Convert units2. Calculate the change in temperature3. Calculate the energy required

STEP 3

STEP 4

STEP 5

STEP 6

Now for the grand finale!We can **calculate** the energy required (represented by qqq) using the formula: q=mcΔTq = mc\Delta Tq=mcΔT, where mmm is the mass, ccc is the specific heat capacity, and ΔT\Delta TΔT is the change in temperature.

STEP 7

STEP 8

**Multiplying** these values, we get q=6.05832 Jq = 6.05832 \mathrm{~J}q=6.05832 J.

STEP 9

Finally, we need to **round** our answer to **3 significant digits**.So, our **final answer** is 6.06 J6.06 \mathrm{~J}6.06 J.Boom!

STEP 10

The energy required to heat the ethanol is 6.06 J6.06 \mathrm{~J}6.06 J.

Get Studdy

Studdy solves anything!

Start learning now

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

What is this asking? How much energy do we need to warm up this little bit of ethanol? Watch out! Don't forget to convert milligrams to grams and Celsius to Kelvin!
Also, keep track of your units throughout the calculation.

1. Convert units
2. Calculate the change in temperature
3. Calculate the energy required

Now for the grand finale!
We can calculate the energy required (represented by $q$ ) using the formula: $q = mc\Delta T$ , where $m$ is the mass, $c$ is the specific heat capacity, and $\Delta T$ is the change in temperature.

Multiplying these values, we get $q = 6.05832 \mathrm{~J}$ .

Finally, we need to round our answer to 3 significant digits.
So, our final answer is $6.06 \mathrm{~J}$ .
Boom!

The energy required to heat the ethanol is $6.06 \mathrm{~J}$ .