Ethanol (C2H5OH) melts at -114 degrees Celsius. The enthalpy of fusion is 5.02 kj/mol. The specific heats of solid and liquid ethanol are 0.97 J/g-K, respectively. How much heat (kJ) is needed to convert 25.0 g of solid ethanol at -135 degrees Celsius to liquid ethanol at -50 degrees Celsius?

6.91

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To calculate the amount of heat needed to convert solid ethanol to liquid ethanol, we need to consider the following steps:

1. Calculate the heat required to raise the temperature of the solid ethanol from -135°C to its melting point, -114°C.
2. Calculate the heat required to melt the solid ethanol at its melting point.
3. Calculate the heat required to raise the temperature of the liquid ethanol from its melting point, -114°C, to -50°C.

Let's use the following formula to calculate the heat:

Q = m * ΔH

where Q is the heat (in Joules), m is the mass (in grams), and ΔH is the enthalpy of fusion (in Joules per mole).

Step 1: Calculating the heat required to raise the temperature of the solid ethanol from -135°C to -114°C.

First, we need to calculate the temperature difference:

ΔT = final temperature - initial temperature
ΔT = -114°C - (-135°C)
ΔT = 21°C

Now, we can calculate the heat using the specific heat of solid ethanol:

Q1 = m * c * ΔT

where c is the specific heat of solid ethanol (in J/g-K).

Q1 = 25.0 g * 0.97 J/g-K * 21°C
Q1 = 492.75 J

Since we need the answer in kilojoules (kJ), we divide the result by 1000:

Q1 = 492.75 J / 1000
Q1 = 0.49275 kJ

Step 2: Calculating the heat required to melt the solid ethanol.

We use the enthalpy of fusion to calculate the heat required:

Q2 = m * ΔH

Q2 = 25.0 g * 5.02 kJ/mol

To convert the grams to moles, we need the molar mass of ethanol (C2H5OH):

Molar mass of C2H5OH = (2 * atomic mass of C) + (6 * atomic mass of H) + atomic mass of O
= (2 * 12.01 g/mol) + (6 * 1.01 g/mol) + 16.00 g/mol
= 46.07 g/mol

Now, we can calculate the heat:

Q2 = (25.0 g / 46.07 g/mol) * 5.02 kJ/mol
Q2 = 2.7211 kJ

Step 3: Calculating the heat required to raise the temperature of the liquid ethanol from -114°C to -50°C.

We first calculate the temperature difference:

ΔT = final temperature - initial temperature
ΔT = -50°C - (-114°C)
ΔT = 64°C

Now, we can calculate the heat using the specific heat of liquid ethanol:

Q3 = m * c * ΔT

where c is the specific heat of liquid ethanol (in J/g-K).

Q3 = 25.0 g * 0.97 J/g-K * 64°C
Q3 = 1568 J

Converting the result to kilojoules:

Q3 = 1568 J / 1000
Q3 = 1.568 kJ

Finally, we add up the three calculated amounts of heat to get the total heat required:

Total heat = Q1 + Q2 + Q3
Total heat = 0.49275 kJ + 2.7211 kJ + 1.568 kJ
Total heat ≈ 4.781 kJ

Therefore, approximately 4.781 kJ of heat is needed to convert 25.0 g of solid ethanol at -135°C to liquid ethanol at -50°C.

The easiest way to do these problems is in parts and you need three formulas to do it.

q at a phase change; i.e., solid to liquid or liquid to gas.
For solid to liquid (melting point) it is
q = mass solid x heat fusion.

For liquid to gas (boiling point) it is
q = mass liquid x heat vaporization.

All other points you have "within" a phase; i.e., change temperature from a lower point to a higher point but solid all the way, or liquid all the way, or vapor all the way. That formula is
q = mass x specific heat in that phase x (Tfinal-Tinitial).
Then add each of the q values together to find the total heat required.
For ethanol at -135 up to -50.
So you will need q for solid from -135 to -114
Then melt it at -114
Then heat from -114 to -50.
Add the three.