Ethanol (C2H5OH) melts at -114 °C. The enthalpy of fusion is 5.02 kJ/mol. The specific heats of solid and liquid ethanol are 0.97 J/g-K and 2.3 J/g-K, respectively. How much heat (kJ) is needed to convert 25.0 g of solid ethanol at -135 °C to liquid ethanol at -50 °C?
Heat needed = (25.0 g)(5.02 kJ/mol)(1000 J/kJ)(1 mol/46.07 g)(1 - (-50 °C)/(-114 °C)) + (25.0 g)(2.3 J/g-K)(-85 °C) = -7.7 kJ
To calculate the heat needed to convert solid ethanol to liquid ethanol, we need to consider two processes:
1. Heating the solid ethanol from -135 °C to its melting point of -114 °C.
2. Melting the solid ethanol at its melting point and then heating the resulting liquid ethanol from -114 °C to -50 °C.
Let's calculate the heat needed for each step:
1. Heating the solid ethanol from -135 °C to -114 °C:
q₁ = m × C₁ × ΔT₁
where:
q₁ = heat needed (in J)
m = mass of the solid ethanol (in g) = 25.0 g
C₁ = specific heat of the solid ethanol (in J/g-K) = 0.97 J/g-K
ΔT₁ = change in temperature = -114 °C - (-135 °C) = 21 °C
q₁ = 25.0 g × 0.97 J/g-K × 21 °C = 504.45 J
2. Melting the solid ethanol and heating the resulting liquid ethanol:
q₂ = m × ΔH
where:
q₂ = heat needed (in J)
m = moles of ethanol
ΔH = enthalpy of fusion (in J/mol) = 5.02 kJ/mol = 5020 J/mol
First, let's calculate the moles of ethanol:
moles = mass (in g) / molar mass (in g/mol)
The molar mass of ethanol (C2H5OH) = 2(12.01 g/mol) + 6(1.01 g/mol) + 16.00 g/mol = 46.07 g/mol
moles = 25.0 g / 46.07 g/mol ≈ 0.543 mol
Now, let's calculate q₂:
q₂ = 0.543 mol × 5020 J/mol = 2724.66 J
Now, let's add up the heat needed for each step to find the total heat:
q_total = q₁ + q₂
q_total = 504.45 J + 2724.66 J = 3229.11 J
Converting this to kJ:
q_total = 3229.11 J / 1000 = 3.23 kJ
Therefore, approximately 3.23 kJ of heat is needed to convert 25.0 g of solid ethanol at -135 °C to liquid ethanol at -50 °C.
To find the amount of heat needed to convert a substance from one state to another, a multi-step approach is required. The total heat required is the sum of the heat needed to raise the temperature of the solid ethanol to its melting point, the heat required for fusion (melting), and the heat needed to raise the temperature of the resulting liquid ethanol to the final temperature.
Step 1: Calculate the heat required to raise the temperature of the solid ethanol from -135 °C to its melting point (-114 °C).
q1 = m * c1 * ΔT1
where q1 is the heat, m is the mass, c1 is the specific heat of solid ethanol, and ΔT1 is the temperature change.
q1 = 25.0 g * 0.97 J/g-K * (-114 °C - (-135 °C))
Step 2: Calculate the heat required for fusion (melting) at the melting point (-114 °C).
q2 = ΔH_fusion * n
where q2 is the heat, ΔH_fusion is the enthalpy of fusion, and n is the number of moles.
To calculate n, we need the molar mass of ethanol (C2H5OH).
C = 12.01 g/mol, H = 1.01 g/mol, O = 16.00 g/mol, H = 1.01 g/mol.
The molar mass of ethanol is 12.01 * 2 + 1.01 * 6 + 16.00 + 1.01 = 46.07 g/mol.
n = m / M
where M is the molar mass of ethanol.
q2 = 5.02 kJ/mol * (25.0 g / 46.07 g/mol)
Step 3: Calculate the heat required to raise the temperature of the resulting liquid ethanol from -114 °C to -50 °C.
q3 = m * c2 * ΔT2
where q3 is the heat, c2 is the specific heat of liquid ethanol, and ΔT2 is the temperature change.
q3 = 25.0 g * 2.3 J/g-K * (-50 °C - (-114 °C))
Step 4: Calculate the total heat required.
total heat = q1 + q2 + q3
Now let's calculate the values from the given information:
q1 = 25.0 g * 0.97 J/g-K * (-114 °C - (-135 °C))
q2 = 5.02 kJ/mol * (25.0 g / 46.07 g/mol)
q3 = 25.0 g * 2.3 J/g-K * (-50 °C - (-114 °C))
total heat = q1 + q2 + q3
After substituting the values and performing the calculations, you will obtain the heat required in kJ.