Question 5.3.5: Calculate the energy change for a pure substance over a temp......
Calculate the energy change for a pure substance over a temperature range.
The following information is given for ethanol at 1 atm:
boiling point = 78.40 °C         ∆H_{vap} (78.40 ºC) = 837.0 J/g
melting point = −114.5 ºC       ∆H_{fus}(−114.5 ºC) = 109.0 J/g
specific heat (gas) = 1.430 J/g · ºC     specific heat (liquid) = 2.460 J/g · ºC
A 33.50-g sample of liquid ethanol is initially at 13.50 ºC. If the sample is heated at constant pressure (P = 1 atm), calculate the amount of energy needed to raise the temperature of the sample to 94.50 ºC.
You are asked to calculate the total energy required to heat a sample of liquid ethanol.
You are given the mass of the ethanol sample, its initial and final temperature, and physical data for ethanol in the liquid and vapor phases.
This is a three step process: (1) raise the temperature of the liquid to its boiling point, (2) vaporize the liquid at its boiling point, and (3) raise the temperature of the resulting gas to the final temperature.
1. Calculate the amount of energy required to heat the liquid from 13.50 ºC to 78.40 ºC
q(1) = m × c_{liquid} × ∆T = (33.50 g)(2.460 J/g · ºC)(78.40 ºC − 13.50 ºC) = 5348 J
2. Calculate the amount of energy required to vaporize the liquid at its boiling point.
q(2) = m × ∆H_{vap} = (33.50 g)(837.0 J/g) = 2.804 × 10^{4} J
3. Calculate the amount of energy required to heat the vapor from 78.40 ºC to 94.50 ºC.
q(3) = m × c_{gas} × ∆T = (33.50 g)(1.430 J/g · ºC)(94.50 ºC − 78.40 ºC) = 771.3 J
The total amount of energy needed to heat the sample of ethanol is the sum of the energy required for the three steps.
q_{total} = q(1) + q(2) + q(3) = 5348 J + (2.804 × 10^{4} J) + 771.3 J = 3.416 × 10^{4} J = 34.16 kJ
Is your answer reasonable? Over the temperature range in this problem, ethanol is heated, converted from liquid to vapor phase and then heated in the vapor phase. All three changes involve a great deal of energy, especially the conversion of a liquid to a vapor. The total amount of energy required to raise the temperature of the ethanol is expected to be large.