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Chapter 9: Problem 99

What quantity of energy does it take to convert 0.500 kg ice at \(-20 .^{\circ} \mathrm{C}\) to steam at \(250 .^{\circ} \mathrm{C} ?\) Specific heat capacities: ice, \(2.03 \mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C} ;\) liquid, \(4.2 \mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C} ;\) steam, \(2.0 \mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C} ; \Delta H_{\mathrm{vap}}=\) \(40.7 \mathrm{kJ} / \mathrm{mol} ; \Delta H_{\mathrm{fus}}=6.02 \mathrm{kJ} / \mathrm{mol}.\)

Short Answer

Expert verified
It takes \(1,744,560 \, J\) of energy to convert \(0.500 \, kg\) of ice at \(-20°C\) to steam at \(250°C\).

Step by step solution

01

Convert mass to grams and calculate the number of moles of water in the sample

Since the specific heat capacities and enthalpies are given in terms of grams and moles, we need to convert the given mass of ice from kilograms to grams and calculate the number of moles: \(m = 0.500 \, kg = 500 \, g\) To calculate the number of moles, we will use the molar mass of water, which is 18 g/mol: \(n = \frac{500 \, g}{18 \, g/mol} = 27.78 \, mol\)
02

Calculate the energy required to heat the ice from -20°C to 0°C

Using the specific heat capacity of ice and the given mass and temperature difference, we can calculate the energy required: \(Q_1 = mcΔT\) \(Q_1 = (500 \, g)(2.03 \, J/g°C)(0°C - (-20°C)) = 20300 \, J\)
03

Calculate the energy required to convert the ice at 0°C to water

Using the enthalpy of fusion and the number of moles, we can now calculate the energy needed to turn the ice into water: \(Q_2 = nΔH_{fus}\) \(Q_2 = (27.78 \, mol)(6.02 \, kJ/mol) = 167.20 \, kJ = 167200 \, J\)
04

Calculate the energy required to heat the water from 0°C to 100°C

Now we need to find the energy needed to raise the temperature of the water to its boiling point: \(Q_3 = mcΔT\) \(Q_3 = (500 \, g)(4.2 \, J/g°C)(100°C - 0°C) = 210000 \, J\)
05

Calculate the energy required to convert the water at 100°C to steam

Using the enthalpy of vaporization and the number of moles, calculate the energy needed to turn the water into steam: \(Q_4 = nΔH_{vap}\) \(Q_4 = (27.78 \, mol)(40.7 \, kJ/mol) = 1131.06 \, kJ = 1131060 \, J\)
06

Calculate the energy required to heat the steam from 100°C to 250°C

Finally, we need to find the energy needed to raise the temperature of the steam to 250°C: \(Q_5 = mcΔT\) \(Q_5 = (500 \, g)(2.0 \, J/g°C)(250°C - 100°C) = 150000 \, J\)
07

Calculate the total energy required

Now, simply add up all the energy values calculated in the previous steps: \(Q_{total} = Q_1 + Q_2 + Q_3 + Q_4 + Q_5\) \(Q_{total} = 20300 \, J + 167200 \, J + 210000 \, J + 1131060 \, J + 150000 \, J\) \(Q_{total} = 1744560 \, J\) Thus, it takes 1,744,560 J of energy to convert 0.500 kg of ice at -20°C to steam at 250°C.

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Most popular questions from this chapter

The molar enthalpy of vaporization of water at \(373 \mathrm{K}\) and 1.00 atm is \(40.7 \mathrm{kJ} / \mathrm{mol} .\) What fraction of this energy is used to change the internal energy of the water, and what fraction is used to do work against the atmosphere? (Hint: Assume that water vapor is an ideal gas.)

A plot of \(\ln \left(P_{\text {vap }}\right)\) versus \(1 / T(\mathrm{K})\) is linear with a negative slope. Why is this the case?

Which of the following statements about intermolecular forces is(are) true? a. London dispersion forces are the only type of intermolecular force that nonpolar molecules exhibit. b. Molecules that have only London dispersion forces will always be gases at room temperature \(\left(25^{\circ} \mathrm{C}\right).\) c. The hydrogen-bonding forces in \(\mathrm{NH}_{3}\) are stronger than those in \(\mathrm{H}_{2} \mathrm{O}\). d. The molecules in \(\mathrm{SO}_{2}(g)\) exhibit dipole-dipole intermolecular interactions. e. \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{3}\) has stronger London dispersion forces than does \(\mathrm{CH}_{4}.\)

Compare and contrast the structures of the following solids. a. diamond versus graphite b. silica versus silicates versus glass

The structure of the compound \(\mathrm{K}_{2} \mathrm{O}\) is best described as a cubic closest packed array of oxide ions with the potassium ions in tetrahedral holes. What percent of the tetrahedral holes are occupied in this solid?
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