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Chapter 6: Problem 137

You have a 1.00 -mole sample of water at \(-30 .^{\circ} \mathrm{C}\) and you heat it until you have gaseous water at \(140 .^{\circ} \mathrm{C}\) . Calculate \(q\) for the entire process. Use the following data. $$ \begin{aligned} \text { Specific heat capacity of ice } &=2.03 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g} \\ \text { Specific heat capacity of water } &=4.18 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g} \\ \text { Specific heat capacity of steam } &=2.02 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g} \end{aligned} $$ $$ \mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) \quad \Delta H_{\mathrm{fision}}=6.02 \mathrm{kJ} / \mathrm{mol}\left(\mathrm{at} 0^{\circ} \mathrm{C}\right) $$ $$ \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g) \quad \Delta H_{\mathrm{vaporization}}=40.7 \mathrm{kJ} / \mathrm{mol}\left(\mathrm{at} 100 .^{\circ} \mathrm{C}\right) $$

Short Answer

Expert verified
The total heat, \(q\), required for the entire process can be calculated by following the steps: 1. Heat ice from -30°C to 0°C: \(q_1 = (1 \cdot 18.02\,g)(2.03\,\frac{J}{g\cdot °C})(0 - (-30))\,°C\) 2. Melt ice at 0°C: \(q_2 = (1\,mol)(6.02\,\frac{kJ}{mol})\) 3. Heat water from 0°C to 100°C: \(q_3 = (1 \cdot 18.02\,g)(4.18\,\frac{J}{g\cdot °C})(100 - 0)\,°C\) 4. Evaporate water at 100°C: \(q_4 = (1\,mol)(40.7\,\frac{kJ}{mol})\) 5. Heat steam from 100°C to 140°C: \(q_5 = (1 \cdot 18.02\,g)(2.02\,\frac{J}{g\cdot °C})(140 - 100)\,°C\) Add the heat calculated in each of the five steps to find the total heat required for the entire process: \(q = q_1 + q_2 + q_3 + q_4 + q_5\)

Step by step solution

01

Heating ice to 0°C

First, calculate the heat required to heat the ice from -30°C to 0°C. Use the specific heat capacity of ice and the mass of water, which is 1 mole × molar mass of water (18.02 g/mol). \(q_1 = mc\Delta T\) Where: \(m = 1 \cdot 18.02\,g\) \(c = 2.03\,\frac{J}{g\cdot °C}\) \(\Delta T = (0 - (-30))\,°C\) Calculate heat required, \(q_1\).
02

Melting ice

Next, calculate the heat needed to melt the ice at 0°C. Use the molar enthalpy of fusion. \(q_2 = n\Delta H_{fusion}\) Where: \(n = 1\,mol\) \(\Delta H_{fusion} = 6.02\,\frac{kJ}{mol}\) Convert \(\Delta H_{fusion}\) to J/mol for consistency. Calculate heat required, \(q_2\).
03

Heating water to 100°C

Calculate the heat needed to heat the liquid water from 0°C to 100°C. Use the specific heat capacity of water and mass of water. \(q_3 = mc\Delta T\) Where: \(m = 1 \cdot 18.02\,g\) \(c = 4.18\,\frac{J}{g\cdot °C}\) \(\Delta T = (100 - 0)\,°C\) Calculate heat required, \(q_3\).
04

Evaporating water

Next, calculate the heat required to evaporate the water at 100°C. Use the molar enthalpy of vaporization. \(q_4 = n\Delta H_{vaporization}\) Where: \(n = 1\,mol\) \(\Delta H_{vaporization} = 40.7\,\frac{kJ}{mol}\) Convert \(\Delta H_{vaporization}\) to J/mol for consistency. Calculate heat required, \(q_4\).
05

Heating steam to 140°C

Lastly, calculate the heat required to heat the steam from 100°C to 140°C. Use the specific heat capacity of steam and mass of water. \(q_5 = mc\Delta T\) Where: \(m = 1 \cdot 18.02\,g\) \(c = 2.02\,\frac{J}{g\cdot °C}\) \(\Delta T = (140 - 100)\,°C\) Calculate heat required, \(q_5\).
06

Total heat for the entire process

Calculate the total heat, \(q\), required for the entire process by adding the heat calculated in each of the five steps. \(q = q_1 + q_2 + q_3 + q_4 + q_5\)

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Most popular questions from this chapter

In a coffee-cup calorimeter, 50.0 \(\mathrm{mL}\) of 0.100\(M \mathrm{AgNO}_{3}\) and 50.0 \(\mathrm{mL}\) of 0.100 \(\mathrm{M} \mathrm{HCl}\) are mixed to yield the following reaction: $$\mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \longrightarrow \mathrm{AgCl}(s)$$ The two solutions were initially at \(22.60^{\circ} \mathrm{C}\) , and the final temperature is \(23.40^{\circ} \mathrm{C}\) Calculate the heat that accompanies this reacture in kJ/mol of AgCl formed. Assume that the combined solution has a mass of 100.0 \(\mathrm{g}\) and a specific heat capacity of 4.18 \(\mathrm{J} / \rho \mathrm{C} \cdot \mathrm{g} .\)

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A system undergoes a process consisting of the following two steps: Step 1: The system absorbs 72 \(\mathrm{J}\) of heat while 35 \(\mathrm{J}\) of work is done on it. Step \(2 :\) The system absorbs 35 \(\mathrm{J}\) of heat while performing 72 \(\mathrm{J}\) of work. Calculate \(\Delta E\) for the overall process.
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