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Chapter 10: Problem 102

What quantity of energy is needed to heat a 1.00 -mole sample of \(\mathrm{H}_{2} \mathrm{O}\) from \(-30.0^{\circ} \mathrm{C}\) to $140.0^{\circ} \mathrm{C} ?\( (see Exercise 101\))$

Short Answer

Expert verified
The total energy required to heat a 1.00-mole sample of \(\mathrm{H}_{2} \mathrm{O}\) from \(-30.0^{\circ} \mathrm{C}\) to \(140.0^{\circ}\mathrm{C}\) can be found by calculating the energy needed for three temperature ranges, using specific heat capacities of ice, water, and steam. First, find the mass of water with \(m = n \times M\). Then, calculate the temperature changes and use the equation \(Q = mc\Delta T\) for each range. Finally, sum the energies for each range to find the total energy: \(Q_\text{total} = Q_1 + Q_2 + Q_3\).

Step by step solution

01

Determine the constants required for calculations

For each heating range, we need to know the heat capacities and specific heats of water. The specific heat capacities (c) for H₂O at different phases are as follows: - Solid (ice): \(c_i = 2.093 J/g·K\) - Liquid (water): \(c_w = 4.186 J/g·K\) - Gas (steam): \(c_s = 1.996 J/g·K\) The molar mass of water (M) is 18.02 g/mol.
02

Calculate masses and temperature changes

Since we are given the number of moles (n) of water, we can use the molar mass (M) of water to calculate the mass (m) of the sample: \(m = n × M\) Next, calculate the temperature changes (∆T) for each range: 1. Range 1: -30.0°C to 0°C, ∆T₁ = (0°C - (-30.0°C)) = 30.0 K 2. Range 2: 0°C to 100°C, ∆T₂ = (100°C - 0°C) = 100.0 K 3. Range 3: 100°C to 140.0°C, ∆T₃ = (140°C - 100°C) = 40.0 K
03

Calculate energy for each range

We will use the specific heat capacity (c) and temperature change (∆T) to calculate the energy (Q) required for each range: \(Q = mc∆T\) 1. Range 1: Q₁ = m × c_i × ∆T₁ (Energy to heat ice from -30.0°C to 0°C) 2. Range 2: Q₂ = m × c_w × ∆T₂ (Energy to heat liquid water from 0°C to 100°C) 3. Range 3: Q₃ = m × c_s × ∆T₃ (Energy to heat steam from 100°C to 140.0°C)
04

Calculate the total energy required

Add the energies (Q) required for all three ranges to determine the total energy required to heat the 1.00-mole sample of H₂O from -30.0°C to 140.0°C: Total energy (Q_total) = Q₁ + Q₂ + Q₃

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Most popular questions from this chapter

A 0.250 -g chunk of sodium metal is cautiously dropped into a mixture of 50.0 \(\mathrm{g}\) water and 50.0 \(\mathrm{g}\) ice, both at \(0^{\circ} \mathrm{C}\) . The reaction is $$ 2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_{2}(g) \quad \Delta H=-368 \mathrm{kJ} $$ Assuming no heat loss to the surroundings, will the ice melt? Assuming the final mixture has a specific heat capacity of 4.18 $\mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C}$ , calculate the final temperature. The enthalpy of fusion for ice is 6.02 \(\mathrm{kJ} / \mathrm{mol}\) .

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Some of the physical properties of \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{D}_{2} \mathrm{O}\) are as follows: Account for the differences. (Note: D is a symbol often used for \(^{2} \mathrm{H},\) the deuterium isotope of hydrogen.)

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