Question

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

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

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.

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

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)

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

Key concepts

Specific Heat Capacity

Specific heat capacity is a key concept in understanding how much energy is needed to change the temperature of a substance. It defines the amount of heat required to change the temperature of 1 gram of a substance by 1 degree Celsius or Kelvin.
The formula used is:

• Q = mcΔT

where Q is the heat added, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
In this problem, we use different values of specific heat capacities for water in its different states:

• Solid (ice): 2.093 J/g·K
• Liquid (water): 4.186 J/g·K
• Gas (steam): 1.996 J/g·K

This means that different amounts of heat are required for changing the temperature of water depending on whether it is in solid, liquid, or gas form. Understanding these values is crucial for making accurate energy calculations in thermodynamics.

Phase Transitions

Phase transitions involve the change of a substance from one state of matter to another, such as from a solid to a liquid or from a liquid to a gas. These transitions occur at specific temperatures and require significant amounts of energy without a change in temperature, known as latent heat.
When heating water from -30°C to 140°C, several phase transitions occur:

• From solid (ice) at -30°C to liquid (water) at 0°C
• From liquid (water) to gas (steam) at 100°C

During these transitions, energy is required to break intermolecular forces, allowing the phase change without a temperature change, a concept known as latent heat.
For precise solutions, the energy required for these transitions must be added to the energy calculated using specific heat capacities for heating only. This ensures the full thermodynamic changes are accounted for.

Thermodynamics

Thermodynamics is the study of energy and heat transfer processes. In the context of this problem, it helps us understand how energy is used to change water from one temperature or phase to another.
Key principles include:

• The conservation of energy, meaning energy cannot be created or destroyed, only transformed from one form to another
• The insights from the Second Law of Thermodynamics, which explains how energy tends to spread out or diffuse across systems

When calculating energy requirements in thermodynamics, it's critical to consider:

• Changes in temperature and phase
• The energy required for each state transition

Using these principles, we can systematically calculate the total energy required for all stages in heating the water, ensuring energy conservation with consistent unit management.

Water Heating

Water heating is a common experiment in thermodynamics, involving extending energy to water to raise its temperature across different phases.
This involves multiple stages:

• Heating ice from -30°C to 0°C using the specific heat of ice
• Melting the ice into water, requiring latent heat
• Heating liquid water from 0°C to 100°C using the specific heat of water
• Boiling water into steam, requiring latent heat of vaporization
• Heating steam from 100°C to 140°C using the specific heat of steam

Each stage and its energy requirement reflect the versatility of water's heat capacities and how it can store and transfer large amounts of energy. Accurately calculating these requirements involves understanding the distinct steps of water state transitions and their unique energy needs.