Problem 78 - Chemistry: An Atoms-Focused Approach

Question

List the steps you would take to convert a \(10.0 \mathrm{g}\) sample of water at \(25^{\circ} \mathrm{C}\) and 2 atm pressure to ice at 1 atm pressure. At what temperature would the water freeze?

Step-by-Step Solution

Answer

Answer: The steps to convert a 10 g sample of water at 25°C and 2 atm pressure to ice at 1 atm pressure are: 1) cool the water down to its freezing point under 2 atm pressure, which is approximately -1°C, 2) reduce the pressure from 2 atm to 1 atm, with the freezing point at 0°C, and 3) calculate the change in enthalpy required to convert the water to ice. The total energy required for these steps is 4421.8 J. The water would freeze at 0°C, which is the freezing point at 1 atm pressure.

Step 1: Cool the water down to its freezing point

First, you would need to cool the water down from 25°C to the freezing point under constant pressure of 2 atm. We know that the standard freezing point (at 1 atm pressure) of water is 0°C. However, since we're at 2 atm pressure, we need to find the freezing point depression caused by the increased pressure. For this exercise, we will assume that the freezing point decreases linearly with increasing pressure, so the freezing point at 2 atm pressure will be around -1°C. To calculate the energy required to cool the water, use the following formula: \(Q = mc\Delta T\) where \(Q\) is the energy required, \(m\) is the mass of water (10 grams), \(c\) is the specific heat capacity of water (4.18 J/g°C), and \(\Delta T\) is the change in temperature (25 - (-1) = 26°C). Plug in the values and get: \(Q = 10 \times 4.18 \times 26 \mathrm{J} = 1086.8 \mathrm{J}\)

Step 2: Reduce pressure from 2 atm to 1 atm

In this step, we will consider that the cooling process continues as the pressure is decreased from 2 atm to 1 atm. As previously mentioned, we will assume that the freezing point change per atm pressure change is linear, so the change in freezing point will also be -1°C. At this new pressure, the freezing point of water is back to 0°C. Note that the energy required to change pressure is negligible in this case, so no calculation is needed.

Step 3: Find the enthalpy change required to convert water to ice

Now, you need to change the water into ice at 0°C and 1 atm pressure. This is a phase change, and the energy required can be calculated using: \(Q = m\Delta H_f\) where \(Q\) is the energy required, \(m\) is the mass of water (10 grams), and \(\Delta H_f\) is the enthalpy of fusion of water (333.5 J/g). Plug in the values and get: \(Q = 10 \times 333.5 \mathrm{J/g} = 3335 \mathrm{J}\) To find the total energy required for all steps, simply add up the energy required in each step: \(Q_\text{total} = 1086.8 \mathrm{J} + 3335 \mathrm{J} = 4421.8 \mathrm{J}\) In conclusion, follow these steps to convert a 10 g sample of water at 25°C and 2 atm pressure to ice at 1 atm pressure: 1. Cool the water down to its freezing point under 2 atm pressure, which will be approximately -1°C. This requires 1086.8 J of energy. 2. Reduce the pressure from 2 atm to 1 atm, with the freezing point at 0°C. 3. Calculate the change in enthalpy required to convert the water to ice, which is 3335 J. The water would freeze at 0°C, which is the freezing point at 1 atm pressure.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a key concept in understanding how substances absorb or release heat. It's defined as the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. The specific heat capacity of water is quite high at 4.18 J/g°C, meaning it takes a lot of energy to change its temperature.

In the given exercise, we calculated the energy required to cool down water from 25°C to approximately \(-1°C\) under a pressure of 2 atm. The formula used to calculate this energy is:

• \(Q = mc\Delta T\)
  where \(Q\) is the energy in joules, \(m\) is the mass in grams, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature.

By plugging in the values (mass = 10 g, change in temperature = 26°C), we calculated that 1086.8 J of energy is needed. This illustrates the significant energy associated with specific heat capacity when transitioning temperatures.

Specific heat capacity is crucial in many scientific and engineering fields, helping to design systems that effectively manage temperatures, such as heating and cooling systems, weather forecasting, and designing everyday products like pots and pans.

Enthalpy of Fusion

Enthalpy of fusion is essential for understanding phase transitions, especially as liquids turn into solids. It represents the amount of energy needed to change a substance from a liquid to a solid without changing its temperature. For water, the enthalpy of fusion is 333.5 J/g.

In the exercise, converting water at 0°C to ice at 1 atm required calculating this energy change:

• \(Q = m\Delta H_f\)\
  where \(Q\) is energy, \(m\) is mass in grams, and \(\Delta H_f\) is the enthalpy of fusion.

Using 10 g of water as our mass, the formula returns 3335 J required to freeze the water into ice. This concept helps quantify the energy involved in phase changes, critical for refrigeration, manufacturing, and food preservation processes.

Understanding enthalpy of fusion explains why ice takes time to form even in freezing temperatures and why ice melts consistently at 0°C, ensuring predictability in natural and industrial processes.

Pressure Effects on Freezing Point

Pressure can significantly impact the freezing point of a substance. Generally, increased pressure lowers the freezing point of liquids. For water, this means ice forms at temperatures below 0°C when under greater pressure.

The exercise mentioned that at 2 atm, water's freezing point dips to around \(-1°C\). This is due to the impact of pressure on the intermolecular forces within water. While water is unusual in expanding upon freezing, typical materials decrease in volume, changing how pressure affects their phase transitions.

• The freezing point depression is often linear with pressure increases for each atm.

Understanding pressure's role in freezing is crucial in industries like ice making, high-pressure environments, and climates where pressure changes are part of daily weather patterns. Although the effects are subtle at usual atmospheric levels, they become vital when dealing with larger scales or engineering precise freezing conditions.

In summary, phase transitions like freezing and boiling aren't affected by temperature alone but also significantly influenced by pressure changes. This explains why, under higher pressures, you might observe unusual phenomena like supercooled liquids.