Question

The molar heats of fusion and vaporization for water are \(6.02 \mathrm{kJ} / \mathrm{mol}\) and \(40.6 \mathrm{kJ} / \mathrm{mol}\), respectively, and the specific heat capacity of liquid water is \(4.18 \mathrm{J} / \mathrm{g}\) C. What quantity of heat energy is required to melt \(25.0 \mathrm{g}\) of ice at \(0^{\circ} \mathrm{C} ?\) What quantity of heat is required to vaporize \(37.5 \mathrm{g}\) of liquid water at \(100^{\circ} \mathrm{C}\) ? What quantity of heat is required to warm \(55.2 \mathrm{g}\) of liquid water from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C} ?\)

Short answer

The heat energy required to melt \(25.0 \mathrm{g}\) of ice at \(0^{\circ} \mathrm{C}\) is \(8.35 \mathrm{kJ}\), to vaporize \(37.5 \mathrm{g}\) of liquid water at \(100^{\circ} \mathrm{C}\) is \(84.48 \mathrm{kJ}\), and to warm \(55.2 \mathrm{g}\) of liquid water from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) is \(23.07 \mathrm{kJ}\).

Step-by-step solution

Find the number of moles of water in the ice

We'll use the formula \(moles = \frac{mass}{molar mass}\). The molar mass of water is \(18.015 \mathrm{g/mol}\). moles = \( \frac{25.0 \mathrm{g}}{18.015 \mathrm{g/mol}}\) moles = \( 1.387 \mathrm{mol} \)

Calculate heat energy for melting ice

Now, use the heat energy formula with the molar heat of fusion for water. \(q = moles \cdot \Delta{}H_{fusion}\) \(q = 1.387 \mathrm{mol} \cdot 6.02 \mathrm{kJ/mol}\) \(q = 8.35 \mathrm{kJ} \) For the second scenario, vaporizing \(37.5 \mathrm{g}\) of liquid water at \(100^{\circ} \mathrm{C}\):

Find the number of moles of liquid water

Like before, use the moles formula: moles = \( \frac{37.5 \mathrm{g}}{18.015 \mathrm{g/mol}}\) moles = \( 2.081 \mathrm{mol} \)

Calculate heat energy for vaporizing water

Now, use the heat energy formula with the molar heat of vaporization for water. \(q = moles \cdot \Delta{}H_{vaporization}\) \(q = 2.081 \mathrm{mol} \cdot 40.6 \mathrm{kJ/mol}\) \(q = 84.48 \mathrm{kJ} \) For the third scenario, warming \(55.2 \mathrm{g}\) of liquid water from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\):

Calculate the change in temperature

The difference between the initial and final temperatures is: \(\Delta{}T = 100^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C}\) \(\Delta{}T = 100^{\circ} \mathrm{C} \)

Calculate heat energy for warming liquid water

Now, use the heat energy formula with the specific heat capacity of liquid water. \(q = mass \cdot C \cdot \Delta{}T\) \(q = 55.2 \mathrm{g} \cdot 4.18 \mathrm{J/g^{\circ}C} \cdot 100^{\circ} \mathrm{C}\) \(q = 23074.4 \mathrm{J} \) So, the quantity of heat energy required for the three scenarios are: - To melt \(25.0 \mathrm{g}\) of ice at \(0^{\circ} \mathrm{C}\): \( 8.35 \mathrm{kJ} \) - To vaporize \(37.5 \mathrm{g}\) of liquid water at \(100^{\circ} \mathrm{C}\): \( 84.48 \mathrm{kJ} \) - To warm \(55.2 \mathrm{g}\) of liquid water from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\): \( 23074.4 \mathrm{J} \) or \( 23.07 \mathrm{kJ} \)

Key concepts

Molar Heat of Fusion

The molar heat of fusion is the amount of energy needed to change one mole of a solid into a liquid at its melting point. This process occurs without changing the temperature of the substance. For water, the molar heat of fusion is given as 6.02 kJ/mol. This means that to melt 1 mole of ice at 0°C, 6.02 kJ of energy is required.

To calculate the energy needed for other amounts, you simply multiply the molar heat of fusion by the number of moles you have. For example, if you have 25 grams of ice, you first convert this to moles by using the molar mass of water (18.015 g/mol). You then multiply the moles by 6.02 kJ/mol to find the total energy needed for melting. This principle allows you to determine the heat needed to melt any quantity of ice.

Molar Heat of Vaporization

The molar heat of vaporization refers to the energy required to turn one mole of a liquid into a vapor at its boiling point, without a temperature change. For water, the molar heat of vaporization is 40.6 kJ/mol. This value helps us understand how much energy is absorbed when water vaporizes at 100°C.

Using this value for calculations involves similar steps to those used with the heat of fusion. Once you determine the number of moles of your liquid water sample using its weight and the molar mass, you can multiply this number by the molar heat of vaporization to get the total energy required to vaporize the water. Such calculations are crucial in understanding and predicting the energy changes involved in phase transitions.

Specific Heat Capacity

Specific heat capacity is a property of a substance that tells us how much energy, in joules, is needed to raise the temperature of one gram of the substance by one degree Celsius. For liquid water, the specific heat capacity is 4.18 J/g°C.

This property allows us to calculate how much heat energy is required to increase the temperature of a given mass of water. The formula to use is:

• q = mass \( \cdot \) specific heat capacity \( \cdot \) temperature change \((\Delta T)\)

For example, to find out how much heat is needed to warm 55.2 grams of water from 0°C to 100°C, you multiply the mass by the specific heat capacity and the change in temperature (which is 100°C in this scenario). This formula is useful whenever heat energy changes involve only temperature changes and not phase changes.

Phase Changes

Phase changes describe the transformations between solid, liquid, and gas states. During these changes, the temperature remains constant even though heat energy is added or removed. The energy is used to break or form bonds between molecules instead of increasing temperature.

The melting of ice and vaporization of water are examples of phase changes. When ice melts, it turns from solid to liquid, and this requires energy equivalent to the molar heat of fusion. Similarly, during vaporization, water turns into vapor, consuming energy as given by the molar heat of vaporization. Understanding phase changes is crucial because it explains why and how substances absorb or release energy without changing temperatures.

Temperature Change

Temperature change involves a change in the thermal energy of a substance, directly affecting its heat state but not causing a phase change. Calculating the heat energy required for a temperature change depends on the mass of the substance, its specific heat capacity, and the change in temperature.

The formula used is:

• q = mass \( \cdot \) specific heat capacity \( \cdot \) \(\Delta T\)

This is applicable when, for instance, heating water from 0°C to 100°C. You need to know how much of the substance you have and how much energy is required to increase its temperature. Recognizing the factors involved in temperature change enables more accurate predictions and calculations regarding energy use and heat management in various contexts, from cooking to industrial processes.