Question

Calculate the heat required to convert \(115 \mathrm{~g}\) of ice at \(0.0^{\circ} \mathrm{C}\) to steam at \(100.0^{\circ} \mathrm{C}\).

Short answer

The total heat required is 346,380 J.

Step-by-step solution

Calculate the heat to melt the ice

First, we need to calculate the heat required to melt 115 g of ice at 0.0°C into water at 0.0°C. We use the formula for heat transfer: \( q_1 = mL_f \) where \( m \) is the mass in grams and \( L_f \) is the latent heat of fusion for ice, which is \( 334 \, \text{J/g} \). Therefore, \( q_1 = 115 \, \text{g} \times 334 \, \text{J/g} = 38,410 \, \text{J} \).

Calculate the heat to raise water temperature

Next, we calculate the heat required to raise the temperature of the resulting water from 0.0°C to 100.0°C using \( q_2 = mc\Delta T \), where \( c \) is the specific heat capacity of water (\( 4.18 \, \text{J/g} \cdot \text{°C} \)) and \( \Delta T = 100.0\, \text{°C} - 0.0\, \text{°C} = 100.0\, \text{°C} \). Thus, \( q_2 = 115 \, \text{g} \times 4.18 \, \text{J/g} \cdot \text{°C} \times 100.0 \text{°C} = 48,070 \, \text{J} \).

Calculate the heat to convert water to steam

Lastly, we calculate the heat needed to convert water at 100.0°C to steam at 100.0°C. We use the formula \( q_3 = mL_v \), where \( L_v \) is the latent heat of vaporization for water, which is \( 2260 \, \text{J/g} \).Therefore, \( q_3 = 115 \, \text{g} \times 2260 \, \text{J/g} = 259,900 \, \text{J} \).

Sum the total heat required

Finally, we sum all the amounts of heat calculated to find the total heat required: \( q_{\text{total}} = q_1 + q_2 + q_3 = 38,410 \, \text{J} + 48,070 \, \text{J} + 259,900 \, \text{J} = 346,380 \, \text{J} \).

Key concepts

Latent Heat of Fusion

When a solid turns into a liquid, it absorbs energy without a change in temperature. This energy absorption is known as the latent heat of fusion. In simpler terms, it's the energy needed to melt a solid. For ice, this value is about 334 Joules per gram.

• "Latent" means hidden, indicating that this heat is not used to increase temperature.
• "Fusion" refers to melting, the phase change from solid to liquid.

In the context of the exercise, converting 115 grams of ice at 0.0°C to water at the same temperature requires this latent heat of fusion. Using the formula:\[ q_1 = m \times L_f \]where \( m \) represents the mass and \( L_f \) the latent heat. Thus, \( q_1 = 115\, \text{g} \times 334\, \text{J/g} = 38,410\, \text{J} \) of energy is needed.

Specific Heat Capacity

Specific heat capacity is the measure of a substance's ability to absorb heat energy. It defines how much energy is required to raise the temperature of one gram of a substance by one degree Celsius. For water, this value is 4.18 Joules per gram per degree Celsius. This means that water requires more heat to change its temperature compared to many other substances.

• Higher specific heat capacity = needs more energy to heat up.
• Crucial for understanding temperature changes within a fluid system.

Looking at the example, the water needs heating after the ice has melted:\[ q_2 = m \times c \times \Delta T \]where \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. Given \( \Delta T \) is 100°C, the required heat is:\( q_2 = 115\, \text{g} \times 4.18\, \text{J/g°C} \times 100 °C = 48,070 \text{J} \).

Latent Heat of Vaporization

When a liquid turns into a gas, it's called vaporization. During this phase change, the substance absorbs a significant amount of energy without a change in temperature, known as the latent heat of vaporization. For water, it takes 2260 Joules per gram to transition from liquid to vapor.

• This process involves breaking intermolecular forces.
• Energy intake is entirely utilized for the phase transition.

In the exercise, converting 115 grams of water at 100°C to steam at the same temperature involves:\[ q_3 = m \times L_v \]where \( L_v \) is the latent heat of vaporization. Thus, the calculation becomes:\( q_3 = 115\, \text{g} \times 2260\, \text{J/g} = 259,900\, \text{J} \) to complete the phase change.