Question

A vessel whose walls are thermally insulated contains 2.40 kg of water and 0.450 kg of ice, all at 0.0${}^{\circ }$C. The outlet of a tube leading from a boiler in which water is boiling at atmospheric pressure is inserted into the water. How many grams of steam must condense inside the vessel (also at atmospheric pressure) to raise the temperature of the system to 28.0${}^{\circ }$C? You can ignore the heat transferred to the container.

Short answer

214.2 grams of steam must condense.

Step-by-step solution

Calculate the heat needed to melt the ice

First, we find the heat needed to melt the 0.450 kg of ice at 0.0°C. The latent heat of fusion of ice is 334,000 J/kg. We calculate the heat ($Q_1$) required: $$Q_1=m_{\text{ice}}\times L_f=0.450\text{kg}\times 334,000\text{J/kg}=150,300\text{J}$$

Calculate the heat needed to raise the temperature of water from the melted ice and existing water

Next, we find the heat required to raise the temperature of the resulting water (0.450 kg from ice and 2.40 kg existing) from 0.0°C to 28.0°C. The specific heat capacity of water is 4,186 J/kg°C. So, the heat ($Q_2$) required is: $$Q_2=(m_{\text{ice o water}}+m_{\text{water}})\times c\times \mathrm{\Delta }T=(0.450\text{ kg}+2.40\text{ kg})\times 4,186\text{J/kg°C}\times 28.0°\text{C}=333,724.8\text{J}$$

Calculate the total heat required for the process

Add the heat needed to melt the ice and raise the temperature to 28.0°C. $$Q_{\text{total}}=Q_1+Q_2=150,300\text{J}+333,724.8\text{J}=484,024.8\text{J}$$

Calculate the mass of steam required

To find the mass of steam needed, use the latent heat of vaporization of water, which is 2,260,000 J/kg. Rearrange the equation for heat to solve for the mass of steam ($m_{\text{steam}}$): $$Q_{\text{total}}=m_{\text{steam}}\times L_v$$ Thus, $$m_{\text{steam}}=\frac{Q_{\text{total}}}{L_v}=\frac{484,024.8\text{J}}{2,260,000\text{J/kg}}\approx 0.2142\text{kg}$$ Convert this to grams: $$0.2142\text{kg}=214.2\text{g}$$

Key concepts

Latent Heat

Latent heat is the energy absorbed or released by a substance during a change of state without changing its temperature. For example, when ice melts into water, it absorbs a certain amount of heat, known as the latent heat of fusion. This energy helps break the molecular bonds in the ice without raising the temperature. It is important to recognize that the amount of latent heat varies depending on the material and the phase change it undergoes.

For ice, the latent heat of fusion is 334,000 J/kg. This means that to melt 1 kg of ice at 0°C into 1 kg of water at 0°C, you need 334,000 joules of heat. No temperature change occurs during this phase; the energy goes into breaking the bonds between water molecules instead of increasing the kinetic energy of the molecules.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of 1 kilogram of a substance by 1°C. It tells us how a given material responds when it receives heat. For instance, water has a specific heat capacity of 4,186 J/kg°C, which is relatively high compared to many other substances. This high value means water can absorb or lose a lot of heat without experiencing a large temperature change.

Understanding specific heat capacity is crucial in solving thermodynamic problems, such as calculating how much energy is needed to raise the temperature of a mass of water, as we see in thermal exercises where we change the temperature of a system. In our scenario, knowing the specific heat capacity enables us to determine how much energy is necessary not just to melt the ice, but also to warm all water to the desired temperature of 28°C.

Phase Change

A phase change is a transition of matter from one state (solid, liquid, gas) to another. During a phase change, a substance absorbs or releases energy in the form of latent heat, but its temperature remains constant. Common examples of phase changes include melting, freezing, vaporization, and condensation.

Each phase change involves energy exchange without altering the temperature of the substance. In our example, ice melting into water is a phase change where the system absorbs energy to break ice's rigid structure. It’s important to understand the concept of phase changes to calculate how much energy is required to complete such transformations, particularly when combined with temperature changes after melting.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or substance to another. This can occur through conduction, convection, or radiation. In thermodynamics problems, understanding how heat is transferred is vital for solving the energy balance within a system.

In our exercise, the heat generated by the condensing steam is transferred to the ice and water in the vessel. The steam loses energy when it condenses, releasing its latent heat of vaporization as it changes from gas to liquid. That energy is then absorbed by the ice and water, leading them to melt and increase in temperature. It's critical to account for all energy transfers in the system to accurately determine the amount of heat required to achieve a desired outcome, such as raising the system's temperature to 28°C.