Question

What will be the final temperature of the water in an insulated container as the result of passing $5.00\mathrm{g}$ of steam, ${\mathrm{H}}_2\mathrm{O}(\mathrm{g}),$ at ${100.0}^{\circ }\mathrm{C}$ into $100.0\mathrm{g}$ of water at ${25.0}^{\circ }\mathrm{C}?(\mathrm{\Delta }{H}_{\mathrm{vap}}^{\circ }=40.6\mathrm{kJ}/\mathrm{mol}{\mathrm{H}}_2\mathrm{O})$.

Short answer

The final temperature of the water will be 52 °C.

Step-by-step solution

Calculate moles of steam

First, convert the mass of steam to moles using the molar mass of water (18.02 g/mol). For example, $5.00g÷18.02g/mol=0.277mol$

Calculate heat released by steam

Next, calculate the heat released when the steam condenses using the molar enthalpy of vaporization: $q=n\cdot \mathrm{\Delta }{H}_{vap}=0.277mol\times 40.6kJ/mol=11.3kJ$ This is the amount of heat that the steam releases as it cools and condenses.

Calculate the heat absorbed by the water and the final temperature

The heat absorbed by the 100.0 g of water will be the same as the heat released by the steam (due to conservation of energy in an insulated container), so we can use the specific heat equation $q=m\cdot c\cdot \mathrm{\Delta }T$ to solve for the change in temperature: $11.3kJ=100.0g\times 4.18\times {10}^{-3}kJ/g°C\times \mathrm{\Delta }T$ . Solving for delta T, we get $\mathrm{\Delta }T=27°C$. Thus, the final temperature of the water will be the initial temperature plus the change in temperature: $T_{f}inal=T_{i}nitial+\mathrm{\Delta }T=25°C+27°C=52°C$.

Key concepts

Calorimetry

Calorimetry is the science of measuring heat transfer during chemical reactions or physical changes. In our exercise, we dealt with an insulated system where the only heat exchange occurs within the materials inside the container. This is important because it ensures any heat released by one component is absorbed by another, maintaining the law of conservation of energy.
When steam is introduced into water, these two substances reach a new temperature balance, known as thermal equilibrium. By calculating the heat released or absorbed (using calorimetry principles), we can find that equilibrium state.
In practice, calorimetry involves mathematical calculations to determine changes in heat content, often relying on equations that incorporate variables like mass, specific heat, and temperature change. These calculations help predict the behavior of substances under different thermochemical interactions, such as combustion or phase shifts.

Enthalpy of Vaporization

The enthalpy of vaporization tells us how much energy is needed to convert a substance from a liquid to a gas. In reverse, it also lets us calculate how much heat is released when a gas condenses into a liquid. For water, this value is quite high, which is why steam carries a lot of energy.
In this exercise, the steam condenses when it comes into contact with cooler water, releasing its stored energy—a process quantified using the enthalpy of vaporization. This release of energy accounts for the increase in temperature of the water in the container.
By knowing the $\mathrm{\Delta }{H}_{vap}$ for water, you can precisely determine how much heat exchange occurs per mole of water when it transitions between gas and liquid. This knowledge is crucial for various applications in heating and cooling systems, energy production, and understanding weather phenomena like condensation.

Specific Heat

Specific heat is a measure of how much energy is required to raise the temperature of a unit mass of a substance by one degree Celsius. Water, for example, has a high specific heat, meaning it can absorb a lot of heat without changing its temperature significantly.
In our problem, the specific heat of water ($4.18\text{J/g°C})$ allows us to calculate how much energy the 100.0 g of water will absorb from the condensing steam, thus changing its temperature.
When dealing with heat transfer problems, specific heat is a crucial factor because it determines how different substances respond to thermal energy. Substances with high specific heat are often used as coolants or heat reservoirs. Understanding this concept helps in designing systems for temperature regulation, such as climate control in buildings or thermal protection in electronics.