Question

Suppose \(0.010 \mathrm{~kg}\) of steam (at \(100.00^{\circ} \mathrm{C}\) ) is added to \(0.10 \mathrm{~kg}\) of water (initially at \(\left.19.0^{\circ} \mathrm{C}\right)\). The water is inside an aluminum cup of mass \(35 \mathrm{~g}\). The cup is inside a perfectly insulated calorimetry container that prevents heat flow with the outside environment. Find the final temperature of the water after equilibrium is reached.

Short answer

Answer: The final equilibrium temperature of the water after steam is added is approximately \(63.84^{\circ} \mathrm{C}\).

Step-by-step solution

Calculate the Heat Gained by the Water and Aluminum Cup

First, we need the specific heat capacity of water \(c_w = 4.18 \times 10^3 \, \mathrm{J/kg \cdot K} \) and aluminum, \(c_{Al} = 0.897 \times 10^3 \, \mathrm{J/kg \cdot K} \). Let the initial temperature of water and the aluminum cup be \(T_1\), the equilibrium temperature be \(T_f\) and let the mass of water and aluminum cup are \(m_w\) and \(m_{Al}\), respectively. The heat absorbed by the water and the aluminum cup can be written as follows: $$ Q_{water} = m_w \cdot c_w \cdot (T_f - T_1)$$ $$ Q_{Al} = m_{Al} \cdot c_{Al} \cdot (T_f - T_1)$$ Now, let's move on to step 2.

Calculate the Heat Released by the Steam

We are given the mass of steam \(m_s = 0.010 \,\mathrm{kg}\). When steam condenses to become water at \(100^{\circ}\mathrm{C}\), it releases heat. The heat of vaporization of water is \(L_v = 2.26 \times 10^6 \, \mathrm{J/kg}\). So, the heat released during this process can be calculated as follows: $$ Q_{steam} = m_s \cdot L_v $$ In addition, we need to calculate the heat released when the condensed water cools down to the equilibrium temperature, \(T_f\): $$ Q_{s_cooling} = m_s \cdot c_w \cdot (100 - T_f)$$ Now, we will move on to step 3.

Equate the Heat Gained and Heat Released

According to the conservation of energy principle, the heat gained by the water and the aluminum cup must be equal to the heat released by steam: $$ Q_{water} + Q_{Al} = Q_{steam} + Q_{s_cooling}$$ Now, plug in the values calculated in steps 1 and 2: $$ m_w \cdot c_w \cdot (T_f - T_1) + m_{Al} \cdot c_{Al} \cdot (T_f - T_1) = m_s \cdot L_v + m_s \cdot c_w \cdot (100 - T_f) $$

Solve for the Final Equilibrium Temperature

Now, we can plug in the known values (\(m_w, m_s, m_{Al}, c_w, c_{Al}, L_v, T_1\)) and solve for the final equilibrium temperature \(T_f\): $$ 0.10 \cdot 4.18 \times 10^3 \cdot (T_f - 19) + 0.035 \cdot 0.897 \times 10^3 \cdot (T_f - 19) = 0.010 \cdot 2.26 \times 10^6 + 0.010 \cdot 4.18 \times 10^3 \cdot (100 - T_f) $$ Solving for \(T_f\), we obtain: $$ T_f \approx 63.84^{\circ} \mathrm{C} $$ Hence, the final equilibrium temperature of water after steam is added is approximately \(63.84^{\circ} \mathrm{C}\).

Key concepts

Specific Heat Capacity

Specific heat capacity is a central concept in thermodynamics. It denotes the amount of heat energy required to raise the temperature of one kilogram of a substance by one degree Celsius. Every material has its unique specific heat capacity, which tells us how much energy it can store.

The specific heat capacity of water, for instance, is relatively high at 4,180 J/(kg·K). This means it takes 4,180 joules to raise the temperature of one kilogram of water by one Kelvin. Aluminium, a typical cup's material for calorimetry exercises, has a much lower specific heat capacity of 897 J/(kg·K).

• Water's high specific heat capacity means it can absorb a lot of heat without a significant temperature rise. This property is crucial in both natural ecosystems and engineering applications.
• Understanding specific heat capacity helps in determining how different materials will react when mixed in a temperature-changing scenario.

In our exercise, recognizing these values allowed us to calculate the heat absorbed by the materials accurately.

Heat of Vaporization

Heat of vaporization is another key thermodynamic concept that refers to the amount of energy required to change a given mass of a substance from a liquid to a gas at its boiling point, without any change in temperature. This is a form of latent heat, called so because it's "hidden" during a phase change.

In the exercise, the heat of vaporization of water is essential because our steam is undergoing a condensation process, turning into liquid water. This releases a substantial amount of energy, precisely because of the high heat of vaporization of water, which is 2,260,000 J/kg.

• When steam condenses to water, it releases the latent heat of vaporization. This heat contributes to warming the water in the cup.
• The high latent heat of vaporization for water is why steam burns can be severe. When steam touches your skin, it condenses and releases a large amount of energy in the process.

Using these principles, we could calculate the energy exchange involved when the steam condenses and determines the final temperature equilibrium.

Energy Conservation

Energy conservation is one of the fundamental principles in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. In thermodynamics, this principle is known as the First Law of Thermodynamics.

In calorimetry problems like ours, energy conservation means the total energy within the system remains constant, though it transfers between objects. Here, the heat released by steam condensing and cooling down is exactly balanced by the heat absorbed by the water and aluminum cup.

• This concept leads us to set up the equation where the heat gained by the water and cup equals the heat released by the steam.
• Understanding energy conservation helps us understand why systems reach equilibrium.

By ensuring both sides of our energy equation were balanced, we could solve for the equilibrium temperature effectively.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It plays a crucial role in understanding energy changes in thermodynamics. A calorimeter is an insulated device that helps us perform these measurements by minimizing heat exchange with the environment.

In practice, calorimetry allows us to determine specific heat capacities, measure energy transformations, and understand reaction energetics.

• Insulation is key in calorimetry, as it ensures that any heat change is only due to the substances involved, not external factors.
• By measuring the temperature change in water and the calorimeter, we can precisely calculate the energy changes occurring in a reaction or physical change.

In our exercise, an insulated calorimeter ensured that heat from steam and the water was only exchanged between these components, allowing us to find the final temperature with confidence.