Problem 105 - University Physics with Modern Physics

Question

In a container of negligible mass, 0.0400 \(\mathrm{kg}\) of steam at \(100^{\circ} \mathrm{C}\) and atmospheric pressure is added to 0.200 \(\mathrm{kg}\) of water at \(50.0^{\circ} \mathrm{C} .\) (a) If no heat is lost to the surroundings, what is the final temperature of the system?(b) At the final temperature, how many kilograms are there of steam and how many of liquid water?

Step-by-Step Solution

Answer

Final temperature is approximately \(69.6^{\circ} \text{C}\); all \(0.240 \text{ kg}\) is liquid water.

Step 1: Define Heat Transfer Equations

First, we need to understand that the system comprises steam and water at different temperatures. The heat lost by the steam is equal to the heat gained by the water. We'll use the formula for heat transfer: \( Q = mc\Delta T \) and the formula for the phase transition from steam to water \( Q = mL \), where \( L \) is the latent heat of vaporization.

Step 2: Calculate Heat to Condense Steam

The steam must first condense into water before mixing to equilibrium temperature. The latent heat of vaporization of water is \( L_v = 2260 \text{ kJ/kg} \). Calculate the heat released when the steam condenses: \( Q_1 = mL = 0.0400 \, \text{kg} \times 2260 \, \text{kJ/kg} = 90.4 \, \text{kJ} \).

Step 3: Determine Heat Needed to Cool Condensed Steam to Final Temperature

For the condensed water (at \(100^{\circ} \text{C}\)) to reach the equilibrium temperature \( T_f \), use the specific heat of water, \( c = 4.18 \text{ kJ/kg°C}\). The heat released is \( Q_2 = 0.0400 \, \text{kg} \times 4.18 \, \text{kJ/kg°C} \times (100 - T_f) \).

Step 4: Calculate Heat Required to Warm Initial Water to Final Temperature

The heat required to warm up the 0.200 kg of water from \(50.0^{\circ} \text{C}\) to \( T_f \) is given by: \( Q_3 = 0.200 \, \text{kg} \times 4.18 \, \text{kJ/kg°C} \times (T_f - 50.0) \).

Step 5: Set Up Heat Balance Equation

Since no heat is lost to the surroundings, the heat gained by the water is equal to the heat lost by the steam. Set up the equation: \( 90.4 + 0.0400 \times 4.18 \times (100 - T_f) = 0.200 \times 4.18 \times (T_f - 50) \).

Step 6: Solve for Final Temperature \(T_f\)

Simplify and solve the heat balance equation: \( 90.4 + 0.1672 \times (100 - T_f) = 0.836 \times (T_f - 50) \). This results in \( T_f \approx 69.6^{\circ} \text{C}\).

Step 7: Analyze the Final State of the System

Since the final temperature is below \(100^{\circ} \text{C}\), all the steam has condensed into water. Therefore, the entire system consists of liquid water.

Key Concepts

Heat Transfer

In the world of thermodynamics, heat transfer is the process by which thermal energy moves from one substance to another. When we consider a system like the one in our exercise, the heat transfer involves both the steam and the liquid water inside the container. The fundamental principle here is that heat will flow naturally from a warmer object to a cooler one until they reach equilibrium.

To analyze heat transfer, we use key equations such as: - The general formula for heat exchange, which is given by: \( Q = mc\Delta T \) where \( Q \) is the heat gained or lost, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.- Additionally, when dealing with phase changes, another formula is required: \( Q = mL \) Here, \( L \) represents the latent heat involved in the phase transition.

• In this problem, steam loses heat as it condenses and mixes with water, releasing energy in the process.
• It's critical to account for all heat exchanges to find the final temperature of mixed substances.

Latent Heat of Vaporization

Latent heat of vaporization is a term used to describe the amount of energy required to change a substance from its liquid phase to a gaseous one, or vice versa, without altering its temperature. This concept is pivotal in our exercise, as it deals specifically with the steam's conversion to liquid water.

For water, the latent heat of vaporization is particularly high because of the strong hydrogen bonds that need to be broken to turn liquid water into vapor. Conversely, energy is released when vapor condenses back into water.

• In this exercise, the calculation kicked off with finding the heat released during this phase change using: \( Q = mL_v \)
• The latent heat of vaporization for water is known to be approximately 2260 kJ/kg. This value illustrates how much energy will be released per kilogram of steam when it condenses.

Understanding this concept helps explain why such a significant amount of heat is involved when dealing with steam transitioning into liquid form.

Specific Heat Capacity

The specific heat capacity is an intrinsic property of a material that dictates how much energy is needed to change its temperature by one degree Celsius. It's crucial for determining how substances react to heat transfer.

In the exercise, both the steam and the water's temperature changes rely on their specific heat capacity, which for water is \( c = 4.18 \, \text{kJ/kg°C} \). This value is one of the reasons water is so effective at temperature regulation in nature.

• For the 0.200 kg of water, the specific heat capacity helps calculate the heat absorbed until the equilibrium temperature is reached:
• When applying: \( Q = mc\Delta T \), both for cooling and warming processes, \( c \) plays a critical role.

Specific heat capacity can vary widely among different materials, so it is always essential to use accurate values during calculations of heat transfer.

Phase Transition

A phase transition, sometimes referred to as a phase change, describes the transformation of a substance from one state of matter to another. In our exercise, we observe a transition from steam to liquid water. This transition not only involves a change in the physical state but also significant energy exchange.

The process of condensing steam into water is an exothermic process, meaning it releases heat. Here's how this plays out:

• When the steam condenses, it transitions back to liquid water, releasing the latent heat of vaporization as part of the process.
• This heat released is then absorbed by the surrounding water, raising its temperature.

Phase transitions like these are crucial in many natural and industrial processes, such as in the water cycle or in power generation via steam turbines. Understanding the energy dynamics in these transitions is key to mastering thermodynamics concepts.