Consider boiling water in an open container. For this process calculate ∆Q/∆U and ∆Q/∆W and express your answers in terms of γ.

To analyze the process of boiling water in an open container, we need to delve into the concepts of heat transfer, internal energy, and work done. In thermodynamics, the relationships between heat (Q), internal energy (U), and work (W) are crucial for understanding how systems behave. Let's break this down step by step.

Understanding the System

When boiling water in an open container, we are primarily concerned with the heat added to the system (the water) and how it affects the internal energy and work done by the system. In this scenario, the water absorbs heat from an external source, which raises its temperature until it reaches the boiling point.

Key Thermodynamic Relationships

According to the first law of thermodynamics, we have the following relationship:

• ∆U = ∆Q - ∆W

Where:

• ∆U = change in internal energy
• ∆Q = heat added to the system
• ∆W = work done by the system

Calculating ∆Q/∆U and ∆Q/∆W

In the case of boiling water, we can assume that the process occurs at constant pressure (since the container is open to the atmosphere). This means that the work done by the system is primarily due to the expansion of water vapor as it turns into steam.

Work Done by the System

The work done by the system can be expressed as:

• ∆W = P∆V

Where P is the pressure and ∆V is the change in volume. For boiling water, the change in volume is significant as water transitions to steam.

Internal Energy Change

The change in internal energy can be expressed in terms of the specific heat capacity of water and the temperature change:

• ∆U = m * c * ∆T

Where:

• m = mass of the water
• c = specific heat capacity of water
• ∆T = change in temperature

Expressing in Terms of γ

In thermodynamics, γ (gamma) is the ratio of specific heats (Cp/Cv) for an ideal gas. For water transitioning to steam, we can relate the heat added and the work done to γ. The specific heat at constant pressure (Cp) is relevant here, as it relates to the heat added during the boiling process.

Using the relationships we established, we can derive:

• ∆Q = ∆U + ∆W

Substituting the expressions for ∆U and ∆W, we can express the ratios:

• ∆Q/∆U = (∆U + ∆W)/∆U = 1 + (∆W/∆U)
• ∆Q/∆W = (∆U + ∆W)/∆W = (∆U/∆W) + 1

Final Expressions

To express these ratios in terms of γ, we would need to consider the specific heat capacities and the relationships between them. For an ideal gas, we know:

• γ = Cp/Cv

In the case of boiling water, while it’s not an ideal gas, we can still use the concept of specific heats to relate the heat added and the work done. Thus, we can derive:

• ∆Q/∆U = 1 + (P∆V)/(m * c * ∆T)
• ∆Q/∆W = (m * c * ∆T)/(P∆V) + 1

In summary, the ratios ∆Q/∆U and ∆Q/∆W can be expressed in terms of the specific heat capacities and the work done during the boiling process, allowing us to understand the thermodynamic behavior of water as it transitions from liquid to vapor.