Let's break down the two scenarios you've presented regarding the relationship between the coefficients of linear expansion of a vessel and the coefficients of volume expansion of a liquid. Understanding the differences in these situations is crucial for grasping how thermal expansion works in different contexts.

Understanding the First Scenario

In the first case, we have a long cylindrical vessel filled with a liquid. When the vessel is heated, the liquid level remains unchanged. This indicates that the liquid and the vessel expand at the same rate. The key here is that the vessel's expansion is linear, while the liquid's expansion is volumetric.

Key Relationships

The relationship between the coefficients can be expressed as:

• Coefficient of linear expansion of the vessel (α)
• Coefficient of volume expansion of the liquid (β)

Since the liquid level does not change, the volume expansion of the liquid must equal the volume expansion of the vessel. For a cylindrical vessel, the volume expansion can be related to the linear expansion by the formula:

β = 3α

This is because volume expansion is three-dimensional, while linear expansion is one-dimensional. Thus, in this case, the answer is 3.

Diving into the Second Scenario

Now, let's consider the second scenario with a horizontal tube open at both ends. Here, we have a column of liquid that also remains at the same length when heated. The reasoning is slightly different because the tube is open at both ends, allowing for a different interaction between the liquid and the tube.

Analyzing the Relationships

In this case, the volume of the liquid is still affected by the linear expansion of the tube, but since the tube is open, the liquid can expand in length without being constrained by a closed volume. The relationship can be derived as follows:

• Let the initial volume of the liquid be V₀ and its length be l₀.
• When heated, the volume expands as V = V₀(1 + βt) and the length expands as l = l₀(1 + αt).

Since the length of the liquid column remains constant, we can equate the expansions:

l₀(1 + αt) = l₀(1 + 2βt)

From this, we derive that:

β = 2α

Thus, the answer here is 2. The reason we cannot interchange the concepts used in both questions is due to the constraints imposed by the vessel's shape and whether it is open or closed.

Considering Partial Volume Occupation

Your question about whether it's possible to write when the liquid does not occupy almost the full volume of the vessel is intriguing. In such cases, the derivation would indeed change, as the expansion of the liquid would not be fully constrained by the vessel's walls. The relationship would still hold, but the effective volume expansion would need to account for the actual volume of liquid present.

In summary, the different answers arise from the geometric constraints and the dimensional nature of the expansions involved. Each scenario has its own unique conditions that dictate how the expansions relate to one another. Understanding these nuances is key to mastering thermal expansion concepts.