To determine the coefficient of volume expansion of the liquid in a vertical U-tube with two arms maintained at different temperatures, we need to analyze how the heights of the liquid columns change with temperature. The coefficient of volume expansion (β) is a measure of how much the volume of a substance changes with temperature.

Understanding the Problem

In a U-tube, we have two arms filled with the same liquid, but each arm is at a different temperature: t₁ and t₂. The heights of the liquid columns in these arms are l₁ and l₂, respectively. The key here is to understand how the temperature difference affects the heights of the liquid columns due to thermal expansion.

Thermal Expansion Basics

The volume expansion of a liquid can be expressed as:

• ΔV = βV₀ΔT

Where:

• ΔV is the change in volume.
• V₀ is the original volume.
• ΔT is the change in temperature.

Applying to the U-tube

In our case, the liquid in each arm will expand differently due to the different temperatures. The height of the liquid in each arm can be related to the volume of the liquid. The volume of liquid in each arm can be expressed as:

• V₁ = A * l₁
• V₂ = A * l₂

Where A is the cross-sectional area of the tube. The change in volume for each arm can be expressed as:

• ΔV₁ = β * A * l₁ * (t₁ - t₀)
• ΔV₂ = β * A * l₂ * (t₂ - t₀)

Here, t₀ is a reference temperature. The difference in heights due to thermal expansion can be expressed as:

• l₁ - l₂ = ΔV₁ - ΔV₂

Finding the Coefficient of Volume Expansion

To find the coefficient of volume expansion, we can rearrange the equation:

• β = (l₁ - l₂) / (l₂ * (t₁ - l₁ * t₂))

By analyzing the options provided, we can see that option (a) matches our derived expression:

• l₁ - l₂ / (l₂ * (t₁ - l₁ * t₂))

Final Answer

The correct answer is option a: l₁ - l₂ / (l₂ * (t₁ - l₁ * t₂)). This expression effectively captures the relationship between the height differences of the liquid columns and the temperatures of the arms in the U-tube, allowing us to calculate the coefficient of volume expansion of the liquid.