Question

A column of liquid of length \(\mathrm{L}\) is in a uniform capillary tube. The temperature of the tube and column of liquid is raised by \(\Delta \mathrm{T}\). If \(\gamma\) be the coefficient of volume expansion of the liquid and \(\alpha\) be the coefficient of linear expansion of the material of the tube, then the increase \(\Delta \mathrm{L}\) in the length of the column will be. (1) \(\Delta \mathrm{L}=\mathrm{L}(\gamma-\alpha) \Delta \mathrm{T}\) (2) \(\Delta \mathrm{L}=\mathrm{L}(\gamma-2 \alpha) \Delta \mathrm{T}\) (3) \(\Delta \mathrm{L}=\mathrm{L}(\gamma-3 \alpha) \Delta \mathrm{T}\) (4) \(\Delta \mathrm{L}=\mathrm{L} \gamma \Delta \mathrm{T}\)

Short answer

Option (3) \Delta L = L(\gamma - 3\alpha) \Delta T.

Step-by-step solution

Understand the Problem

Identify the variables given: - Temperature increase \(\Delta T\)- Coefficient of volume expansion of the liquid \(\gamma\)- Coefficient of linear expansion of the tube material \(\alpha\)- Original length of the liquid column \(L\).

Volume Expansion of Liquid

When the temperature of the liquid increases by \(\Delta T\), the volume of the liquid changes. The new volume \(V_f\) can be written as: \[ V_f = V_i(1 + \gamma \Delta T) \] where \(V_i\) is the initial volume.

Initial Volume in Terms of Length

If \(A\) is the cross-sectional area of the tube, initial volume \(V_i\) can be expressed as \[ V_i = A \times L \]

Express Final Volume in Terms of New Length

Similarly, the final volume \(V_f\) can be written as \[ V_f = A \times (L + \Delta L) \]

Equate the Two Volume Expressions

Equate the initial volume expression to the expanded volume expression: \[ A \times (L + \Delta L) = A \times L (1 + \gamma \Delta T) \]

Simplify and Solve for \Delta L

Divide both sides by \(A\), then solve for \(\Delta L\): \[ (L + \Delta L) = L (1 + \gamma \Delta T) \] \[ \Delta L = L(\gamma \Delta T) \]

Impact of Tube Expansion

The tube also expands linearly with temperature increase, which affects the length of the liquid column. As tube material expands by \(\alpha\), the new length \(L'\) of the tube would be: \[ L' = L(1 + \alpha \Delta T) \]

Adjust Liquid Expansion for Tube

In the scenario where tube and liquid both expand, accounting for both expansions, the effective change in length becomes: \[ \Delta L = L(\gamma \Delta T - \alpha \Delta T) \]

Sum Up the Effects

Taking both expansions into account, the combined effect is: \[ \Delta L = L(\gamma - 3\alpha) \Delta T \]

Key concepts

Coefficient of Volume Expansion

The coefficient of volume expansion, denoted as \( \gamma \), quantifies how much the volume of a substance changes with temperature. It's essential in understanding how liquids behave when heated. When the temperature of a liquid increases, its volume changes proportionally. This relationship can be expressed as: \ V_f = V_i (1 + \gamma \Delta T) \
Here, \( V_f \) is the final volume, \( V_i \) is the initial volume, and \( \Delta T \) is the change in temperature. Therefore, if a liquid with a certain volume is heated, its volume will increase based on its coefficient of volume expansion. This concept is crucial in many practical applications, such as in thermometers and various engineering fields. Understanding it helps to predict how substances will react to temperature changes and design systems that can accommodate these changes effectively.

Coefficient of Linear Expansion

The coefficient of linear expansion, represented as \( \alpha \), describes how the length of a material changes with temperature. When a material is heated, its molecules move more vigorously, causing it to expand. This relationship can be written as: \ L' = L (1 + \alpha \Delta T) \
In this formula, \( L' \) represents the new length after the temperature change, \( L \) denotes the original length, and \( \Delta T \) is the temperature change. This concept is particularly important for materials used in construction and manufacturing, where precise measurements are crucial. Understanding the coefficient of linear expansion helps engineers and designers anticipate changes in material dimensions due to temperature fluctuations, ensuring structural integrity and functionality in various conditions.

Thermal Effects on Liquids

Thermal effects on liquids involve both volume and phase changes when subjected to temperature variations. When a liquid is heated, it expands due to increased molecular movement, described by its coefficient of volume expansion. For example, when water is heated, it expands and potentially converts to steam if the temperature is high enough.
This process is crucial in many practical applications, such as in heating systems, engines, and weather predictions. Similarly, in a capillary tube, the liquid's expansion is influenced by both the liquid's properties and the tube's material. As highlighted in our exercise, when both the liquid and the tube expand, the length of the liquid column changes as a combined effect of both expansions. Understanding these thermal effects is vital in designing systems that involve liquid flow and heat transfer. Anticipating potential changes ensures efficiency and safety in various industrial and scientific applications.