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Thermal Physics

Thermal Expansion of Liquids

Thermal expansion of liquids is the fractional change in volume per unit change in temperature at constant pressure. Since liquids lack a definite shape, they exhibit only volume expansion characterized by real and apparent coefficients of expansion.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of real and apparent expansion of liquids, relation between coefficients (γ, γ<sub>r</sub>, γ<sub>a</sub>), expansion of vessels, and simple numerical problems on overflow of liquids.

PGT extension

Detailed treatment of Dulong and Petit's method for determining real expansion, anomalous expansion of water, temperature dependence of coefficients, and differential expansion in complex manometric systems.

Prerequisites

Concept of Temperature
Thermal expansion of solids (linear, superficial, and volume expansion)
Equation of continuity and hydrostatic pressure

Real and Apparent Expansion

When a liquid contained in a vessel is heated, both the vessel and the liquid expand simultaneously.
Real expansion (γ_r) is the actual true expansion of the liquid per unit original volume per degree rise in temperature.
Apparent expansion (γ_a) is the observed expansion of the liquid, uncorrected for the expansion of the container vessel.

Relation Between Expansion Coefficients

The fundamental relation governing liquid expansion in a solid vessel is given by γ_r = γ_a + γ_g, where γ_g is the volume expansion coefficient of the glass or container vessel.
Since γ_g = 3α_g (where α_g is the coefficient of linear expansion of the solid vessel), the relation becomes γ_r = γ_a + 3α_g.
If γ_a is positive, the liquid level rises upon heating; if it equals 3α_g, the apparent expansion is zero.

Anomalous Expansion of Water

Unlike most liquids that expand monotonically upon heating, water contracts when heated from 0°C to 4°C.
At 4°C, water possesses its maximum density of 1000 kg m^-3 (or 1 g cm^-3).
This unique behavior is crucial for aquatic life survival in cold climates, as ice floats and bottom water remains at 4°C.

Variation of Density with Temperature

As temperature increases, the volume of a given mass of liquid increases, causing its density to decrease according to the relation ρ_t = ρ₀ / (1 + γΔt).
For small temperature changes, using the binomial approximation yields ρ_t ≈ ρ₀ (1 - γΔt).
This temperature-induced density gradient is the primary driver of natural convection currents in fluids.

Derivations

Relation Between Real and Apparent Expansion Coefficients

Conditions: A liquid of initial volume V₀ is heated in a solid container of initial volume V₀ through a temperature rise ΔT.

Let the initial volume of the liquid and the container be V₀ at temperature T₁.
Upon heating to T₂ (where T₂ - T₁ = ΔT), let the true volume of the liquid become V_L and the inner volume of the container become V_C.
By definition of real expansion: V_L = V₀ (1 + γ_rΔT).
By definition of solid volume expansion: V_C = V₀ (1 + γ_gΔT), where γ_g = 3α_g.
The apparent volume of the liquid overflow or observed volume in the graduated vessel is V_a = V_L - V_C.
Substituting the expressions: V_a = V₀ (1 + γ_rΔT) - V₀ (1 + γ_gΔT) = V₀ (γ_r - γ_g)ΔT.
Since apparent volume is also defined as V_a = V₀ γ_a ΔT, equating both gives γ_a = γ_r - γ_g.

Result: gamm$a_r = \gamma_a + 3 \alpha_g$

Variation of Liquid Density with Temperature

Conditions: Mass m of the liquid remains constant while temperature changes by ΔT.

Mass of the liquid $m = V_{0}\rho_{0} at initial temperature$, where V₀ is initial volume and ρ₀ is initial density.
At final temperature T + ΔT, the volume becomes $V = V_{0} (1 + \gamma\DeltaT)$.
The final density ρ is given by $\rho = m / V$.
Substitute $m = V_{0}\rho_{0}$ and $V = V_{0} (1 + \gamma\DeltaT) into the density equation$.
Cancel V₀ from numerator and denominator: $\rho = \rho_{0} / (1 + \gamma\DeltaT)$.
Apply binomial theorem for small γΔT: $\rho = \rho_{0} (1 + \gamma\DeltaT)^{-1} \approx \rho_{0} (1 - \gamma\DeltaT)$.

Result: rh$o = \rho_0 (1 - \gamma \delta T)$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Real Coefficient of Volume Expansion	$$\gamma_r = \frac{V_2 - V_1}{V_0 \Delta T}$$	$V_0 = initial volume$, V_1, $V_2 = volumes at T_1$, T_2, \Delta $T = T_2 - T_1$ K^{-1} or ^\circ\text{C}^{-1}	Constant pressure
Apparent Coefficient of Volume Expansion	$$\gamma_a = \frac{V_{apparent}}{V_0 \Delta T}$$	$V_{apparent} = observed change in volume$, $V_0 = initial volume$ K^{-1}	Measured in an expanding vessel
Core Expansion Relation	$$\gamma_r = \gamma_a + 3 \alpha_g$$	\gamm$a_r = real expansion$, \gamm$a_a = apparent expansion$, \alph$a_g = linear expansion of vessel$ K^{-1}	Isotropic solid container
Density Temperature Dependence	$$\rho_T = \frac{\rho_0}{1 + \gamma \Delta T}$$	\rh$o_0 = initial density$, \rh$o_T = density at T$, \gamma = real volume expansion coefficient kg m^{-3}	Mass conservation
Apparent Mass Overflow	$$M_{overflow} = V_0 \rho_0 (\gamma_r - 3 \alpha_g) \Delta T$$	$M_{overflow} = mass spilled out$, $V_0 = initial volume$, \rh$o_0 = initial density$ kg	Vessel filled to brim initially
Volume at Temperature T	$$V_T = V_0 (1 + \gamma \Delta T)$$	$V_T = volume at T$, $V_0 = volume at 0^\circ\text{C}$ m^3	Moderate temperature range
Differential Height in Manometer/U-Tube	$$h_1 \rho_1 = h_2 \rho_2$$	h_1, $h_2 = liquid column heights$, \rho_1, \rh$o_2 = corresponding densities at different temperatures$ m	Hydrostatic equilibrium (e.g., Dulong and Petit apparatus)
Volume Expansion Coefficient via Linear Expansion (Solids reference)	$$\gamma = 3 \alpha$$	\gamma = volume expansion coefficient, \alpha = linear expansion coefficient K^{-1}	Valid for isotropic substances

Graphs and experiments

Volume vs Temperature Graph for Water

Draw: A curve plotting volume (or specific volume) on the y-axis against temperature on the x-axis, showing a dip/minimum at 4°C.

Interpret: The volume of a given mass of water decreases from 0°C to 4°C, reaching a minimum at 4°C, and increases for temperatures above 4°C, reflecting maximum density at 4°C.

Exam use: Tested conceptually in TGT/PGT regarding why lakes freeze from top down and anomalous behavior interpretation.

Dulong and Petit's Apparatus Principle

Draw: Two vertical balanced liquid columns connected at the base by a horizontal tube, with one column maintained at 0°C in ice bath and the other heated to t°C in a steam/oil bath.

Interpret: At hydrostatic equilibrium, the product of height and density remains constant: h₀ρ₀ = h_tρ_t. This allows precise determination of absolute liquid expansion without vessel interference.

Exam use: Direct PGT theoretical questions on experimental determination of real expansion coefficient of mercury.

Apparent vs Real Expansion Demonstration

Draw: A glass flask filled to the brim with colored liquid attached to a capillary tube, showing initial fall of liquid level followed by a continuous rapid rise upon heating.

Interpret: The initial drop occurs because the glass flask expands first before heat reaches the liquid. The subsequent rapid rise represents the combined real expansion of the liquid exceeding the vessel expansion.

Exam use: Conceptual multiple-choice questions explaining the apparent initial drop in liquid level.

Solved numericals

TGT core

A glass vessel of volume 1000 cm³ is filled completely with mercury at 0°C. What volume of mercury will overflow when the temperature is raised to 80°C? Given: coefficient of linear expansion of glass α_g = 9 × 10^-6 K^-1 and coefficient of real expansion of mercury γ_r = 1.8 × 10^-4 K^-1.

Identify given parameters: V₀ = 1000 cm³, Δ$T = 80°C$, γ_r = 1.8 × 10^-4 K^-1, α_g = 9 × 10^-6 K^-1.
Calculate volume expansion coefficient of glass vessel: γ_g = 3α_g = 3 × (9 × 10^-6) = 2.7 × 10^-5 K^-1.
Calculate apparent expansion coefficient of mercury: γ_a = γ_r - γ_g = 1.8 × 10^-4 - 0.27 × 10^-4 = 1.53 × 10^-4 K^-1.
Calculate overflow volume V_a = V₀ γ_a Δ$T = 1000 \times (1.53 \times 10^{-4}) \times 80$.
Perform arithmetic: V_a = 1000 × $0.01224 = 12.24 cm^{3}$.

Answer: 12.24 cm^3

PGT extension

The density of a liquid at 0°C is 1.06 g cm^-3 and its coefficient of real expansion is 0.00021 K^-1. Calculate its density at 100°C.

Identify given data: ρ₀ = 1.06 g cm^-3, $\gamma = 0.00021 K^{-1}$, Δ$T = 100°C - 0°C = 100 K$.
Use the density-temperature relation: ρ_T = ρ₀ / (1 + γΔT).
Substitute values into the expression: ρ₁₀₀ = 1.06 / (1 + (0.00021 × 100)).
Evaluate denominator: 1 + 0.021 = 1.021.
Perform division: ρ₁₀₀ = 1.06 / 1.021 ≈ 1.0382 g cm^-3.

Answer: 1.038 g cm^{-3}

PGT extension

A solid of density 5000 kg m^-3 is weighed in a liquid whose coefficient of real expansion is 8 × 10^-4 K^-1. If the temperature of the system is raised by 20 K, find the percentage change in the apparent weight of the solid, ignoring the expansion of the solid.

Apparent weight W' = Actual Weight (W) - Upthrust (F_u).
Upthrust F_u = V_solid × ρ_liquid × g.
When temperature increases by ΔT, the liquid density changes as ρ_T = ρ₀ / (1 + γΔT) ≈ ρ₀ (1 - γΔT).
Since solid expansion is ignored, V_solid remains constant, meaning Upthrust F_u becomes F_u' = F_u (1 - γΔT).
Change in upthrust ΔF_u = F_u × γΔT.
Since apparent weight decreases when upthrust increases, percentage change in apparent weight depends on the relative magnitude. The fractional change in upthrust is γΔ$T = 8 \times 10^{-4} \times 20 = 0.016 or 1.6%$.

Answer: 1.6% decrease in apparent weight (increase in upthrust)

Assumptions and limitations

Coefficients of expansion (γ and α) are assumed to be constant over moderate temperature ranges (ΔT < 100°C).
Container vessels are assumed to be isotropic and undergo uniform homogeneous expansion.
Pressure is assumed constant during thermal expansion experiments unless explicit compressibility effects are integrated.

Common misconceptions

Common error: Liquids have only one coefficient of thermal expansion like solids.

Correct: Liquids exhibit both real and apparent coefficients of expansion because the container holding the liquid also expands.

The observed expansion of a liquid depends entirely on the expansion properties of the vessel used to contain it.

Common error: Water expands continuously as temperature is lowered to freezing point.

Correct: Water contracts from 4°C to 0°C due to anomalous expansion, reaching maximum density at 4°C.

Hydrogen bonding and cage-like molecular structuring in cold water cause volume to increase despite cooling.

Common error: When a heated liquid-filled vessel is first placed in a heat source, the liquid level immediately rises.

Correct: The liquid level initially drops before rising.

The solid container receives thermal energy first and expands, increasing its internal volume before the heat reaches the liquid inside.

Quick revision

Real expansion of liquid equals the sum of apparent expansion and volume expansion of the container: gamm$a_r = \gamma_a + 3 \alpha_g$.
Water has maximum density and minimum volume at exactly 4°C.
Density of a liquid decreases inversely with temperature rise according to rh$o = \rho_0 / (1 + \gamma \delta T)$.

Must remember

Real expansion of a liquid is the sum of apparent expansion and volume expansion of the container: $\gamma_r = \gamma_a + 3\alpha_g$.
Water exhibits anomalous expansion between $0^\circ\text{C}$ and $4^\circ\text{C}$, achieving maximum density of $1000\text{ kg m}^{-3}$ at $4^\circ\text{C}$.
When a liquid-filled vessel is heated, the liquid level initially drops because the solid vessel expands first before the heat reaches the liquid.
Density varies with temperature according to the relation $\rho_t \approx \rho_0 (1 - \gamma\Delta t)$ for small temperature variations.
The volume expansion coefficient of an isotropic solid container is related to its linear expansion coefficient by $\gamma_g = 3\alpha_g$.

Glossary

Real Expansion ($\gamma_r$)The actual true expansion of a liquid per unit original volume per degree rise in temperature.

Apparent Expansion ($\gamma_a$)The observed expansion of a liquid contained in a vessel, uncorrected for the container's own expansion.

Anomalous ExpansionThe unusual behavior of water where it contracts upon heating from $0^\circ\text{C}$ to $4^\circ\text{C}$ instead of expanding.

Linear Expansion Coefficient ($\alpha$)The fractional change in length per degree rise in temperature for a solid material.

Real vs Apparent Expansion

Feature	Real Expansion ($\gamma_r$)	Apparent Expansion ($\gamma_a$)
Definition	Actual expansion of liquid independent of container	Observed expansion relative to the container
Dependence on Vessel	Independent of container material	Depends directly on the expansion coefficient of the vessel
Relation	$\gamma_r = \gamma_a + 3\alpha_g$	$\gamma_a = \gamma_r - 3\alpha_g$

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following relations correctly connects the real expansion coefficient of a liquid ($\gamma_r$), apparent expansion coefficient ($\gamma_a$), and the linear expansion coefficient of the container ($\alpha_g$)?

Explanation: The fundamental relation governing liquid expansion in a solid vessel is $\gamma_r = \gamma_a + \gamma_g$, where $\gamma_g = 3\alpha_g$, leading to $\gamma_r = \gamma_a + 3\alpha_g$.

2. At what temperature does water possess its maximum density?

Explanation: Water contracts when heated from $0^\circ\text{C}$ to $4^\circ\text{C}$, possessing its maximum density of $1000\text{ kg m}^{-3}$ (or $1\text{ g cm}^{-3}$) at $4^\circ\text{C}$.

3. When a liquid-filled vessel is heated, why does the liquid level initially drop before starting to rise?

Explanation: The solid container receives thermal energy directly from the heat source and expands first, increasing its internal volume before the heat is conducted to the liquid inside.

4. How does the density of a liquid generally vary with an increase in temperature?

Explanation: As temperature increases, the volume of a given mass of liquid increases, causing its density to decrease according to $\rho_t = \rho_0 / (1 + \gamma \Delta t)$.

5. If the apparent expansion coefficient of a liquid equals the volume expansion coefficient of the container ($3\alpha_g$), what will be observed when the system is heated?

Explanation: Since $\gamma_a = \gamma_r - 3\alpha_g$, if $\gamma_a = 0$ (i.e., $\gamma_r = 3\alpha_g$), the apparent expansion is zero and no change in liquid level is observed.

6. What is the approximate expression for liquid density at temperature $t$ for small temperature changes using binomial expansion?

Explanation: Using the binomial approximation on $\rho_t = \rho_0 (1 + \gamma \Delta t)^{-1}$, for small temperature changes we get $\rho_t \approx \rho_0 (1 - \gamma \Delta t)$.

7. Which coefficient of expansion is an intrinsic property of the liquid alone, independent of the container?

Explanation: Real expansion ($\gamma_r$) is the actual true expansion of the liquid per unit original volume per degree rise, completely independent of the vessel.

8. Why is the anomalous expansion of water important in nature?

Explanation: Water's maximum density at $4^\circ\text{C}$ ensures that bottom water stays at $4^\circ\text{C}$ while ice floats on top, allowing aquatic life to survive beneath frozen surfaces.

9. What is the unit of the volume expansion coefficient of a liquid?

Explanation: Expansion coefficients represent fractional change in volume per unit temperature change, giving units of $\text{K}^{-1}$ or $^\circ\text{C}^{-1}$.

10. What primary phenomenon drives natural convection currents in fluids?

Explanation: The temperature-induced density gradient (warmer fluid becoming less dense and rising) is the primary driver of natural convection currents in fluids.

11. If a glass vessel ($\alpha_g = 9 \times 10^{-6} \text{ K}^{-1}$) is filled with a liquid, what is the volume expansion coefficient of the glass container ($\gamma_g$)?

Explanation: The volume expansion coefficient for an isotropic solid is $\gamma_g = 3\alpha_g = 3 \times (9 \times 10^{-6}) = 27 \times 10^{-6} \text{ K}^{-1}$.

12. Which of the following statements about real and apparent expansion is correct?

Explanation: By definition, $\gamma_r = \gamma_a + \gamma_g$, meaning real expansion is the sum of apparent expansion and container expansion.

13. What happens to the volume of water when it is heated from $0^\circ\text{C}$ to $4^\circ\text{C}$?

Explanation: Due to anomalous expansion, water contracts when heated from $0^\circ\text{C}$ to $4^\circ\text{C}$, reaching its minimum volume (and maximum density) at $4^\circ\text{C}$.

14. In hydrostatic equilibrium for temperature-dependent liquid columns (such as in the Dulong and Petit apparatus), what relation holds between heights and densities?

Explanation: Hydrostatic pressure equilibrium requires $P = h_1 \rho_1 g = h_2 \rho_2 g$, which simplifies to $h_1 \rho_1 = h_2 \rho_2$.

15. Which misconception about liquid thermal expansion is scientifically invalid?

Explanation: It is a common misconception that liquids have only one coefficient; because containers also expand, liquids exhibit both real and apparent expansion coefficients.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A vessel of volume $V_0$ is completely filled with a liquid at $0^\circ\text{C}$. If the coefficients of real expansion of the liquid and linear expansion of the vessel are $\gamma_r$ and $\alpha_g$ respectively, the mass of liquid overflowing when heated through $\Delta T$ is:

Explanation: The overflow mass corresponds to apparent volume change $V_{apparent} = V_0(\gamma_r - 3\alpha_g)\Delta T$. Multiplying by initial density $\rho_0$ gives $M_{overflow} = V_0 \rho_0 (\gamma_r - 3\alpha_g) \Delta T$.

2. The apparent coefficient of volume expansion of a liquid in a brass vessel is $15 \times 10^{-6} \text{ K}^{-1}$. If the linear expansion coefficient of brass is $18 \times 10^{-6} \text{ K}^{-1}$, find the real coefficient of volume expansion of the liquid.

Explanation: Using $\gamma_r = \gamma_a + 3\alpha_g$, we get $\gamma_r = 15 \times 10^{-6} + 3(18 \times 10^{-6}) = 15 \times 10^{-6} + 54 \times 10^{-6} = 69 \times 10^{-6} \text{ K}^{-1}$.

3. A density bottle contains a certain mass of liquid at $0^\circ\text{C}$. When heated to $100^\circ\text{C}$, $10\%$ of the liquid's mass overflows. What is the average apparent volume expansion coefficient of the liquid?

Explanation: Overflow mass fraction is $\frac{\Delta M}{M} = \gamma_a \Delta T = 0.10$. For $\Delta T = 100^\circ\text{C}$, $\gamma_a = \frac{0.10}{100} = 10^{-3}$, accounting for exact density conservation gives approximately $1.11 \times 10^{-3} \text{ K}^{-1}$.

4. Two liquids have real expansion coefficients $\gamma_1$ and $\gamma_2$. When heated in identical vessels, their apparent expansions are equal. Which of the following must be true regarding the container's linear expansion coefficient $\alpha_g$?

Explanation: Since $\gamma_{a1} = \gamma_{a2}$, $\gamma_1 - 3\alpha_g = \gamma_2 - 3\alpha_g'$ which links the real coefficients and container expansion properties directly.

5. If the density of a liquid at $0^\circ\text{C}$ is $\rho_0$ and its volume expansion coefficient is $\gamma$, what is its density at temperature $T$ ($^\circ\text{C}$)?

Explanation: Mass conservation implies $\rho_T V_T = \rho_0 V_0$. Since $V_T = V_0(1 + \gamma T)$, we get $\rho_T = \frac{\rho_0}{1 + \gamma T}$.

6. In the Dulong and Petit method for determining real expansion of liquids, two vertical columns maintained at temperatures $T_1$ and $T_2$ have heights $h_1$ and $h_2$. The real expansion coefficient $\gamma_r$ is given by:

Explanation: Using $h_1 \rho_1 = h_2 \rho_2$ and $\rho_T = \rho_0 / (1 + \gamma_r T)$, equating heights at $0^\circ\text{C}$ and $T$ yields $h_1 = h_2(1 + \gamma_r \Delta T)$, leading to $\gamma_r = \frac{h_1 - h_2}{h_2 \Delta T}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Define the relationship between real expansion coefficient ($\gamma_r$), apparent coefficient ($\gamma_a$), and volume expansion of the container ($\gamma_g$).

Explanation: Real expansion equals the observed apparent expansion plus the expansion of the container vessel: $\gamma_r = \gamma_a + \gamma_g$.

2. A liquid is heated in a glass flask. If the apparent volume expansion is zero, what does this indicate about the liquid's real expansion coefficient?

Explanation: If apparent expansion $\gamma_a = 0$, then $\gamma_r = \gamma_a + 3\alpha_g = 3\alpha_g$.

3. What is the physical reason behind water's anomalous expansion between $0^\circ\text{C}$ and $4^\circ\text{C}$?

Explanation: Hydrogen bonding creates an open, tetrahedral cage-like structure in ice and near-freezing water that collapses upon heating up to $4^\circ\text{C}$, causing contraction.

4. If a liquid's density at $0^\circ\text{C}$ is $1.0\text{ g cm}^{-3}$ and its volume expansion coefficient is $5 \times 10^{-4}\text{ K}^{-1}$, what is its density at $50^\circ\text{C}$?

Explanation: $\rho_{50} = \frac{1.0}{1 + (5 \times 10^{-4})(50)} = \frac{1.0}{1 + 0.025} = \frac{1.0}{1.025} \approx 0.9756\text{ g cm}^{-3}$.

5. Which of the following describes the behavior of a liquid container when heat is initially applied?

Explanation: Solid containers receive direct thermal energy from the external source, expanding their internal volume before heat reaches the contained liquid.

6. What happens to the coefficient of volume expansion of most liquids as temperature increases over a moderate range?

Explanation: For most liquids, the volume expansion coefficient increases slightly as temperature rises.

7. In an experiment to find $\gamma_a$, a student forgets to account for the expansion of the measuring flask. The calculated expansion coefficient will be:

Explanation: Uncorrected observed expansion is precisely the definition of the apparent expansion coefficient.

8. If $\alpha_g$ is the linear expansion coefficient of a cubical vessel, what is its total surface area expansion coefficient?

Explanation: Area expansion coefficient (superficial expansion) is $2\alpha_g$, while volume expansion coefficient is $3\alpha_g$.

9. Why does ice float on water?

Explanation: Water expands upon freezing below $4^\circ\text{C}$, making ice less dense ($\approx 0.917\text{ g cm}^{-3}$) than liquid water at $4^\circ\text{C}$, causing it to float.

10. Which mathematical approximation is valid for calculating density at temperature $T$ when $\gamma \Delta T \ll 1$?

Explanation: Using binomial theorem expansion on $\rho_T = \rho_0(1 + \gamma \Delta T)^{-1}$, for $\gamma \Delta T \ll 1$, we get $\rho_T \approx \rho_0(1 - \gamma \Delta T)$.

Thermal Expansion of Liquids

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Real and Apparent Expansion

Relation Between Expansion Coefficients

Anomalous Expansion of Water

Variation of Density with Temperature

Derivations

Relation Between Real and Apparent Expansion Coefficients

Variation of Liquid Density with Temperature

Formula sheet

Graphs and experiments

Volume vs Temperature Graph for Water

Dulong and Petit's Apparatus Principle

Apparent vs Real Expansion Demonstration

Solved numericals

A glass vessel of volume 1000 cm3 is filled completely with mercury at 0°C. What volume of mercury will overflow when the temperature is raised to 80°C? Given: coefficient of linear expansion of glass αg = 9 × 10-6 K-1 and coefficient of real expansion of mercury γr = 1.8 × 10-4 K-1.

The density of a liquid at 0°C is 1.06 g cm-3 and its coefficient of real expansion is 0.00021 K-1. Calculate its density at 100°C.

A solid of density 5000 kg m-3 is weighed in a liquid whose coefficient of real expansion is 8 × 10-4 K-1. If the temperature of the system is raised by 20 K, find the percentage change in the apparent weight of the solid, ignoring the expansion of the solid.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Real vs Apparent Expansion

Practice Quiz

1. Which of the following relations correctly connects the real expansion coefficient of a liquid ($\gamma_r$), apparent expansion coefficient ($\gamma_a$), and the linear expansion coefficient of the container ($\alpha_g$)?

2. At what temperature does water possess its maximum density?

3. When a liquid-filled vessel is heated, why does the liquid level initially drop before starting to rise?

4. How does the density of a liquid generally vary with an increase in temperature?

5. If the apparent expansion coefficient of a liquid equals the volume expansion coefficient of the container ($3\alpha_g$), what will be observed when the system is heated?

6. What is the approximate expression for liquid density at temperature $t$ for small temperature changes using binomial expansion?

7. Which coefficient of expansion is an intrinsic property of the liquid alone, independent of the container?

8. Why is the anomalous expansion of water important in nature?

9. What is the unit of the volume expansion coefficient of a liquid?

10. What primary phenomenon drives natural convection currents in fluids?

11. If a glass vessel ($\alpha_g = 9 \times 10^{-6} \text{ K}^{-1}$) is filled with a liquid, what is the volume expansion coefficient of the glass container ($\gamma_g$)?

12. Which of the following statements about real and apparent expansion is correct?

13. What happens to the volume of water when it is heated from $0^\circ\text{C}$ to $4^\circ\text{C}$?

14. In hydrostatic equilibrium for temperature-dependent liquid columns (such as in the Dulong and Petit apparatus), what relation holds between heights and densities?

15. Which misconception about liquid thermal expansion is scientifically invalid?

PYQ Patterns

1. A vessel of volume $V_0$ is completely filled with a liquid at $0^\circ\text{C}$. If the coefficients of real expansion of the liquid and linear expansion of the vessel are $\gamma_r$ and $\alpha_g$ respectively, the mass of liquid overflowing when heated through $\Delta T$ is:

2. The apparent coefficient of volume expansion of a liquid in a brass vessel is $15 \times 10^{-6} \text{ K}^{-1}$. If the linear expansion coefficient of brass is $18 \times 10^{-6} \text{ K}^{-1}$, find the real coefficient of volume expansion of the liquid.

3. A density bottle contains a certain mass of liquid at $0^\circ\text{C}$. When heated to $100^\circ\text{C}$, $10\%$ of the liquid's mass overflows. What is the average apparent volume expansion coefficient of the liquid?

4. Two liquids have real expansion coefficients $\gamma_1$ and $\gamma_2$. When heated in identical vessels, their apparent expansions are equal. Which of the following must be true regarding the container's linear expansion coefficient $\alpha_g$?

5. If the density of a liquid at $0^\circ\text{C}$ is $\rho_0$ and its volume expansion coefficient is $\gamma$, what is its density at temperature $T$ ($^\circ\text{C}$)?

6. In the Dulong and Petit method for determining real expansion of liquids, two vertical columns maintained at temperatures $T_1$ and $T_2$ have heights $h_1$ and $h_2$. The real expansion coefficient $\gamma_r$ is given by:

Timed Topic Test

1. Define the relationship between real expansion coefficient ($\gamma_r$), apparent coefficient ($\gamma_a$), and volume expansion of the container ($\gamma_g$).

2. A liquid is heated in a glass flask. If the apparent volume expansion is zero, what does this indicate about the liquid's real expansion coefficient?

3. What is the physical reason behind water's anomalous expansion between $0^\circ\text{C}$ and $4^\circ\text{C}$?

4. If a liquid's density at $0^\circ\text{C}$ is $1.0\text{ g cm}^{-3}$ and its volume expansion coefficient is $5 \times 10^{-4}\text{ K}^{-1}$, what is its density at $50^\circ\text{C}$?

5. Which of the following describes the behavior of a liquid container when heat is initially applied?

6. What happens to the coefficient of volume expansion of most liquids as temperature increases over a moderate range?

7. In an experiment to find $\gamma_a$, a student forgets to account for the expansion of the measuring flask. The calculated expansion coefficient will be:

8. If $\alpha_g$ is the linear expansion coefficient of a cubical vessel, what is its total surface area expansion coefficient?

9. Why does ice float on water?

10. Which mathematical approximation is valid for calculating density at temperature $T$ when $\gamma \Delta T \ll 1$?

A glass vessel of volume 1000 cm³ is filled completely with mercury at 0°C. What volume of mercury will overflow when the temperature is raised to 80°C? Given: coefficient of linear expansion of glass α_g = 9 × 10^-6 K^-1 and coefficient of real expansion of mercury γ_r = 1.8 × 10^-4 K^-1.

The density of a liquid at 0°C is 1.06 g cm^-3 and its coefficient of real expansion is 0.00021 K^-1. Calculate its density at 100°C.

A solid of density 5000 kg m^-3 is weighed in a liquid whose coefficient of real expansion is 8 × 10^-4 K^-1. If the temperature of the system is raised by 20 K, find the percentage change in the apparent weight of the solid, ignoring the expansion of the solid.