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Thermal Physics

Thermal Expansion of Gases

Thermal expansion of gases governs the volumetric change of gaseous systems as temperature varies under specific thermodynamic constraints. Unlike solids and liquids, gases exhibit large coefficients of volume expansion that depend directly on whether the pressure or the volume is held constant.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Ideal gas laws, Boyle's Law, Charles's Law, Gay-Lussac's Law, coefficients of volume and pressure expansion, absolute zero temperature.

PGT extension

Real gas behavior, van der Waals equation corrections, coefficient of volume expansion for real gases, thermodynamic relations involving partial derivatives for expansivity.

Prerequisites

Concept of Temperature
Kinetic theory of gases
Law of equipartition of energy

Coefficient of Volume Expansion for Ideal Gases

The coefficient of volume expansion (gamma_v) represents the fractional change in volume per degree change in temperature at constant pressure.
For an ideal gas, gamma_v is equal to the reciprocal of the absolute temperature (1/T).
At standard temperature (273.15 K), the volume expansion coefficient is approximately 1/273 per Kelvin.

Coefficient of Pressure Expansion

The coefficient of pressure expansion (gamma_p) defines the fractional change in pressure per unit rise in temperature at constant volume.
For an ideal gas, gamma_p is also equal to 1/T, obeying Charles's and Gay-Lussac's laws.
This equality between volume and pressure expansion coefficients reflects the fundamental symmetry of ideal gas behavior.

Absolute Zero of Temperature

Extrapolating the linear volume-temperature or pressure-temperature relationship to zero volume or pressure yields absolute zero (-273.15 deg C).
At absolute zero, the kinetic energy of ideal gas molecules theoretically vanishes.
Real gases liquefy and solidify before reaching absolute zero, hence it remains an asymptotic limit.

Real Gas Deviations

Real gases deviate from ideal expansion behavior due to intermolecular forces and finite molecular volumes.
The van der Waals equation of state must be utilized to derive accurate thermal expansion coefficients for high-pressure or low-temperature regimes.
Inversion temperatures and compressibility factors dictate real gas volume responses.

Derivations

Derivation of Volume Expansion Coefficient for an Ideal Gas

Conditions: Constant pressure process, fixed mass of ideal gas

Start with the ideal gas equation: PV = nRT.
Differentiate implicitly with respect to temperature T at constant pressure P: P(dV/dT)_P = nR.
Substitute n$R = P V / T into the differentiated equation$ to get P(dV/dT)_$P = (PV)/T$.
Divide by V to isolate the coefficient of volume expansion: gamm$a_v = (1/V)(dV/dT)_P = 1/T$.

Result: gamm$a_v = 1/T$

Derivation of Pressure Expansion Coefficient for an Ideal Gas

Conditions: Constant volume process, fixed mass of ideal gas

Start with the ideal gas equation: PV = nRT.
Differentiate implicitly with respect to temperature T at constant volume V: V(dP/dT)_V = nR.
Substitute n$R = P V / T into the differentiated equation$ to get V(dP/dT)_$V = (PV)/T$.
Divide by P to isolate the coefficient of pressure expansion: gamm$a_p = (1/P)(dP/dT)_V = 1/T$.

Result: gamm$a_p = 1/T$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Coefficient of Volume Expansion	$$\gamma_v = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$$	V = Volume, T = Absolute Temperature, P = Pressure K^-1	Constant pressure
Coefficient of Pressure Expansion	$$\gamma_p = \frac{1}{P} \left(\frac{\partial P}{\partial T}\right)_V$$	P = Pressure, T = Absolute Temperature, V = Volume K^-1	Constant volume
Ideal Gas Thermal Expansion Law	$$V_T = V_0 (1 + \gamma_v \Delta T)$$	$V_T = Final volume$, $V_0 = Initial volume at 0 deg C$, Delta T = Temperature change m^3	Low pressure, moderate temperature
Charles's Law in Terms of Volume Expansion	$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$	V_1, $V_2 = Volumes at absolute temperatures T_1$, T_2 m^3, K	Constant pressure
Gay-Lussac's Law	$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$	P_1, $P_2 = Pressures at absolute temperatures T_1$, T_2 Pa, K	Constant volume
Van der Waals Equation of State	$$\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT$$	a, b = van der Waals constants, n = moles, P, V, T = pressure, volume, temperature Pa, m^3, mol, K	Real gases at moderate to high density
Volume Expansivity via Partial Derivatives	$$\gamma_v = \frac{1}{V} \frac{n R}{P - \frac{a n^2}{V^2} \left(1 - \frac{2b}{V}\right) ... }$$	Applicable for non-ideal expansions K^-1	Real gas approximation
Coefficient of Isothermal Compressibility	$$\kappa_T = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T$$	V = volume, P = pressure, T = temperature Pa^-1	Isothermal process

Graphs and experiments

Volume vs Temperature (Isobaric)

Draw: Straight lines passing through the origin when plotted on an absolute Kelvin scale, or intercepting -273.15 deg C on the Celsius scale.

Interpret: Direct proportionality between volume and absolute temperature; slope gives (nR/P).

Exam use: Used to determine absolute zero graphically by linear extrapolation.

Pressure vs Temperature (Isochoric)

Draw: Linear plot of pressure against temperature with positive slope intersecting the temperature axis at absolute zero.

Interpret: Confirms Gay-Lussac's law and uniform pressure coefficient across ideal gases.

Exam use: Tested via constant-volume gas thermometer principles.

Regnault's Apparatus Experiment

Draw: Schematic setup showing bulb, mercury manometer, and constant temperature baths.

Interpret: Measures expansion of gases by tracking pressure changes at constant volume to eliminate container expansion errors.

Exam use: Conceptual questions on experimental corrections for bulb expansion.

Solved numericals

TGT core

A given mass of an ideal gas occupies 100 mL at 27 deg C under constant pressure. Find its volume at 327 deg C.

Convert temperatures to Kelvin: $T_1 = 27 + 273 = 300 K$, $T_2 = 327 + 273 = 600 K$.
Apply Charles's law: $V_2 = V_1 * (T_2 / T_1)$.
Substitute values: $V_2 = 100 mL * (600 / 300) = 200 mL$.

Answer: 200 mL

TGT core

Calculate the coefficient of volume expansion of an ideal gas at 300 K.

Recall the formula for volume expansion coefficient of an ideal gas: gamm$a_v = 1/T$.
Substitute $T = 300 K: \gamma_v = 1 / 300 K^-1$.

Answer: 3.33 x 10^-3 K^-1

PGT extension

A real gas obeys the van der Waals equation with negligible 'b'. If 1 mole of the gas is kept at constant pressure P, express its volume expansion coefficient taking 'a' into account.

Write van der Waals equation for 1 mole with $b=0: (P + a/V^2)V = RT$.
Expand: PV + a/V = RT.
Differentiate with respect to T at constant pressure: P(dV/dT)_P - (a/$V^2)(dV/dT)_P = R$.
Solve for (dV/dT)_$P = R / (P - a/V^2)$.
Multiply by 1/V to find gamm$a_v = (1/V) * [ R / (P - a/V^2) ]$.

Answer: gamm$a_v = R / [ V(P - a/V^2) ]$

Assumptions and limitations

Gases are assumed to obey the ideal gas equation PV = nRT unless real gas corrections are explicitly applied.
Intermolecular forces and molecular volumes are neglected in basic ideal thermal expansion models.
Container expansion is assumed negligible or corrected for in experimental setups like Regnault's apparatus.

Common misconceptions

Common error: The coefficient of volume expansion for all gases is constant and independent of temperature.

Correct: For ideal gases, gamma_v depends inversely on temperature (gamm$a_v = 1/T)$.

Expansion coefficient decreases as temperature increases because it is inversely proportional to absolute temperature.

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

Correct: Gases have distinct volume and pressure expansion coefficients depending on thermodynamic constraints.

Gases are compressible fluids requiring specification of either constant pressure or constant volume.

Common error: Absolute zero can be attained by continuously cooling any gas.

Correct: Real gases liquefy and solidify well above absolute zero.

Intermolecular attractive forces cause phase transitions before absolute zero is experimentally reached.

Quick revision

The volume expansion coefficient for an ideal gas is gamm$a_v = 1/T$.
Coefficient of Volume Expansion: \gamm$a_v = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$
Coefficient of Pressure Expansion: \gamm$a_p = \frac{1}{P} \left(\frac{\partial P}{\partial T}\right)_V$

Must remember

For an ideal gas, the coefficient of volume expansion $\gamma_v$ and the coefficient of pressure expansion $\gamma_p$ are both equal to the reciprocal of absolute temperature, i.e., $1/T$.
At standard temperature ($273.15\text{ K}$), the volume expansion coefficient of an ideal gas is approximately $1/273\text{ K}^{-1}$.
Absolute zero ($-273.15^\circ\text{C}$) is obtained by extrapolating the linear volume-temperature or pressure-temperature relationship to zero volume or pressure.
Real gases deviate from ideal behavior due to intermolecular forces and finite molecular volumes, requiring the van der Waals equation for accurate modeling.
Gases possess both a coefficient of volume expansion ($\gamma_v$) and a coefficient of pressure expansion ($\gamma_p$) depending on whether pressure or volume is held constant.

Glossary

Coefficient of Volume Expansion ($\gamma_v$)The fractional change in volume per degree change in temperature at constant pressure, given by $\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$.

Coefficient of Pressure Expansion ($\gamma_p$)The fractional change in pressure per unit rise in temperature at constant volume, given by $\frac{1}{P}\left(\frac{\partial P}{\partial T}\right)_V$.

Absolute ZeroThe theoretical temperature ($-273.15^\circ\text{C}$ or $0\text{ K}$) at which the volume or pressure of an ideal gas extrapolates to zero and molecular kinetic energy vanishes.

Van der Waals EquationAn equation of state for real gases that incorporates finite molecular volume ($b$) and intermolecular attractive forces ($a$): $\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT$.

Isothermal Compressibility ($\kappa_T$)The fractional decrease in volume per unit increase in pressure at constant temperature: $-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$.

Ideal Gas Volume vs. Pressure Expansion Coefficients

Parameter	Volume Expansion ($\gamma_v$)	Pressure Expansion ($\gamma_p$)
Definition	Fractional change in volume with temperature at constant $P$	Fractional change in pressure with temperature at constant $V$
Mathematical Form	$\frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$	$\frac{1}{P} \left(\frac{\partial P}{\partial T}\right)_V$
Ideal Gas Value	$1/T$	$1/T$
Symmetry	Reflects ideal gas state equation symmetry	Reflects ideal gas state equation symmetry

Ideal Gases vs. Real Gases

Feature	Ideal Gases	Real Gases
Intermolecular Forces	Negligible or zero	Significant (attractive and repulsive forces)
Molecular Volume	Negligible (point masses)	Finite and non-negligible volume
Expansion Coefficient	Strictly $\gamma_v = 1/T$	Modified by van der Waals corrections and compressibility
Behavior at Low Temperature	Volume approaches zero at $0\text{ K}$	Liquefy or solidify before reaching absolute zero

Practice Quiz

Choose an option to reveal the explanation.

1. What is the coefficient of volume expansion $\gamma_v$ for an ideal gas at absolute temperature $T$?

Explanation: For an ideal gas, the coefficient of volume expansion is defined as $\gamma_v = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P = \frac{1}{T}$.

2. At standard temperature ($273.15\text{ K}$), what is the approximate value of the volume expansion coefficient for an ideal gas?

Explanation: Since $\gamma_v = 1/T$, substituting $T = 273.15\text{ K}$ yields approximately $1/273\text{ K}^{-1}$.

3. How does the coefficient of pressure expansion $\gamma_p$ compare to $\gamma_v$ for an ideal gas?

Explanation: For an ideal gas, both coefficients are equal to $1/T$, reflecting the fundamental symmetry of ideal gas laws (Charles's and Gay-Lussac's laws).

4. Extrapolating the linear volume-temperature relationship of gases to zero volume yields absolute zero, which is approximately:

Explanation: Extrapolating volume or pressure to zero yields absolute zero, located at $-273.15^\circ\text{C}$ (or $0\text{ K}$).

5. Why do real gases deviate from ideal thermal expansion behavior?

Explanation: Real gases experience intermolecular forces and have finite molecular volumes, causing deviations from ideal expansion described by the van der Waals equation.

6. Which equation of state is utilized to derive accurate thermal expansion coefficients for real gases?

Explanation: The van der Waals equation incorporates corrections for real gas volume and pressure, allowing accurate derivation of thermal expansion coefficients.

7. What happens to real gases when they are cooled continuously toward absolute zero?

Explanation: Real gases undergo phase transitions (liquefaction and solidification) due to intermolecular forces before reaching absolute zero.

8. Which law states that pressure is directly proportional to absolute temperature at constant volume?

Explanation: Gay-Lussac's Law governs the relationship $P/T = \text{constant}$ at constant volume, leading directly to the coefficient of pressure expansion.

9. What is the formula for the coefficient of isothermal compressibility $\kappa_T$?

Explanation: Isothermal compressibility is defined as $\kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$.

10. In the van der Waals equation $\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT$, what does parameter $b$ account for?

Explanation: Parameter $b$ represents the co-volume or finite volume occupied by the gas molecules themselves.

11. If an ideal gas undergoes thermal expansion at constant pressure such that $V_T = V_0(1 + \gamma_v \Delta T)$, what does $\Delta T$ represent?

Explanation: In the linear expansion law for gases, $\Delta T$ represents the temperature change from the initial reference temperature.

12. Why do gases have two distinct coefficients of expansion (volume and pressure) whereas solids generally have linear, area, and volume expansion?

Explanation: Gases are compressible fluids whose volume and pressure both depend on temperature, requiring distinct constraints ($P$ constant or $V$ constant).

13. What is the unit of the coefficient of volume expansion $\gamma_v$?

Explanation: Since $\gamma_v$ represents fractional change in volume per unit temperature change, its SI unit is per Kelvin ($\text{K}^{-1}$).

14. In the van der Waals equation, what correction is applied to the measured pressure $P$ to account for intermolecular attractions?

Explanation: Intermolecular attractive forces reduce inward pressure, so the correction term $\frac{an^2}{V^2}$ must be added to $P$.

15. How does the coefficient of volume expansion change as an ideal gas is heated to higher temperatures at constant pressure?

Explanation: Since $\gamma_v = 1/T$, as absolute temperature $T$ increases, the coefficient of volume expansion decreases inversely.

PYQ Patterns

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

1. For an ideal gas, the coefficient of volume expansion at temperature $T$ is given by:

Explanation: By definition, $\gamma_v = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$. For an ideal gas $V = \frac{nRT}{P}$, yielding $\gamma_v = 1/T$.

2. At standard temperature ($273.15\text{ K}$), the coefficient of volume expansion of a confined ideal gas is:

Explanation: Substituting $T = 273.15\text{ K}$ into $\gamma_v = 1/T$ gives approximately $1/273\text{ K}^{-1}$.

3. Gay-Lussac's law of pressure expansion for an ideal gas is expressed mathematically as:

Explanation: Gay-Lussac's law states that pressure is directly proportional to absolute temperature at constant volume, leading to $\frac{P_1}{T_1} = \frac{P_2}{T_2}$.

4. The theoretical temperature at which the volume of an ideal gas becomes zero according to Charles's law is:

Explanation: Extrapolating volume to zero gives absolute zero at $-273.15^\circ\text{C}$ (or $0\text{ K}$).

5. Which of the following corrections is introduced in the van der Waals equation for real gases to account for intermolecular attractive forces?

Explanation: The pressure correction term added to observed pressure $P$ is $\frac{an^2}{V^2}$ to account for attractive forces.

6. What are the dimensions or SI units of the coefficient of pressure expansion $\gamma_p$?

Explanation: Being a fractional change in pressure per unit temperature change, its unit is Kelvin inverse ($\text{K}^{-1}$).

Timed Topic Test

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

1. For a real gas obeying the van der Waals equation, how does the thermal expansion behavior differ from an ideal gas at very high pressures?

Explanation: At high pressures, the volume $V$ approaches the co-volume $nb$, restricting further compression and altering the thermal expansion response.

2. If the temperature of an ideal gas is increased from $27^\circ\text{C}$ to $327^\circ\text{C}$ at constant pressure, its volume will:

Explanation: Absolute temperatures are $T_1 = 300\text{ K}$ and $T_2 = 600\text{ K}$. Since $V \propto T$, doubling the absolute temperature doubles the volume.

3. The coefficient of volume expansion $\gamma_v$ and pressure expansion $\gamma_p$ for an ideal gas satisfy which relation?

Explanation: For an ideal gas, both $\gamma_v$ and $\gamma_p$ equal $1/T$, making them equal to each other.

4. What physical phenomenon prevents real gases from reaching absolute zero temperature?

Explanation: Real gases condense into liquids and solids due to intermolecular attractive forces long before $0\text{ K}$ is attained.

5. Using Charles's Law in terms of volume expansion, if an initial volume $V_0$ at $0^\circ\text{C}$ is heated by $\Delta T$, the final volume is $V_T = V_0(1 + \gamma_v \Delta T)$. What is $\gamma_v$ at $0^\circ\text{C}$ ($273.15\text{ K}$)?

Explanation: At $T = 273.15\text{ K}$, $\gamma_v = 1/T = 1/273.15\text{ K}^{-1}$.

6. Which thermodynamic coefficient is defined as $-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$?

Explanation: This expression is the exact definition of isothermal compressibility $\kappa_T$.

7. In the study of real gas deviations, what does the compressibility factor $Z = \frac{PV}{nRT}$ indicate when $Z > 1$?

Explanation: When $Z > 1$, repulsive intermolecular forces dominate, making the gas harder to compress than an ideal gas.

8. How does the pressure of a given mass of gas change when heated from $0^\circ\text{C}$ to $273^\circ\text{C}$ at constant volume?

Explanation: Absolute temperature increases from $273\text{ K}$ to $546\text{ K}$ (doubles), so pressure doubles according to Gay-Lussac's law.

9. Which partial derivative expression correctly defines the coefficient of volume expansion $\gamma_v$?

Explanation: Volume expansion coefficient is defined as $\gamma_v = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$.

10. Why is the study of real gas volume expansion via the van der Waals equation considered a PGT-level extension over basic UP TGT topics?

Explanation: PGT physics extends beyond ideal gas laws to include thermodynamic partial derivatives, real gas corrections, and equations of state like van der Waals.