Thermodynamics: Phase Transitions & Heat Capacity

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Questions and Answers

A quantity of water is completely converted into steam by boiling. Which of the following statements are correct? (Select all that apply)

The boiling temperature would increase if the boiling were done under increased pressure (correct)

This is a second order phase transition (correct)

At the boiling point, the chemical potential in the vapor phase is less than that in the liquid phase

The entropy of the steam is greater than that of the water (correct)

The ratio $C_p / C_v$ of H2 gas is 7/5 at room temperature, and 5/3 at low temperatures (around 50 K). What is the reason for this change?

Suppression of rotational degrees of freedom (correct)

Liquefaction of the gas

Dissociation of H2 molecules

Formation of random clusters of molecules

If $v$, $v_p$, and $v_{rms}$ denote the average, most probable, and root mean square values respectively of the molecular speeds of a gas at room temperature under a Maxwellian velocity distribution, which of the following relations is true?

$v_{rms} < v_p < v$

$v < v_{rms} < v_p$

$v_p < v < v_{rms}$

$v_{rms} < v < v_p$ (correct)

For an isolated thermodynamical system, which of the following relations is true among $p$, $V$, $T$, $U$, $S$, and $F$?

${wT} imes {S} = {T}$ Signup and view all the answers

If the entropy of a system remains constant in a thermodynamic process, what type of process is it?

Isothermal Signup and view all the answers

Which of the following is not an exact differential?

dQ (Q = heat absorbed) Signup and view all the answers

An ideal gas in a cylinder is compressed adiabatically to one-third its volume, during which 20J of work is done on the gas. Which of the following statements is true?

The internal energy increases by 20J Signup and view all the answers

Mean total energy of a classical three-dimensional harmonic oscillator in equilibrium with a heat reservoir at temperature T is:

$\frac{3}{2} k_B T$ Signup and view all the answers

The root mean square speed of a particle of mass m in kinetic theory is given by:

$\frac{2 k_B T}{m}$ Signup and view all the answers

The sublimation curve of solid ammonia is given by $p = 23 - \frac{3750}{T}$ and the vaporization curve of the liquid ammonia is given by $\ln p = 19.5 - \frac{3050}{T}$. What is the temperature of the triple point of ammonia?

3050 K Signup and view all the answers

An engine absorbs heat at a temperature of 1000 K and rejects heat at 600 K. If the engine operates at maximum possible efficiency, what is the amount of work performed by the engine for 2000 J in heat input?

1200 J Signup and view all the answers

Boyle’s law can be expressed in a different form as:

$\frac{dV}{dP} = \frac{P}{V}$ Signup and view all the answers

Study Notes

Phase Transitions and Thermodynamics

Phase Transitions: The conversion of water into steam through boiling is a second-order phase transition.
Chemical Potential: During boiling, the chemical potential in the vapor phase is less than that in the liquid phase.
Boiling Point and Pressure: The boiling temperature increases if the boiling process occurs under increased pressure.
Entropy and Phase Change: The entropy of steam is greater than that of water.

Degrees of Freedom and Heat Capacity

Degrees of Freedom and Heat Capacity: The ratio of heat capacities (Cp/Cv) for H2 gas changes with temperature because of changes in the degrees of freedom of the molecules.
Low Temperatures: At low temperatures (around 50 K), rotational degrees of freedom are suppressed, leading to a Cp/Cv ratio of 5/3.
Room Temperature: At room temperature, all degrees of freedom are active, resulting in a Cp/Cv ratio of 7/5.

Molecular Speeds in Gases

Molecular Speeds: The average speed (v), most probable speed (vp), and root-mean-square speed (vrms) of gas molecules are related in the following order: vrms > v > vp.

Thermodynamic Relations

Thermodynamic Relations: For an isolated thermodynamic system, the following relation holds true: (∂F/∂T)v = -S, where F is the free energy, T is the temperature, and S is the entropy.

Thermodynamic Processes

Isothermal Process: A thermodynamic process where the entropy of the system remains constant is called an isothermal process.

Exact Differentials

Exact Differentials: dU (internal energy), dS (entropy), and dV (free energy) are exact differentials, while dQ (heat absorbed) is not.

Adiabatic Compression

Adiabatic Compression: During adiabatic compression, no heat exchange occurs with the surroundings.
Work and Internal Energy: When an ideal gas is compressed adiabatically, work is done on the gas, and its internal energy increases.
Temperature Change: In adiabatic compression, the temperature of the gas increases.

Classical Harmonic Oscillator

Mean Total Energy: The mean total energy of a classical three-dimensional harmonic oscillator in equilibrium with a heat reservoir at temperature T is 3kBT, where kB is the Boltzmann constant.

Root Mean Square Speed

Root Mean Square Speed: The root-mean-square speed of a particle of mass m in the kinetic theory is given by √(3kBT/m), where kB is the Boltzmann constant and T is the temperature.

Triple Point of Ammonia

Triple Point: The triple point of ammonia is the temperature and pressure at which the solid, liquid, and vapor phases coexist in equilibrium.
Triple Point Temperature: The triple point temperature of ammonia can be determined by finding the intersection of the sublimation curve and vaporization curve equations.

Engine Efficiency

Carnot Efficiency: The efficiency of a reversible engine cycle is determined by the temperature difference between the heat source and heat sink.
Maximum Efficiency: The maximum possible efficiency of an engine is given by the Carnot efficiency: η = 1 - (Tc/Th), where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.

Boyle's Law

Boyle's Law: Boyle's law states that the product of the pressure and volume of an ideal gas is constant at a constant temperature. It can be expressed in different forms such as dV/dP = V/P.

Boiling Point and Pressure

Clausius-Clapeyron Equation: The boiling point of a liquid is dependent on pressure. This relationship can be described using the Clausius-Clapeyron equation.
The boiling point T at pressure P can be determined by considering the latent heat of vaporization, the molar volume of the liquid phase, and the ideal gas equation for the vapor phase.

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Description

Test your understanding of thermodynamics with this quiz focused on phase transitions, chemical potential, and heat capacity ratios. Explore the behavior of molecules during changes in temperature and pressure, and challenge your knowledge about the properties of gases. Perfect for students studying chemical thermodynamics and related subjects.