Thermodynamics: Internal Energy of Ideal Gas

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Questions and Answers

What does the Stefan-Boltzmann law state about the relationship between energy radiated and temperature?

Energy radiated is directly proportional to the fourth power of temperature. (correct)

Energy radiated is inversely proportional to the fourth power of temperature.

Energy radiated is directly proportional to the square root of temperature.

Energy radiated does not depend on temperature.

Which of the following represents the correct SI unit for the irradiance j*?

Joules per square meter per Kelvin

Watts per cubic meter

Watt per square meter (correct)

Joules per second per cubic meter

What is the value of the Stefan-Boltzmann constant σ in SI units?

5.67 × 10^-9 Js^-1m^-2K^-4

5.67 × 10^-8 Js^-1m^-2K^-4 (correct)

5.67 × 10^-7 Js^-1m^-2K^-5

5.67 × 10^-10 Js^-1m^-2K^-3

What dimension does the irradiance j* have?

Power density Signup and view all the answers

In the context of the Stefan-Boltzmann law, what does the variable ε represent?

The emissivity of the blackbody. Signup and view all the answers

What is the relationship between the molar heat capacities at constant pressure and constant volume for an ideal gas?

CP = CV + R Signup and view all the answers

In an isothermal process, what remains constant?

Temperature Signup and view all the answers

During an adiabatic process, what is true about the heat transfer?

No heat transfer occurs. Signup and view all the answers

How does the internal energy change in a cyclic process?

It remains constant. Signup and view all the answers

Which statement is true about an isochoric process?

Volume remains constant, and no work is done. Signup and view all the answers

In the first law of thermodynamics, which equation correctly relates heat added, internal energy change, and work done?

dQ = dU + dW Signup and view all the answers

What happens in an adiabatic expansion of an ideal gas?

Temperature decreases. Signup and view all the answers

Which equation represents the relationship of work done when a gas expands at constant pressure?

dW = PdV Signup and view all the answers

What is the first Maxwell relation derived from the energy equation dU?

$\frac{\partial P}{\partial S} = -\frac{\partial T}{\partial V}$ Signup and view all the answers

Which expression correctly represents the second Maxwell relation derived from the enthalpy equation dH?

$\frac{\partial V}{\partial S} = \frac{\partial T}{\partial P}$ Signup and view all the answers

From the equation dF = -SdT - PdV, what is the relationship denoted by the third Maxwell relation?

$\frac{\partial P}{\partial T} = -\frac{\partial S}{\partial V}$ Signup and view all the answers

What does the last Maxwell relation, derived from dG = VdP - SdT, express?

$\frac{\partial S}{\partial T} = -\frac{\partial V}{\partial P}$ Signup and view all the answers

In the context of the Maxwell relations, what is the implication of the relation $\frac{\partial C_P}{\partial P} = -T \frac{\partial^2 V}{\partial T^2}$?

Heat capacity changes with temperature and pressure changes Signup and view all the answers

What key characteristic connects the partial derivatives in Maxwell relations?

Each derivative represents a state function Signup and view all the answers

What can be inferred about the derivatives in the expression of entropy, S, from the equation dF?

Entropy is negatively correlated with temperature changes. Signup and view all the answers

Which of the following statements about the Maxwell relations is correct?

They highlight the interdependence of thermodynamic variables. Signup and view all the answers

In thermodynamics, which equation corresponds to the calculation of heat capacities using Maxwell relations?

$C_P = T \frac{\partial S}{\partial T}$ Signup and view all the answers

What does the term 'state function' signify in the context of Maxwell relations?

Properties that are determined by the state of a system. Signup and view all the answers

What happens to the momentum of a gas molecule after it strikes the wall of the box?

It reverses direction with the same magnitude. Signup and view all the answers

How is the change in momentum of a molecule determined?

By subtracting its final momentum from its initial momentum. Signup and view all the answers

What does the temperature of the gas signify in terms of molecular motion?

It is proportional to the average kinetic energy of the molecules. Signup and view all the answers

What is the formula for calculating pressure based on the momentum change of a molecule?

Pressure = mv²x / (Area * L) Signup and view all the answers

Which of the following correctly describes the relationship between pressure and molecular impacts?

Pressure is proportional to the square of the average velocity of molecules. Signup and view all the answers

How is the time taken for a molecule to travel across the box determined?

It is calculated as distance travelled divided by the velocity. Signup and view all the answers

What defines the total pressure exerted by gas molecules on the walls of the container?

The total momentum of all molecules as they collide with the walls. Signup and view all the answers

In terms of molecular dynamics, what can be inferred about potential energy in an ideal gas?

It is negligible and not associated with any molecule. Signup and view all the answers

What is the condition for identifying a critical point in the given model?

$rac{ rac{ igg( rac{ igg( rac{ igg( rac{ igg( }{ rac{ igg( rac{ }{V} igg( P igg( {T} igg(}{ } igg( }{ = 0$ Signup and view all the answers

What characterizes a first order phase change?

Change in specific volume and accompanied by latent heat Signup and view all the answers

In the context of viral expansion, what does the term 'Vm' represent?

Molar volume Signup and view all the answers

What does a second order phase change indicate?

It occurs without accompanying latent heat or change in volume Signup and view all the answers

What is the Gibbs free energy in a phase transition?

Is minimized at thermodynamic equilibrium Signup and view all the answers

How does temperature influence phase changes?

It can induce phase changes between solid and liquid states Signup and view all the answers

What does the Clausius-Clapeyron equation primarily relate to?

Pressure and temperature during first order phase changes Signup and view all the answers

Which of the following is an example of a second order phase change?

Ferro-magnet to paramagnet transition Signup and view all the answers

Study Notes

Internal Energy of an Ideal Gas

The change in internal energy (dU) of an ideal gas during any process involving a temperature change (dT) is given by dU = CV dT, where CV is the molar heat capacity at constant volume.
When an ideal gas is heated at constant pressure, it expands against a movable piston and does work (dW = PdV), where P is the pressure and dV is the change in volume.
The heat added at constant pressure (dQ) is given by dQ = CP dT, where CP is the molar heat capacity at constant pressure.
For an ideal gas expanding against a constant pressure, PdV = RdT, where R is the ideal gas constant.
Applying the first law of thermodynamics (dU = dQ - PdV) to both an isochoric (constant volume) and an isobaric (constant pressure) process leads to the conclusion that CP = CV + R.

Thermodynamic Processes

A thermodynamic cycle is a process where a system returns to its initial state, with a net change in internal energy of zero.
During a cycle, the heat absorbed (dQ) equals the work done (dW): dQ = dW = PdV.
An adiabatic process is one where the heat transfer is zero (dQ = 0). In this case, the change in internal energy equals the negative work done: dU = -dW = -PdV.
An isothermal process is one where the temperature remains constant (dT = 0), and the heat absorbed equals the work done: dQ = dW = PdV.
An isochoric process is one where the volume remains constant (dV = 0). Here, the heat absorbed equals the change in internal energy: dQ = dU.
An isobaric process is one where the pressure remains constant (dP = 0). In this case, the heat absorbed equals the change in internal energy plus the work done: dQ = dU + PdV.

Adiabatic Expansion of an Ideal Gas

An adiabatic process is both adiathermal (no heat flow) and reversible.
The average kinetic energy of gas molecules is proportional to the temperature.
The force exerted by a gas molecule due to its collision with a wall of a container is proportional to the square of its velocity in the direction perpendicular to the wall.
The pressure exerted by the gas is directly proportional to the average kinetic energy of the molecules.

Maxwell Relations

Maxwell relations are derived from the fundamental thermodynamic equations for internal energy (U), enthalpy (H), Helmholtz free energy (F), and Gibbs free energy (G).
The four Maxwell relations are:
-  P   T       S V  V S
-  V   T       S P  P S
-  P   S       T V  V T
-  V   S        T P  P T
These relations provide relationships between measurable and unmeasurable thermodynamic quantities.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature.
The law is expressed as J* = eσT^4, where J* is the blackbody irradiance, e is the emissivity, σ is the Stefan-Boltzmann constant, and T is the absolute temperature.
The Stefan-Boltzmann constant is derived from fundamental constants like Boltzmann's constant (kB), Planck's constant (h), and the speed of light (c).

Viral Expansion

Viral expansion is a method to model real gases by modifying the ideal gas equation with a power series in 1/Vm (where Vm is the molar volume).
The viral expansion is written as: PVm = RT (1 + B/Vm + C/Vm^2 + ...), where B, C, etc. are called viral coefficients.

Phase Changes

A phase change occurs when a system transitions between distinct states.
Phase changes can be caused by factors like temperature, pressure, or applied magnetic fields.
A first-order phase change is characterized by a change in specific volume and the release or absorption of latent heat.
Examples of first-order phase changes include melting, boiling, and the transition between superconductor and normal conductor in an applied magnetic field.
A second-order phase change is characterized by no change in specific volume or latent heat.
Examples of second-order phase changes include the transition from ferromagnet to paramagnet at the Curie temperature, the transition between superconductor and normal conductor in a zero magnetic field, and the change from normal liquid helium to superfluid helium.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the relationship between the pressure and temperature at which two phases of a substance coexist in equilibrium.
At thermodynamic equilibrium, the Gibbs free energy is minimized.
Along the transition line on a PT diagram, the specific Gibbs energy is the same for both phases.

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Description

Explore the properties and processes of thermodynamics related to ideal gases, focusing on the internal energy changes and heat capacities in isochoric and isobaric processes. This quiz will challenge your understanding of the first law of thermodynamics and thermodynamic cycles.