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Chapter 3: Problem 14

Compute the heat capacity at constant magnetic field \(C_{H, n}\), the susceptibilities \(X_{T}, \mathrm{n}\) and \(X_{S, n}\) ' and the thermal expansivity \(\alpha_{H, n}\) for a magnetic system, given that the mechanical equation of state is \(M=\mathrm{n} D H / T\) and the heat capacity is \(C_{M, n}=\pi c_{\prime}\) where \(M\) is the magnetization, \(H\) is the magnetic field, \(\mathrm{n}\) is the number of moles, \(D\) is a constant, \(c\) is the molar heat capacity, and \(T\) is the temperature.

Short Answer

Expert verified
The heat capacity at constant magnetic field is ; the susceptibilities are and ; and the thermal expansivity is .

Step by step solution

01

Understand the Given Information

From the exercise, the following parameters and equations are provided: - Mechanical equation of state: - Heat capacity at constant magnetization: We will use these to find the heat capacity at constant magnetic field, susceptibilities, and thermal expansivity.
02

Set up the expression for heat capacity at constant magnetic field

Using the relation between different heat capacities, we know: a) where Using the given data to find
03

Calculate the susceptibilities

The susceptibility can be derived from the mechanical equation of state: Similarly, the isothermal susceptibility:
04

Determine the thermal expansivity

Thermal expansivity can be found from Given the mechanical equation of state , we derive:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

heat capacity in magnetic systems
When we talk about heat capacity in magnetic systems, it's essential to understand how it changes under different conditions. For our exercise, we look at the heat capacity at constant magnetic field, denoted as \(C_{H,n}\).
Given the heat capacity at constant magnetization, \(C_{M,n}=\frac{\beta}{n}\),
where \(\beta\) is some constant, we can relate these two heat capacities using thermodynamic relations.
The heat capacity at constant magnetic field considers the energy absorbed by the system when the magnetic field is kept constant. This specifically influences the internal energy, changing how the system reacts to temperature changes.
susceptibilities
Susceptibility measures how a material reacts to an external magnetic field. In our exercise, we have two types:
  • Magnetic susceptibility at constant temperature, \(\text{X}_{T,n}\)
  • Isothermal susceptibility, \(\text{X}_{S,n}\)
Using the given mechanical equation of state, \(M=\frac{nDH}{T}\), where \(M\) is magnetization, \(H\) is the magnetic field, \(n\) is the number of moles, \(D\) is a constant, and \(T\) is the temperature,
We can derive both susceptibilities based on how they modify the magnetization and field interactions.
For magnetic susceptibility at constant temperature:\( \text{X}{T,n}=nD/T\).
For isothermal susceptibility,\(\text{X}_{S,n}=nD\)
Each susceptibility sheds light on the material's responsiveness to changes in magnetic field under different thermal conditions.
thermal expansivity
Thermal expansivity, denoted \(\text{alpha}{H,n}\), describes how much a material tends to expand when heated, specifically under a constant magnetic field.
From our mechanical equation of state, \(M=\frac{nDH}{T}\), we can understand how the magnetization and temperature affect the material's expansion.
Thermal expansivity is crucial in understanding the interplay between a material's volume and temperature during thermal processes in magnetic fields.
Using our given data, we differentiate with respect to temperature to derive the thermal expansivity:
\(\text{alpha}{H,n} = \frac{\text{d}V}{V\text{d}T}\),which helps us see the proportional change in volume with temperature.
mechanical equation of state
The mechanical equation of state for our system is \(M=\frac{nDH}{T}\).
This relation connects magnetization \(M\), the number of moles\(n\), the magnetic field\(H\), a constant \(D\), and the temperature \(T\).
Understanding this equation is vital as it explains how magnetization varies under different field strengths and temperatures.
From this, we can derive other properties like susceptibilities and thermal expansivity by manipulating and differentiating with respect to the necessary variables.
By knowing the mechanical equation of state, we obtain a foundational understanding of how the physical properties of magnetic systems behave under varying thermal and magnetic conditions.

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Most popular questions from this chapter

A stochastic process, involving three fluctuating quantities, \(x_{1}, x_{2}\), and \(x_{3}\), has a probability distribution $$ P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left[-\frac{1}{2}\left(2 x_{1}^{2}+2 x_{1} x_{2}+4 x_{2}^{2}+2 x_{1} x_{3}+2 x_{2} x_{3}+2 x_{3}^{2}\right)\right] $$ where \(C\) is the normalization constant. (a) Write probability distribution in the form \(P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left(-1 / 2 x^{T} \cdot g+x\right)\), where \(g\) is a \(3 \times 3\) symmetric matrix, \(x\) is a column matrix with matrix elements \(x_{i}, i=1,2,3\), and \(x^{T}\) is its transpose. Obtain the matrix \(\boldsymbol{g}\) and its inverse \(g^{-1}\). (b) Find the eigenvalues \(\lambda_{i}(i=1,2,3)\) and orthonormal eigenvectors of \(\boldsymbol{g}\) and obtain the \(3 \times 3\) orthogonal matrix \(\boldsymbol{O}\) that diagonalizes the matrix \(\boldsymbol{g}\) (get numbers for all of them). Using this orthogonal matrix, we can write \(x^{\mathrm{T}} \cdot g \cdot x=x^{\mathrm{T}} \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}^{\mathrm{T}} \cdot \bar{\Lambda} \cdot \boldsymbol{a}=\sum_{i=1}^{3} \lambda_{i} a_{i}^{2}\) where \(\boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}}=\bar{\Lambda}\) is a \(3 \mathrm{x}\) 3 diagonal matrix with matrix elements \((\bar{A})_{i, j}=\lambda_{i} \delta_{i, j}\) and \(\boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}\) is a column matrix with elements, \(a_{i}(i=1,2,3)\). (c) Compute the normalization constant, C. (d) Compute the moments \(\left(x_{i}\right)(i=1,2,3),\left\langle x_{i} x_{j}\right\rangle(i=1,2,3, j=1,2,3)\left(x_{1}^{2} x_{2} x_{3}\right)\) and \(\left\langle x_{1} x_{2}^{2} x_{3}\right\rangle+\) (Note that Exercises \(\mathrm{A.7}\) and \(\mathrm{A} .8\) might be helpful.)

A Carnot engine uses a paramagnetic substance as its working substance. The equation of state is \(M=n D H / T\), where \(M\) is the magnetization, \(H\) is the magnetic field, \(n\) is the number of moles, \(D\) is a constant determined by the type of substance, and \(T\) is the temperature. (a) Show that the internal energy \(U\), and therefore the heat capacity \(C_{M}\), can only depend on the temperature and not the magnetization. Let us assume that \(C_{M}=C=\) constant. (b) Sketch a typical Carnot cycle in the \(M-H\) plane. (c) Compute the total heat absorbed and the total work done by the Carnot engine. (d) Compute the efficiency of the Carnot engine.

A monatomic fluid in equilibrium is contained in a large insulated box of volume \(V\). The fluid is divided (conceptually) into \(m\) cells, each of which has an average number of particles \(N_{0}\), where \(N_{0}\) is large (neglect coupling between cells). Compute the variance in fluctuations of internal energy per particle \(u=U / N,\left\langle\left(\Delta u_{i}\right)^{2}\right\rangle\), in the ith cell. (Hint: Use temperature \(T\) and volume per particle \(v=V / N\) as independent variables.)

A biological molecule of unknown mass can be prepared in pure powdered form. If 15 \(\mathrm{g}\) of this powder is added to a container with \(1 \mathrm{~L}\) of water at \(T=300 \mathrm{~K}\), which is initially at atmospheric pressure, the pressure inside the container increases to \(P=1.3 \mathrm{~atm}\). (a) What is the molecular weight of the biological molecules? (b) What is the mass of each molecule expressed in atomic units?

Electromagnetic radiation in an evacuated vessel of volume \(V\) at equilibrium with the walls at temperature \(T\) (blackbody radiation) behaves like a gas of photons having internal energy \(U=a V T^{4}\) and pressure \(P=1 / 3 a T^{4}\), where \(a\) is Stefan's constant. (a) Plot the closed curve in the \(P-V\) plane for a Carnot cycle using blackbody radiation. (b) Derive explicitly the efficiency of a Carnot engine which uses blackbody radiation as its working substance.
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