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'The susceptibility of a paramagnetic material is \(K\) at \(27.9 \mathrm{C}\). At what temperature will its susceptibility be \(K / 2 ?\) (a) \(600^{\circ} \mathrm{C}\) (b) \(287^{\circ} \mathrm{C}\) (c) \(54^{\circ} \mathrm{C}\) (d) \(327^{\circ} \mathrm{C}\)

Short Answer

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None of the given options match the result.

Step by step solution

01

Understand the Concept

The susceptibility of a paramagnetic material can be described by Curie's law, which states that the susceptibility \( \chi \) is inversely proportional to the absolute temperature \( T \). This can be expressed as \( \chi = \frac{C}{T} \), where \( C \) is a constant. This implies that \( \chi \) is inversely proportional to the temperature in Kelvin. Hence, if the susceptibility reduces by half, the temperature should double.
02

Convert Celsius to Kelvin

The initial temperature given is 27.9°C. To work with Curie's law, we need the temperature in Kelvin. The conversion formula from Celsius to Kelvin is \( T(K) = T(°C) + 273.15 \). Thus, the initial temperature in Kelvin is \( 27.9 + 273.15 = 301.05 \text{ K} \).
03

Setup Curie's Law Equation

To find the new temperature \( T_2 \) when the susceptibility becomes half, we use the relationship \( \chi_1 T_1 = \chi_2 T_2 \) with \( \chi_2 = \frac{\chi_1}{2} \). Thus, we have the equation \( T_1 = 2 T_2 \).
04

Solve for New Temperature

Using the equation from the previous step, substitute the initial temperature: \( 301.05 = 2 T_2 \). Therefore, solve for \( T_2 \): \( T_2 = \frac{301.05}{2} = 150.525 \text{ K} \).
05

Convert Kelvin Back to Celsius

Now, convert the temperature back to Celsius using \( T(°C) = T(K) - 273.15 \). This gives: \( 150.525 - 273.15 = -122.625 \text{ °C} \). Since the choices provided are all positive, the question might need adjustments or additional interpretations are needed, such as absolute values being ignored.

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Key Concepts

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

Paramagnetism
Paramagnetism is a type of magnetism that occurs in materials that are weakly attracted to a magnetic field. It is a form of magnetism that is only present when an external magnetic field is applied. Unlike ferromagnetism, paramagnetic materials do not retain magnetization in the absence of an external magnetic field. Here's what happens inside these materials: paramagnetic substances have unpaired electrons, which have magnetic moments that tend to align with an external magnetic field. However, the thermal motion in the material generally disorients these moments, so the overall effect is weak compared to ferromagnetism.
In the context of Curie's Law, paramagnetism is significant because the strength of paramagnetic behavior decreases with increasing temperature, as the thermal agitation overcomes the magnetic alignment of the unpaired electrons.
Susceptibility
Magnetic susceptibility denotes how much a material will become magnetized in an applied magnetic field. It measures the degree of magnetization of a paramagnetic or ferromagnetic material in response to an external magnetic field.The susceptibility, represented by the Greek letter chi (\( \chi \)), is an essential factor in understanding how materials interact with magnetic fields. According to Curie's Law, susceptibility for a paramagnetic material is inversely proportional to its temperature. This can be expressed mathematically through the equation:\[ \chi = \frac{C}{T} \]where \( C \) is the Curie constant, which is specific to the material. This means that as the temperature increases, susceptibility decreases, indicating that the magnetic response of the material weakens. This inverse relationship is crucial when determining changes in susceptibility relative to temperature variations.
Temperature Conversion
Temperature conversion is a fundamental part of applying Curie's Law. The relationship between susceptibility and temperature is often given in Kelvin, as Kelvin is the standard unit of temperature in scientific calculations.To convert from Celsius to Kelvin, which is necessary when working with equations like Curie's Law, you use the formula:\[ T(\text{K}) = T(\text{°C}) + 273.15 \]This formula ensures that any calculated changes in temperature adhere to absolute standards. In our example problem, the temperature given was converted from 27.9°C to 301.05K to make accurate calculations regarding susceptibility possible. Conversely, converting back from Kelvin to Celsius uses:\[ T(\text{°C}) = T(\text{K}) - 273.15 \]This approach keeps the equations consistent when comparing temperatures or solving problems involving temperature-dependent properties like magnetism.
Magnetism
Magnetism is a broad term that covers various types of magnetic behavior in materials. It arises due to the motion of electric charges, particularly electrons with magnetic moments. In the context of paramagnetism, magnetism refers to the weak attraction that paramagnetic materials exhibit towards an external magnetic field. This occurs due to the alignment of magnetic dipoles in the material. Tackling Curie's Law involves understanding how these magnetic moments interact with thermal energy and magnetic fields. * Essential points to remember about magnetism in paramagnetic materials include: * Only exhibit magnetism when external fields are applied. * The strength of magnetic alignment is minimal compared to strong magnets like ferromagnets. * The alignment and therefore the magnetism become weaker as temperature rises, because the increased thermal agitation disturbs the alignment. This magnetic behavior is key in analyzing changes in susceptibility and how temperature influences it. Understanding magnetism in the context of temperature helps scientists and engineers design better materials for various applications, including sensors and magnetic storage devices.

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Most popular questions from this chapter

The points \(A\) and \(B\) are situated perpendicular to the axis of \(2 \mathrm{~cm}\) long bar magnet at large distances \(x\) and \(3 x\) from the centre on opposite sides. The ratio of magnetic fields at \(A\) and \(B\) will be approximately equal to (a) \(27: 1\) (b) \(1: 27\) (c) \(9: 1\) (d) \(1: 9\)

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