91Ó°ÊÓ

The ammonia molecule \(\left(\mathrm{NH}_{3}\right)\) has a dipole moment of \(5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .\) Ammonia molecules in the gas phase are placed in a uniform electric field \(\overrightarrow{\boldsymbol{E}}\) with magnitude \(1.6 \times 10^{6} \mathrm{N} / \mathrm{C}\) . (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to \(\overrightarrow{\boldsymbol{E}}\) from parallel to perpendicular?(b) At what absolute temperature \(\boldsymbol{T}\)is the average translational kinetic energy \(\frac{3}{2} k T\) of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

Step by step solution

01

Define Relevant Concepts and Formulas

To solve the problem, we need to understand the concept of electric potential energy of a dipole in an electric field. The potential energy of a dipole with moment \( \boldsymbol{p} \) in an electric field \( \overrightarrow{E} \) is given by \( U = - \boldsymbol{p} \cdot \overrightarrow{E} \) or \( U = - pE\cos(\theta) \), where \( \theta \) is the angle between \( \boldsymbol{p} \) and \( \overrightarrow{E} \).

02

Calculate Initial and Final Potential Energies

When the dipole is parallel to the field, \( \theta = 0 \) and \( \cos(\theta) = 1 \). So, \( U_{\text{initial}} = -pE \). When the dipole is perpendicular to the field, \( \theta = 90° \) and \( \cos(\theta) = 0 \), so \( U_{\text{final}} = 0 \).

03

Find the Change in Potential Energy

The change in potential energy, \( \Delta U \), when the dipole changes from parallel to perpendicular orientation is given by \( \Delta U = U_{\text{final}} - U_{\text{initial}} = 0 - (-pE) = pE \). Substitute the given values: \( p = 5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} \) and \( E = 1.6 \times 10^{6} \mathrm{N} / \mathrm{C} \). Thus, \( \Delta U = (5.0 \times 10^{-30})(1.6 \times 10^{6}) = 8.0 \times 10^{-24} \mathrm{J} \).

04

Set the Change in Potential Energy Equal to the Average Kinetic Energy

For part (b), we equate the change in potential energy \( \Delta U \) to the average translational kinetic energy of a molecule given by \( \frac{3}{2} k T \), where \( k = 1.38 \times 10^{-23} \mathrm{J/K} \) (Boltzmann constant). Thus, \( 8.0 \times 10^{-24} = \frac{3}{2} (1.38 \times 10^{-23}) T \).

05

Solve for Absolute Temperature

Rearranging the equation for \( T \) gives \( T = \frac{8.0 \times 10^{-24}}{\frac{3}{2} \times 1.38 \times 10^{-23}} = \frac{8.0 \times 10^{-24}}{2.07 \times 10^{-23}} \). Calculating, we find \( T \approx 0.39 \text{ K} \).

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Key Concepts

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

Dipole Moment

The dipole moment is a fundamental concept in chemistry and physics. It is a vector quantity representing the separation of positive and negative charges within a molecule. For ammonia ( H_3), this dipole moment is essential because it affects how the molecule interacts with electric fields.
A dipole moment arises when there is an uneven distribution of electrons between atoms in a molecule. In the case of ammonia, the nitrogen atom pulls shared electrons more strongly than the hydrogen atoms, creating a dipole.
Key properties of a dipole moment:

• Measured in units of coulombs meters (C·m).
• Indicates the polarity of a molecule.
• Aids in predicting how the molecule behaves in an electric field.

When placed in an electric field, the dipole moment causes the molecule to experience potential energy changes due to the alignment or misalignment with the external field.

Electric Field

An electric field is a region around a charged particle or object within which a force would be experienced by other charges. In this exercise, a uniform electric field with a magnitude of \(1.6 \times 10^{6} \text{ N/C}\) is applied to the ammonia molecule.
The force exerted by the electric field on a dipole can cause it to align with the field direction, minimizing its potential energy. The formula for the potential energy \(U\) of a dipole in an electric field is given by \(U = -pE \cos(\theta)\), where:

• \(p\) is the dipole moment.
• \(E\) is the electric field strength.
• \(\theta\) is the angle between the dipole moment and the field.

These interactions are critical because they determine how energy changes as the molecule rotates through different orientations with the field.

Translational Kinetic Energy

Translational kinetic energy is a form of energy associated with the motion of molecules as they move from one place to another. It's an important quantity when examining the temperature of gases and their behavior.
The average translational kinetic energy of a molecule is calculated using the formula \(\frac{3}{2} k T\), where:

• \(k\) is the Boltzmann constant \(1.38 \times 10^{-23} \text{ J/K}\).
• \(T\) is the absolute temperature in Kelvins.

This formula shows that as temperature increases, molecular motion increases, leading to greater translational kinetic energy. In this exercise, we equate this kinetic energy to the potential energy of an oriented dipole, helping us find at what temperature the thermal agitation of molecules overcomes their alignment with the electric field.

Boltzmann Constant

The Boltzmann constant \(k\) bridges the macroscopic scale of temperature with the microscopic scale of kinetic energy for particles in thermal motion. It's a fundamental constant in physics and is crucial for understanding phenomena in thermodynamics and statistical mechanics.
\(k\) plays a key role in calculating the average kinetic energy of molecules in a gas, expressed in the equation for translational kinetic energy \(\frac{3}{2} k T\).
Important points about the Boltzmann constant:

• Its value is \(1.38 \times 10^{-23} \text{ J/K}\).
• Links temperature (a macroscopic variable) to energy at the molecular level.
• Central to the distribution and behavior of particles at given temperatures in statistical mechanics.

In this type of exercise, using the Boltzmann constant allows us to relate the energy changes within the dipole to broader thermal behaviors and temperature influences.