(II) The HCl molecule has a dipole moment of about 3.4 x 10^-30 Cm. The two atoms are separated by about 1.0 x 10^-10 m. (c) What maximum torque would this dipole experience in a 2.5 x 10⁴ N/C electric field? (d) How much energy would be needed to rotate one molecule 45° from its equilibrium position of lowest potential energy?

Verified step by step guidance

1. Step 1: Understand the problem. The dipole moment of the HCl molecule is given as 3.4 × 10⁻³⁰ Cm, and the separation between the atoms is 1.0 × 10⁻¹⁰ m. We are tasked with calculating the maximum torque experienced by the dipole in a uniform electric field of 2.5 × 10⁴ N/C and the energy required to rotate the dipole by 45° from its equilibrium position.
2. Step 2: Recall the formula for torque on a dipole in an electric field. The torque (τ) is given by: $\tau =p\cdot E\cdot \sin \left(\theta \right)$, where $p$ is the dipole moment, $E$ is the electric field strength, and $\theta$ is the angle between the dipole moment and the electric field. For maximum torque, $\sin \left(\theta \right)$ is equal to 1, which occurs when $\theta =90°$. Substitute the given values into the formula.
3. Step 3: Recall the formula for potential energy of a dipole in an electric field. The potential energy ($U$) is given by: $U=-p\cdot E\cdot \cos \left(\theta \right)$. To calculate the energy required to rotate the dipole by 45° from its equilibrium position, find the difference in potential energy between the initial position ($\theta =0°$) and the final position ($\theta =45°$). Substitute the given values into the formula.
4. Step 4: Perform symbolic substitution for torque. Using the formula for torque, substitute $p=3.4\times {10}^{-}$ Cm and $E=2.5\times {10}^4$ N/C into the formula for maximum torque: $\tau =p\cdot E$. This gives the symbolic expression for the maximum torque.
5. Step 5: Perform symbolic substitution for energy. Using the formula for potential energy, substitute $p=3.4\times {10}^{-}$ Cm, $E=2.5\times {10}^4$ N/C, and the angles $\theta =0°$ and $\theta =45°$ into the formula for potential energy. Calculate the difference in symbolic terms to find the energy required to rotate the dipole.

Key Concepts

Dipole Moment

The dipole moment is a vector quantity that represents the separation of positive and negative charges in a molecule. It is calculated as the product of the charge and the distance between the charges. In the case of HCl, the dipole moment indicates the molecule's polarity, which affects how it interacts with electric fields.

Torque on a Dipole in an Electric Field

When a dipole is placed in an electric field, it experiences a torque that tends to align it with the field. The torque ( au) can be calculated using the formula τ = pE sin(θ), where p is the dipole moment, E is the electric field strength, and θ is the angle between the dipole moment and the electric field direction. This concept is crucial for understanding how dipoles behave in external fields.

Potential Energy of a Dipole

The potential energy (U) of a dipole in an electric field is given by the equation U = -pE cos(θ), where p is the dipole moment, E is the electric field strength, and θ is the angle between the dipole moment and the field. This energy changes as the dipole rotates, and the work done to rotate the dipole can be calculated from the change in potential energy, which is essential for understanding energy requirements in molecular rotations.