A carbon monoxide molecule can be modeled as a carbon atom and an oxygen atom connected by a spring. If a displacement of the carbon by \(1.6 \times 10^{-12} \mathrm{m}\) from its equilibrium position relative to the oxygen increases the molecule's potential energy by \(0.015 \mathrm{eV},\) what's the spring constant?

In physics, potential energy is the stored energy of an object because of its position or configuration. One of the common examples of potential energy in action involves springs. The potential energy in a spring is a result of its elasticity and is calculated when the spring is either compressed or stretched from its original equilibrium position.

The formula for potential energy (\( PE \) in springs is given by \( PE = 0.5 \times k \times x^2 \), where \( k \) is the spring constant and \( x \) is the displacement of the spring from its equilibrium position. This type of potential energy is also known as elastic potential energy because it comes from the deformation of an elastic object, like the stretching of a carbon monoxide molecule as described in the exercise.

Understanding the conversion between potential energy stored in springs and other forms of energy can be important for practical applications, like the design of molecular machinery or the calibration of sensitive instruments.

Hooke's Law is a principle of physics that relates the force exerted by a spring to the distance it is stretched or compressed. This fundamental law is mathematically expressed as \( F = -kx \), where \( F \) represents the force applied to the spring, \( k \) is the spring constant, which is a measure of the spring's stiffness, and \( x \) is the displacement of the spring from its rest position.

Spring Constant and Hooke's Law

The spring constant, \( k \) in Hooke's Law, is a unique value for each spring, indicating how much force is needed to extend or compress the spring by a unit distance. A higher spring constant means that the spring is more resistant to deformation. By rearranging Hooke's Law to \( k = -F/x \), we see the direct relationship the spring constant has with force and displacement - the two critical quantities you need to determine the stiffness of a spring.

In the context of the problem given, Hooke's Law helps us understand how the molecular 'spring' connecting the carbon and oxygen atoms resists the displacement and stores potential energy as a result.

The electron volt (\( eV \) is a unit of energy commonly used in the field of particle physics and chemistry, particularly when dealing with atomic and molecular scales. One electron volt is defined as the amount of kinetic energy gained or lost by a single electron when it moves across an electric potential difference of one volt.

To convert electron volts to joules, which is the standard International System of Units (SI) for energy, we use the conversion factor \( 1 eV = 1.6 \times 10^{-19} J \). This enables us to translate our understanding of energy from the microscopic scales where electrons and atoms interact, up to more intuitive scales that we can relate to in everyday experiences involving joules.

By applying this conversion to the exercise, we can understand the molecular potential energy variation in terms more suitable for broader physics applications. For example, converting the \( 0.015 eV \) change in potential energy to joules allows us to use classical formulas, such as the one for a spring's potential energy, to determine the spring constant for the molecule.