Hey guys! Today, we are diving into a fascinating physics problem involving the oscillation of an oxygen-18 atom. Specifically, we're going to figure out the force constant of the bonds that connect this atom to others. This problem is a fantastic way to understand how the microscopic world of atoms and their vibrations relate to macroscopic properties.
Decoding the Problem: Oxygen-18 and Simple Harmonic Motion
So, let's break it down. We're told that an oxygen-18 atom (¹⁸O) is oscillating in simple harmonic motion (SHM). Now, ¹⁸O is an isotope of oxygen, meaning it has the same number of protons but a different number of neutrons compared to the more common ¹⁶O. This difference in neutron count affects the atom's mass, which is crucial in determining its oscillatory behavior. Simple harmonic motion, if you remember, is that back-and-forth motion where the restoring force is directly proportional to the displacement – think of a spring being stretched or compressed. The problem also gives us the frequency of this oscillation, which is 7.0 × 10^12 per second. That's a mind-bogglingly fast vibration!
The big question we need to answer is: what's the force constant of the bonds connecting this ¹⁸O atom? The force constant, often denoted as k, is a measure of the stiffness of a bond. A higher force constant means a stiffer bond, requiring more force to stretch or compress it. To solve this, we'll need to dust off our knowledge of SHM, atomic mass, and a little bit of Avogadro's number magic. We need to relate the frequency of oscillation to the force constant and the mass of the atom. The formula that connects these is derived from the physics of SHM and is given by: ω = √(k/m), where ω is the angular frequency, k is the force constant, and m is the mass. However, we're given the regular frequency (f) in Hertz (cycles per second), not the angular frequency (ω) in radians per second. We can easily convert between them using the relationship ω = 2πf. This means we need to find the mass (m) of a single ¹⁸O atom first. This is where Avogadro's constant comes in handy. Avogadro's number (approximately 6.0 × 10^23) tells us how many atoms are in a mole of a substance. We know the molar mass of ¹⁸O is about 18 grams per mole (since it has 18 nucleons – protons and neutrons). So, we can divide the molar mass by Avogadro's number to get the mass of a single ¹⁸O atom. Once we have the mass, we can plug it into our SHM formula, along with the frequency, and solve for the force constant (k). This is where the algebra comes in, but don't worry, it's pretty straightforward. We just need to rearrange the formula and do the calculations carefully. So, the core of this problem lies in understanding the connection between the microscopic vibrations of an atom and the macroscopic property of bond stiffness, which is represented by the force constant. It's a beautiful example of how physics bridges the gap between the tiny world of atoms and the everyday world we experience.
Calculation Steps: Finding the Force Constant
Alright, let's get into the nitty-gritty and calculate the force constant. Remember, the problem gives us the frequency (f) of oscillation as 7.0 × 10^12 per second, and we're armed with Avogadro's constant (6.0 × 10^23) to help us find the mass of a single oxygen-18 atom. Here's the step-by-step process we'll follow:
- Calculate the mass of a single ¹⁸O atom:
- The molar mass of ¹⁸O is approximately 18 grams per mole. To convert this to kilograms per mole (our standard unit in physics), we divide by 1000: 18 g/mol = 0.018 kg/mol.
- Now, we use Avogadro's number to find the mass of a single atom: mass of one ¹⁸O atom = (0.018 kg/mol) / (6.0 × 10^23 atoms/mol).
- This gives us a mass of approximately 3.0 × 10^-26 kg. Remember to keep track of your units; they're super important!
- Convert the frequency to angular frequency:
- We know the frequency f = 7.0 × 10^12 Hz (cycles per second). To get the angular frequency ω (radians per second), we use the formula ω = 2πf.
- So, ω = 2π × (7.0 × 10^12 Hz) ≈ 4.4 × 10^13 rad/s. We've now converted our regular frequency to angular frequency, which is what we need for the SHM formula.
- Use the SHM formula to find the force constant:
- The formula relating angular frequency, force constant, and mass is ω = √(k/m). We want to solve for k, the force constant. First, let's square both sides of the equation: ω² = k/m.
- Now, multiply both sides by m to isolate k: k = mω².
- Plug in the values we calculated: k = (3.0 × 10^-26 kg) × (4.4 × 10^13 rad/s)².
- This gives us a force constant k of approximately 5.8 × 10^1 N/m. That's our answer!
So, we've successfully calculated the force constant of the bond connecting the ¹⁸O atom. It's around 58 N/m, which tells us how much force is required to stretch or compress the bond by a certain amount. This calculation highlights how fundamental physics principles, like simple harmonic motion, can be used to understand the behavior of atoms and molecules.
Interpreting the Result: What Does the Force Constant Tell Us?
Now that we've crunched the numbers and found the force constant, let's take a step back and think about what this result actually means. A force constant of approximately 5.8 × 10^1 N/m, or 58 N/m, gives us a quantitative measure of the bond's stiffness. But what does that really tell us about the behavior of the ¹⁸O atom and its connections to other atoms?
Imagine the bond between the ¹⁸O atom and its neighbors as a spring. The force constant k is like the spring constant – it describes how much force you need to apply to stretch or compress the spring by a certain distance. A larger k means a stiffer spring (or a stiffer bond, in our case), and a smaller k means a more flexible spring (or a more flexible bond). So, a force constant of 58 N/m suggests that the bond connecting the ¹⁸O atom is moderately stiff. It's not an incredibly rigid bond that would be very difficult to stretch or compress, but it's also not a flimsy bond that would easily deform.
This stiffness directly influences the vibrational frequency of the atom. Remember, we started with the fact that the ¹⁸O atom oscillates at a frequency of 7.0 × 10^12 Hz. This high frequency is a direct consequence of the force constant and the mass of the atom. A stiffer bond (higher k) and a lighter atom (lower m) will lead to higher vibrational frequencies. Conversely, a more flexible bond (lower k) or a heavier atom (higher m) will result in lower frequencies. In our case, the specific value of the force constant, combined with the mass of the ¹⁸O atom, dictates the observed frequency of 7.0 × 10^12 Hz. This connection between force constant and frequency is a fundamental principle in molecular vibrations and spectroscopy. The vibrational frequencies of molecules can be measured experimentally, and these measurements can be used to determine the force constants of the bonds within the molecule. This information, in turn, provides insights into the strength and nature of the chemical bonds. Moreover, the force constant is related to the potential energy of the bond. The potential energy stored in a bond that is stretched or compressed from its equilibrium position is given by the formula U = (1/2)kx², where x is the displacement from equilibrium. A higher force constant means that more energy is required to stretch or compress the bond by a given amount. This has implications for the chemical reactivity of the molecule. Stronger bonds (higher force constants) are generally less reactive than weaker bonds (lower force constants) because more energy is required to break them.
Real-World Applications: Why This Matters
Okay, so we've calculated a force constant for an oscillating oxygen-18 atom. But you might be wondering,