Problem 45 - University Physics with Modern Physics

Question

In a period of \(6.00 \mathrm{~s}, 9.00 \cdot 10^{23}\) nitrogen molecules strike a section of a wall with an area of \(2.00 \mathrm{~cm}^{2}\). If the molecules move with a speed of \(400.0 \mathrm{~m} / \mathrm{s}\) and strike the wall head on in elastic collisions, what is the pressure exerted on the wall? (The mass of one \(\mathrm{N}_{2}\) molecule is \(4.68 \cdot 10^{-26} \mathrm{~kg}\).)

Step-by-Step Solution

Answer

Answer: The pressure exerted on the wall is 2.81 x 10^4 Pa.

Step 1: Calculate the momentum change for one molecule after collision

After the elastic collision, the direction of the nitrogen molecule is reversed. The momentum change for one molecule can be calculated by subtracting the initial momentum from the final momentum. This can be expressed as: Momentum change \(= m(v_{final} - v_{initial})\) Since the molecule changes its direction, \(v_{final} = -v_{initial}\), and the momentum change becomes: Momentum change \(= 2mv_{initial}\) Substituting the given values, we have: Momentum change \(= 2(4.68\cdot 10^{-26}\mathrm{~kg})(400.0\mathrm{~m} / \mathrm{s}) = 3.744\cdot 10^{-23}\mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\)

Step 2: Calculate the momentum change for all molecules

We are given that there are \(9.00\cdot 10^{23}\) nitrogen molecules striking the wall. To find the total momentum change for all molecules, multiply the momentum change for one molecule by the number of molecules: Total momentum change \(= \text{number of molecules} \times \text{momentum change per molecule}\) Total momentum change \(= (9.00\cdot 10^{23})(3.744\cdot 10^{-23}\mathrm{~kg} \cdot \mathrm{m} / \mathrm{s})= 3.37\cdot 10^1\mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\)

Step 3: Use the impulse-momentum theorem to find the force

The impulse-momentum theorem states that the force times the time interval is equal to the momentum change: \(F\Delta t = \Delta p\) Where \(F\) is the force, \(\Delta t\) is the time interval, and \(\Delta p\) is the momentum change. We are given a time interval of \(6\mathrm{~s}\). Solving for the force, we have: \(F=(\Delta p)/(\Delta t) = (3.37\cdot 10^1\mathrm{~kg} \cdot \mathrm{m} / \mathrm{s})/(6\mathrm{~s}) = 5.62\mathrm{~N}\)

Step 4: Calculate the pressure

Now, we can find the pressure exerted on the wall by dividing the force by the given area (\(2.00\mathrm{~cm}^2\)). First, convert the area from \(\mathrm{cm}^2\) to \(\mathrm{m}^2\): \(2.00\mathrm{~cm}^{2} = 2.00\cdot 10^{-4}\mathrm{~m}^{2}\) Now, calculate the pressure: Pressure \(= \frac{F}{A} = \frac{5.62\mathrm{~N}}{2.00\cdot 10^{-4}\mathrm{~m}^{2}} = 28100\mathrm{~Pa}\) So, the pressure exerted on the wall is \(2.81\cdot 10^4\mathrm{~Pa}\).

Key Concepts

Elastic Collisions

Elastic collisions occur when two objects collide and there is no loss of kinetic energy in the system. In such a collision, the total kinetic energy before and after the event remains the same. This type of collision is fundamental in various physical interactions, such as the striking of nitrogen molecules against a wall, as described in the given problem.
When a nitrogen molecule collides elastically with the wall, its velocity changes direction, but not its speed. If initially moving with a velocity of \(v\), after the collision, its velocity becomes \(-v\). This means the momentum, which depends on velocity, also changes, specifically by an amount of \(2mv\).
Key aspects of elastic collisions include:

• Conservation of kinetic energy
• Reversal of direction for the colliding particles
• No deformation or energy transformation into forms like heat

Understanding elastic collisions helps in calculating how molecules in a system, such as gases, exert pressure on surfaces they encounter.

Impulse-Momentum Theorem

The impulse-momentum theorem states that the change in an object's momentum is equal to the impulse applied to it. Impulse is the product of force and the time interval over which the force is applied. Mathematically, it is represented as \(F\Delta t = \Delta p\), where \(\Delta p\) is the change in momentum.
In the context of the given problem, as nitrogen molecules strike the wall, each experiences an impulse resulting in a change in momentum of \(2mv\) for each molecule. By knowing the number of molecules and their individual momentum change, we calculate the total momentum change. The impulse-momentum theorem allows us to find the average force these molecules exert on the wall over the collision period. Using this force, we can then determine the pressure applied, which is crucial in studying gases and their interactions with surfaces.
Key points about the impulse-momentum theorem include:

• Relates force, time, and momentum change
• Helpful in understanding how gases exert pressure
• Connecting microscopic molecular behavior to macroscopic forces

Molecular Dynamics

Molecular dynamics is a field in physics and chemistry that focuses on the movement and interaction of molecules. This discipline provides a more detailed view of how molecules behave and interact at the atomic level, helping explain important concepts like pressure and temperature in gases.
In the problem context, molecular dynamics help us understand how nitrogen molecules striking a wall result in pressure. Knowing the speed, mass, and frequency of collision allows us to compute how these interactions culminate in measurable pressure on a wall. This kind of analysis is critical in areas such as thermodynamics and fluid dynamics.
The study of molecular dynamics involves:

• Simulation and modeling of molecular interactions and movements
• Understanding how molecular-level behavior leads to observable physical phenomena
• Application in understanding gas laws, heat flow, and energy transfer

By studying molecular dynamics, we gain insight into how substances behave under different conditions, which is crucial for fields such as material science and engineering.

Momentum

Momentum is a fundamental concept in physics, defined as the product of an object's mass and velocity, represented as \(p = mv\). It is a vector quantity, meaning it has both magnitude and direction, and it describes the motion of an object. Momentum is conserved in isolated systems, which is a cornerstone principle in mechanics.
In the example provided, each nitrogen molecule has a specific momentum based on its mass and velocity. When the molecules strike the wall, their momentum changes, leading to a measurable force over time as calculated through the impulse-momentum theorem. This change is crucial for calculating the pressure the gas exerts on the wall.
Important concepts relating to momentum include:

• Conservation of momentum in isolated systems
• The role of momentum change in force and pressure calculations
• Understanding vector properties of momentum for analyzing directions and subsequently the impact on surfaces

Mastering the concept of momentum allows us to analyze and predict the outcomes of dynamic interactions in both simple and complex systems.