How did they arrive at this equation?

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In summary, the conversation discusses the relationship between n(x)F(x)=d[p(x)]/dx and a collection of gases, where n(x) is the number density, F(x) is the force along the x direction, and p(x) is the pressure of the gas along the x direction. The conversation also brings up the question of whether F(x)=-d[U(x)]/dx is related, with the clarification that U(x) is unknown. The solution involves considering a small rectangular "slab" of gas and calculating the total force based on the pressure and number of particles within the slab. The final solution takes into account the density of the gas to complete the calculation.
  • #1
pivoxa15
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Homework Statement


n(x)F(x)=d[p(x)]/dx in relation to a collection of gases


n(x) is the number density
F(x) is the force along the x direction
p(x) is the pressure of the gas along the x direction



The Attempt at a Solution


Is F(x)=-d[U(x)]/dx related?
 
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  • #2
what is U(x) ?
 
  • #3
pivoxa15 said:

Homework Statement


n(x)F(x)=d[p(x)]/dx in relation to a collection of gases


n(x) is the number density
F(x) is the force along the x direction
p(x) is the pressure of the gas along the x direction



The Attempt at a Solution


Is F(x)=-d[U(x)]/dx related?

Here's what comes mind (without having thought about it too deeply):

Consider a small rectangular "slab" of gas of thickness dx and surface area A. Consider the net pressure on the slab from the inside toward the outside (say). The slab contains N particles. Then the force on the left side will be minus the pressure there times the surface area, whereas the pressure on the right side will be the pressure there times the surface area (I neglect the pressure on the other sides of the slab which is very small).

We get

Total force = P(x+dx) *A - P(x) * A [itex] \approx dV \frac{P(x+dx) - P(x)}{dx} \approx dV \frac{d P(x) }{dx} [/itex].

I think it should be easy now to complete this by taking into account a certain number of particles with a density n(x).


Hope this helps

Patrick
 

1. How did they arrive at this equation?

The process of arriving at an equation involves a combination of mathematical reasoning, experimentation, and theoretical models. Scientists use existing knowledge and data to make predictions and test hypotheses, and then refine and revise their equations based on the results of these experiments.

2. What factors are considered when creating an equation?

When creating an equation, scientists consider a variety of factors such as the laws of physics, existing theories and models, and experimental data. They also take into account any relevant constants or variables that may affect the outcome of the equation.

3. Can equations change over time?

Yes, equations can change over time as new data and evidence is collected. Scientists are constantly refining and improving their equations to better describe and predict the natural world.

4. How do scientists validate their equations?

Scientists validate their equations by testing them against real-world data and comparing the results to their predictions. If the equation accurately reflects the observed data, it is considered valid. Additionally, equations are often peer-reviewed and replicated by other scientists to ensure their accuracy.

5. Are all equations based on mathematical calculations?

No, not all equations are based on mathematical calculations. Some equations are derived from theoretical models or principles, while others are based on empirical data and observations. However, all equations must be supported by evidence and have a logical and consistent structure.

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