a) Write an instantaneous mole balance of helium in bulb 1. Show that the mole balance leads to the
following differential equation
V dPA 1
dt =− πd2
4
DAB
L ( PA 1−PA 2)
where PA1 and PA2 are the partial pressures of helium at time t in bulbs 1 and 2, respectively.

To derive the differential equation, we need to perform a mole balance for helium in **bulb 1** based on the assumptions and conditions of the problem. Here's how the balance and derivation proceed:

### Key Assumptions:
- There are two bulbs connected by a small tube of length \( L \) and diameter \( d \).
- **Bulb 1** has volume \( V \), and **bulb 2** has the same volume.
- The system is filled with helium, and we are considering the **diffusion** of helium between the two bulbs.
- The partial pressures of helium in bulbs 1 and 2 are \( P_{A1}(t) \) and \( P_{A2}(t) \) respectively, as functions of time.
- Diffusion is governed by Fick's law, and no convection is considered.
- The rate of change of moles in **bulb 1** is related to the diffusive transport of helium through the connecting tube.

### Step 1: Define a Mole Balance in Bulb 1

The general mole balance in bulb 1 can be written as:

\[
\text{Rate of change of moles in Bulb 1} = -\text{Rate of moles of helium diffusing out of Bulb 1 into Bulb 2}
\]

The number of moles of helium in **bulb 1** is related to the partial pressure \( P_{A1}(t) \) by the ideal gas law:

\[
n_{A1}(t) = \frac{V P_{A1}(t)}{RT}
\]

So, the rate of change of moles in **bulb 1** is:

\[
\frac{dn_{A1}}{dt} = \frac{V}{RT} \frac{dP_{A1}}{dt}
\]

### Step 2: Diffusive Transport Based on Fick's Law

The rate of diffusive transport of helium from **bulb 1** to **bulb 2** through the connecting tube is described by Fick's first law of diffusion:

\[
\dot{n}_A = -\frac{\pi d^2}{4} \frac{D_{AB}}{L} \left( P_{A1} - P_{A2} \right)
\]

where:
- \( \dot{n}_A \) is the molar flux of helium (rate of moles of helium diffusing),
- \( d \) is the diameter of the tube,
- \( L \) is the length of the tube,
- \( D_{AB} \) is the diffusivity of helium in the gas mixture,
- \( P_{A1} \) and \( P_{A2} \) are the partial pressures of helium in bulbs 1 and 2 respectively.

### Step 3: Setting Up the Mole Balance

The rate of change of moles in **bulb 1** must equal the negative of the diffusive transport from bulb 1 to bulb 2:

\[
\frac{V}{RT} \frac{dP_{A1}}{dt} = -\frac{\pi d^2}{4} \frac{D_{AB}}{L} \left( P_{A1} - P_{A2} \right)
\]

### Step 4: Simplifying the Expression

Multiply both sides of the equation by \( RT \) to cancel the \( \frac{1}{RT} \) on the left side:

\[
V \frac{dP_{A1}}{dt} = -\frac{\pi d^2}{4} \frac{D_{AB}}{L} \left( P_{A1} - P_{A2} \right)
\]

This is the desired differential equation:

\[
V \frac{dP_{A1}}{dt} = -\frac{\pi d^2}{4} \frac{D_{AB}}{L} \left( P_{A1} - P_{A2} \right)
\]

### Summary

We have derived the differential equation that describes the mole balance of helium in bulb 1 based on Fick's law of diffusion. The rate of change of the partial pressure \( P_{A1}(t) \) is driven by the pressure difference between the two bulbs and depends on the diffusivity, geometry of the tube, and the volume of the bulb.

---

Would you like to see the next steps in solving this differential equation, or perhaps more details on diffusion laws? 

Here are some related questions:
1. How does Fick's law of diffusion change if temperature varies between the bulbs?
2. What assumptions were made about the behavior of the gas in this system?
3. Can you explain how the geometry of the tube affects the diffusion rate?
4. How would the differential equation change if a different gas mixture were used?
5. What role does the ideal gas law play in connecting pressure and moles?

**Tip**: When dealing with differential equations involving diffusion, carefully check the boundary conditions to ensure accurate solutions!

Write an instantaneous mole balance of helium in bulb 1. Show that the mole balance leads to the following differential equation: V dPA1/dt = − (πd^2/4) * DAB / L * (PA1 − PA2), where PA1 and PA2 are the partial pressures of helium at time t in bulbs 1 and 2, respectively.

Explore the derivation of a differential equation for the instantaneous mole balance of helium in two connected bulbs using Fick's law of diffusion. This detailed explanation covers key concepts like mole balance, diffusion laws, and the ideal gas law, suitable for undergraduate chemical engineering or physics students.

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