At room temperature, ammonia gas at 1 atm pressure and hydrogen chloride gas at P atm pressure are allowed to effuse through identical pin holes from opposite ends of a glass tube of one metre length and of uniform cross section. Ammonium chloride is first formed at a distance of 60 cm from the end through which HCl gas is sent in. What is the value of P?

Explanation

To solve the problem, we will use Graham's law of effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. We will denote the rates of effusion for ammonia (NH₃) and hydrogen chloride (HCl) and set up the equation accordingly.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Pressure of ammonia (NH₃), \( P_{NH₃} = 1 \, \text{atm} \)
- Pressure of hydrogen chloride (HCl), \( P_{HCl} = P \, \text{atm} \)
- Distance where ammonium chloride forms = 60 cm from HCl end
- Total length of the tube = 100 cm

2. **Define the Rates of Effusion:**
- Let \( R_{NH₃} \) be the rate of effusion of NH₃ and \( R_{HCl} \) be the rate of effusion of HCl.
- From Graham's law, we have:
\[ \frac{R_{NH₃}}{R_{HCl}} = \frac{P_{NH₃}}{P_{HCl}} \cdot \sqrt{\frac{M_{HCl}}{M_{NH₃}}} \]
- Where \( M_{HCl} = 36.5 \, \text{g/mol} \) and \( M_{NH₃} = 17 \, \text{g/mol} \).

3. **Calculate the Distances Traveled:**
- The distance traveled by NH₃ is \( 40 \, \text{cm} \) (since it travels from the opposite end to where the reaction occurs).
- The distance traveled by HCl is \( 60 \, \text{cm} \).

4. **Set Up the Ratios:**
- The time taken for both gases to travel their respective distances is the same, so we can express the rates in terms of distance:
\[ R_{NH₃} = \frac{40}{t} \quad \text{and} \quad R_{HCl} = \frac{60}{t} \]
- Substituting these into Graham's law gives:
\[ \frac{40/t}{60/t} = \frac{1}{P} \cdot \sqrt{\frac{36.5}{17}} \]

5. **Simplify the Equation:**
- The \( t \) cancels out:
\[ \frac{40}{60} = \frac{1}{P} \cdot \sqrt{\frac{36.5}{17}} \]
- Simplifying \( \frac{40}{60} \) gives \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{1}{P} \cdot \sqrt{\frac{36.5}{17}} \]

6. **Calculate the Square Root:**
- Calculate \( \sqrt{\frac{36.5}{17}} \):
\[ \sqrt{\frac{36.5}{17}} \approx \sqrt{2.147} \approx 1.465 \]

7. **Substitute Back to Find P:**
- Rearranging gives:
\[ P = \frac{1.465 \cdot 3}{2} = \frac{4.395}{2} \approx 2.1975 \]

8. **Final Answer:**
- Thus, the value of \( P \) is approximately \( 2.198 \, \text{atm} \).