To solve the problem step by step, we will analyze the situation and apply the concepts of gas laws and equilibrium.
### Step 1: Understand the initial conditions
We have a container with air saturated with water vapor. The total pressure of the system is given as 1230 torr. This total pressure is the sum of the partial pressures of the air and the water vapor.
### Step 2: Identify the pressure of water vapor
The vapor pressure of water (P_water) is given as 30 torr.
### Step 3: Calculate the pressure of air
Using Dalton's Law of Partial Pressures:
\[ P_{total} = P_{air} + P_{water} \]
Substituting the known values:
\[ 1230 \, \text{torr} = P_{air} + 30 \, \text{torr} \]
Now, we can solve for \( P_{air} \):
\[ P_{air} = 1230 \, \text{torr} - 30 \, \text{torr} = 1200 \, \text{torr} \]
### Step 4: Analyze the change in volume
When the volume of the container is doubled, we need to find the new pressure of the air. According to Boyle's Law, pressure is inversely proportional to volume when temperature and the amount of gas are constant:
\[ P_1 V_1 = P_2 V_2 \]
If the volume is doubled (i.e., \( V_2 = 2V_1 \)), then:
\[ P_2 = \frac{P_1}{2} \]
Substituting the initial pressure of air:
\[ P_{air, new} = \frac{1200 \, \text{torr}}{2} = 600 \, \text{torr} \]
### Step 5: Calculate the total pressure at equilibrium
At equilibrium, the total pressure in the container will be the sum of the new pressure of the air and the constant pressure of the water vapor:
\[ P_{total, new} = P_{air, new} + P_{water} \]
Substituting the values:
\[ P_{total, new} = 600 \, \text{torr} + 30 \, \text{torr} = 630 \, \text{torr} \]
Thus, \( x = 630 \, \text{torr} \).
### Step 6: Find the value of \( \frac{x}{126} \)
Now, we need to calculate:
\[ \frac{x}{126} = \frac{630}{126} = 5 \]
### Final Answer:
The value of \( \frac{x}{126} \) is \( 5 \).
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