The vapor pressure of liquid mercury at 25.00 °C is 2.58 ✕ 10−6 atm. Calculate the number of Hg atoms per cubic centimeter in the "empty" space above the top of the column of mercury in a barometer at 25.00 °C.

To calculate the number of mercury (Hg) atoms per cubic centimeter in the "empty" space above the mercury column in a barometer at 25.00 °C, we can use the ideal gas law and the given vapor pressure.

### Step 1: Write the Ideal Gas Law
The ideal gas law is given by:

\[
PV = nRT
\]

Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles
- \( R \) = ideal gas constant \( (0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}) \)
- \( T \) = temperature (in Kelvin)

### Step 2: Convert Temperature to Kelvin
Given the temperature is 25.00 °C:

\[
T = 25.00 + 273.15 = 298.15 \, \text{K}
\]

### Step 3: Rearrange the Ideal Gas Law to Solve for \( n/V \)
Since we want the number of Hg atoms per cubic centimeter, we rearrange the ideal gas law to solve for \( n/V \) (number of moles per unit volume):

\[
\frac{n}{V} = \frac{P}{RT}
\]

### Step 4: Calculate the Moles per Cubic Centimeter
Substituting the given values:

\[
\frac{n}{V} = \frac{2.58 \times 10^{-6} \, \text{atm}}{(0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}) \times 298.15 \, \text{K}}
\]

Let's calculate this.

\[
\frac{n}{V} \approx \frac{2.58 \times 10^{-6}}{24.489} \, \text{mol/L}
\]

\[
\frac{n}{V} \approx 1.05 \times 10^{-7} \, \text{mol/L}
\]

Since 1 L = 1000 cm³, we convert to moles per cm³:

\[
\frac{n}{V} \approx 1.05 \times 10^{-10} \, \text{mol/cm}^3
\]

### Step 5: Convert Moles to Number of Atoms
To find the number of atoms, multiply by Avogadro's number \( N_A = 6.022 \times 10^{23} \, \text{atoms/mol} \):

\[
\text{Number of atoms per cm}^3 = 1.05 \times 10^{-10} \times 6.022 \times 10^{23}
\]

\[
\text{Number of atoms per cm}^3 \approx 6.32 \times 10^{13} \, \text{atoms/cm}^3
\]

### Final Answer:
The number of Hg atoms per cubic centimeter in the "empty" space above the top of the column of mercury in a barometer at 25.00 °C is approximately **\( 6.32 \times 10^{13} \, \text{atoms/cm}^3 \)**.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. How would the number of Hg atoms per cubic centimeter change if the temperature was increased?
2. What is the significance of vapor pressure in this context?
3. How does the ideal gas law apply to real gases, especially near their boiling points?
4. How would the result differ if the vapor pressure was given in a different unit?
5. What is the relationship between vapor pressure and intermolecular forces?

**Tip:** Remember, the ideal gas law is an approximation and works best under conditions of low pressure and high temperature.

Learn how to calculate the number of mercury (Hg) atoms per cubic centimeter above the mercury column in a barometer at 25.00 °C using the ideal gas law. Detailed step-by-step solution with calculations and unit conversions provided.

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