The pressure of sodium vapour in a 1.0 L container is 10 torr at `1000^(@)C`. How many atoms are in the container?

Options

A. `9.7xx10^(17)`
B. `7.6xx10^(19)`
C. `4.2xx10^(17)`
D. `9.7xx10^(19)`

Explanation

To solve the problem of determining how many sodium atoms are in a 1.0 L container with a pressure of 10 torr at 1000°C, we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature is given in Celsius, and we need to convert it to Kelvin using the formula:
\[ T(K) = T(°C) + 273.15 \]
For 1000°C:
\[ T = 1000 + 273.15 = 1273.15 \, K \]

### Step 2: Convert Pressure from Torr to Atmospheres
Pressure is given in torr, and we need to convert it to atmospheres. The conversion factor is:
\[ 1 \, \text{atm} = 760 \, \text{torr} \]
Thus, the pressure in atmospheres is:
\[ P(atm) = \frac{10 \, \text{torr}}{760 \, \text{torr/atm}} \approx 0.01316 \, \text{atm} \]

### Step 3: Use the Ideal Gas Law to Calculate Moles of Sodium
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 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

Rearranging the equation to solve for \( n \):
\[ n = \frac{PV}{RT} \]

Substituting the values:
\[ n = \frac{(0.01316 \, \text{atm}) \times (1.0 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)}) \times (1273.15 \, K)} \]

Calculating:
\[ n \approx \frac{0.01316}{104.123} \approx 1.26 \times 10^{-4} \, \text{moles} \]

### Step 4: Calculate the Number of Atoms
To find the number of atoms, we use Avogadro's number, which is approximately \( 6.022 \times 10^{23} \) atoms/mol. The total number of atoms can be calculated as:
\[ \text{Number of atoms} = n \times N_A \]
Where \( N_A \) is Avogadro's number.

Substituting the values:
\[ \text{Number of atoms} = (1.26 \times 10^{-4} \, \text{moles}) \times (6.022 \times 10^{23} \, \text{atoms/mole}) \]

Calculating:
\[ \text{Number of atoms} \approx 7.58 \times 10^{19} \, \text{atoms} \]

### Final Answer
The number of sodium atoms in the container is approximately \( 7.58 \times 10^{19} \) atoms.

---