An expensive vacuum system can achieve a pressure as low as 1.11 x 10-7 N/m2 at 17.4ºC. How many atoms are there in a cubic centimeter at this pressure and temperature?



To determine the number of atoms in a cubic centimeter at a given pressure and temperature, we can use the **ideal gas law**:

\[
PV = nRT
\]

where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the ideal gas constant, and
- \( T \) is the temperature in Kelvin.

The number of atoms, \( N \), is related to the number of moles by:

\[
N = n \times N_A
\]

where \( N_A \) is Avogadro's number (\(6.022 \times 10^{23} \, \text{atoms/mol}\)).

### Step 1: Convert Temperature to Kelvin
The temperature \( T \) is given as \( 17.4^\circ \text{C} \). Convert this to Kelvin:

\[
T = 17.4 + 273.15 = 290.55 \, \text{K}
\]

### Step 2: Solve for \( n \)
Rearrange the ideal gas law to solve for \( n \):

\[
n = \frac{PV}{RT}
\]

#### Given values:
- \( P = 1.11 \times 10^{-7} \, \text{N/m}^2 \)
- \( V = 1 \, \text{cm}^3 = 1 \times 10^{-6} \, \text{m}^3 \) (since \(1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3\))
- \( R = 8.314 \, \text{J/(mol} \cdot \text{K)} \)
- \( T = 290.55 \, \text{K} \)

Substitute these values:

\[
n = \frac{(1.11 \times 10^{-7}) \times (1 \times 10^{-6})}{8.314 \times 290.55}
\]

Calculating this, we get:

\[
n \approx 4.6 \times 10^{-18} \, \text{moles}
\]

### Step 3: Calculate the Number of Atoms
Now, use Avogadro's number to find the number of atoms:

\[
N = n \times N_A = (4.6 \times 10^{-18}) \times (6.022 \times 10^{23})
\]

\[
N \approx 2.77 \times 10^{6} \, \text{atoms}
\]

### Final Answer
There are approximately \( 2.77 \times 10^6 \) atoms in a cubic centimeter at this pressure and temperature.

Would you like further details or have any questions?

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Here are some related questions:
1. How would the number of atoms change if the temperature increased?
2. What happens to the number of atoms if the pressure is reduced further?
3. How does this calculation change if the gas is not ideal?
4. Can we use the same method if the volume is in liters instead of cubic centimeters?
5. How does the ideal gas constant vary in different unit systems?

**Tip:** When dealing with extremely low pressures, assumptions of ideal gas behavior are generally more accurate, but real gas deviations might still be worth considering at very high or low temperatures.

An expensive vacuum system can achieve a pressure as low as 1.11 x 10^-7 N/m^2 at 17.4ºC. How many atoms are there in a cubic centimeter at this pressure and temperature?

Learn to calculate the number of atoms in a cubic centimeter of gas at extremely low pressure (1.11 x 10^-7 N/m^2) and 17.4ºC using the ideal gas law. This solution covers the ideal gas law, temperature conversion, and the relationship between moles and Avogadro's number. Suitable for students in Grades 10-12.

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