An expensive vacuum system can achieve a pressure as low as 1.00×10 – 7 N/m2 at 20ºC . How many atoms are there in a cubic centimeter at this pressure and temperature?
The Correct Answer and Explanation is :
To calculate the number of atoms in a cubic centimeter at a given pressure and temperature, we use the ideal gas law:
[
PV = nRT
]
Where:
- ( P ) is the pressure ((1.00 \times 10^{-7} \, \text{N/m}^2)),
- ( V ) is the volume ((1 \, \text{cm}^3 = 1 \times 10^{-6} \, \text{m}^3)),
- ( n ) is the number of moles,
- ( R ) is the universal gas constant ((8.314 \, \text{J/mol·K})),
- ( T ) is the temperature ((20^\circ \text{C} = 293.15 \, \text{K})).
Step 1: Solve for ( n ) (moles of gas)
Rearranging the ideal gas law:
[
n = \frac{PV}{RT}
]
Substitute the given values:
[
n = \frac{(1.00 \times 10^{-7})(1 \times 10^{-6})}{(8.314)(293.15)}
]
Perform the calculation:
[
n = \frac{1.00 \times 10^{-13}}{2436.93} \approx 4.1 \times 10^{-17} \, \text{moles}.
]
Step 2: Convert moles to atoms
The number of atoms is found using Avogadro’s number ((6.022 \times 10^{23} \, \text{atoms/mol})):
[
\text{Number of atoms} = n \times N_A = (4.1 \times 10^{-17})(6.022 \times 10^{23}).
]
Perform the calculation:
[
\text{Number of atoms} \approx 2.47 \times 10^7 \, \text{atoms}.
]
Final Answer:
At a pressure of (1.00 \times 10^{-7} \, \text{N/m}^2) and a temperature of (20^\circ \text{C}), there are approximately (2.47 \times 10^7) atoms in a cubic centimeter.
Explanation:
The calculation hinges on the ideal gas law, which relates pressure, volume, temperature, and the number of gas particles. Extremely low pressures, such as (10^{-7} \, \text{N/m}^2), correspond to very few gas particles per unit volume. Despite the small number, vacuum systems still contain atoms, and their count can be determined with high precision by converting from moles to atoms using Avogadro’s number. This process demonstrates the physical reality of gases, even at such extreme conditions.