Question

The lowest pressure attainable using the best available vacuum techniques is about 10⁻¹²N/m².

A. At such a pressure, how many molecules are there per cm³ at 13 ∘C ?
B. Express your answer using two significant figures.

Answer 1

To calculate the number of molecules per cm³ at a given pressure and temperature, we can use the ideal gas law equation. At a pressure of 10⁻¹² N/m² and a temperature of 13°C, there are approximately 4.96×10²¹ molecules per cm³.

Step-by-step explanation:

To calculate the number of molecules per cm³ at a given pressure and temperature, we can use the ideal gas law equation: PV = nRT. Rearranging the equation, we get n/V = PV/RT, where n is the number of moles, V is the volume, P is the pressure, R is the gas constant, and T is the temperature in Kelvin.

For part A, at a pressure of 10⁻¹²N/m² and a temperature of 13°C (which is 13 + 273 = 286K), we can plug in the values into the equation to calculate the number of moles per cm³. Since the pressure is given in N/m², we need to convert it to atm.

Using the conversion factor, 1.0 atm = 1.013×105 N/m², we can conclude that the pressure is equivalent to approximately 9.87×10¹⁴ atm. Plugging the values into the equation, we get n/V = (9.87×10¹⁴ atm)(1 cm³)/ ((0.0821 L·atm/mol·K)(286K)). The volume must be converted to liters to ensure the units are consistent. Solving for n/V gives us the answer of approximately 4.96×10²¹ molecules/cm³.