Final answer:
The pressure in deep space can be found using the ideal gas law with the known density and temperature, resulting in a very low pressure. The volume for 1 mole of gas in space is exceptionally large, and if it were shaped into a cube, the side length in kilometers would be calculated by finding the cube root of the volume and converting meters to kilometers.
Step-by-step explanation:
To solve these problems, we'll use the ideal gas law, which is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature in kelvins.
Answer for (a)
The pressure (P) can be calculated by re-arranging the ideal gas law to P = (nRT)/V. Since we know the density of atoms is 10⁶ atoms/m³, we can convert this to moles (1 mole = 6.022×10²³ atoms). Therefore, n = (10⁶ atoms/m³) / (6.022×10²³ atoms/mol) = 1.66×10⁻ⁱ⁸ moles/m³. Plugging the values into the equation, we have P = (1.66×10⁻ⁱ⁸ moles/m³)(8.314 J/(mol·K))(2.7 K). The pressure in deep space is very low.
Answer for (b)
To find the volume occupied by 1 mole of gas, we use V = nRT/P. However, given the extremely low pressure in space, this volume will be exceedingly large. Since we've found the number of moles per cubic meter, we can find how many cubic meters would contain one mole.
Answer for (c)
If this volume were shaped into a cube, its side length in meters can be found by taking the cube root of the volume calculated previously, and then we can convert this length from meters to kilometers by dividing by 1000.